Algebra Examples

$\begin{array}{c} x & f (x) \\ 25 & 2 \\ 125 & 3 \\ 625 & 4 \end{array}$

Step 1

Check if the function rule is linear.

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Step 1.1

To find if the table follows a function rule, check to see if the values follow the linear form $y = a x + b$ .

$y = a x + b$

Step 1.2

Build a set of equations from the table such that $f (x) = a x + b$ .

$2 = a (25) + b 3 = a (125) + b 4 = a (625) + b$

Step 1.3

Calculate the values of $a$ and $b$ .

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Step 1.3.1

Solve for $b$ in $2 = a (25) + b$ .

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Step 1.3.1.1

Rewrite the equation as $a (25) + b = 2$ .

$a (25) + b = 2$

$3 = a (125) + b$

$4 = a (625) + b$

Step 1.3.1.2

Move $25$ to the left of $a$ .

$25 a + b = 2$

$3 = a (125) + b$

$4 = a (625) + b$

Step 1.3.1.3

Subtract $25 a$ from both sides of the equation.

$b = 2 - 25 a$

$3 = a (125) + b$

$4 = a (625) + b$

$b = 2 - 25 a$

$3 = a (125) + b$

$4 = a (625) + b$

Step 1.3.2

Replace all occurrences of $b$ with $2 - 25 a$ in each equation.

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Step 1.3.2.1

Replace all occurrences of $b$ in $3 = a (125) + b$ with $2 - 25 a$ .

$3 = a (125) + 2 - 25 a$

$b = 2 - 25 a$

$4 = a (625) + b$

Step 1.3.2.2

Simplify $3 = a (125) + 2 - 25 a$ .

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Step 1.3.2.2.1

Simplify the left side.

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Step 1.3.2.2.1.1

Remove parentheses.

$3 = a (125) + 2 - 25 a$

$b = 2 - 25 a$

$4 = a (625) + b$

$3 = a (125) + 2 - 25 a$

$b = 2 - 25 a$

$4 = a (625) + b$

Step 1.3.2.2.2

Simplify the right side.

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Step 1.3.2.2.2.1

Simplify $a (125) + 2 - 25 a$ .

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Step 1.3.2.2.2.1.1

Move $125$ to the left of $a$ .

$3 = 125 a + 2 - 25 a$

$b = 2 - 25 a$

$4 = a (625) + b$

Step 1.3.2.2.2.1.2

Subtract $25 a$ from $125 a$ .

$3 = 100 a + 2$

$b = 2 - 25 a$

$4 = a (625) + b$

$3 = 100 a + 2$

$b = 2 - 25 a$

$4 = a (625) + b$

$3 = 100 a + 2$

$b = 2 - 25 a$

$4 = a (625) + b$

$3 = 100 a + 2$

$b = 2 - 25 a$

$4 = a (625) + b$

Step 1.3.2.3

Replace all occurrences of $b$ in $4 = a (625) + b$ with $2 - 25 a$ .

$4 = a (625) + 2 - 25 a$

$3 = 100 a + 2$

$b = 2 - 25 a$

Step 1.3.2.4

Simplify $4 = a (625) + 2 - 25 a$ .

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Step 1.3.2.4.1

Simplify the left side.

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Step 1.3.2.4.1.1

Remove parentheses.

$4 = a (625) + 2 - 25 a$

$3 = 100 a + 2$

$b = 2 - 25 a$

$4 = a (625) + 2 - 25 a$

$3 = 100 a + 2$

$b = 2 - 25 a$

Step 1.3.2.4.2

Simplify the right side.

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Step 1.3.2.4.2.1

Simplify $a (625) + 2 - 25 a$ .

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Step 1.3.2.4.2.1.1

Move $625$ to the left of $a$ .

$4 = 625 a + 2 - 25 a$

$3 = 100 a + 2$

$b = 2 - 25 a$

Step 1.3.2.4.2.1.2

Subtract $25 a$ from $625 a$ .

$4 = 600 a + 2$

$3 = 100 a + 2$

$b = 2 - 25 a$

$4 = 600 a + 2$

$3 = 100 a + 2$

$b = 2 - 25 a$

$4 = 600 a + 2$

$3 = 100 a + 2$

$b = 2 - 25 a$

$4 = 600 a + 2$

$3 = 100 a + 2$

$b = 2 - 25 a$

$4 = 600 a + 2$

$3 = 100 a + 2$

$b = 2 - 25 a$

Step 1.3.3

Solve for $a$ in $4 = 600 a + 2$ .

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Step 1.3.3.1

Rewrite the equation as $600 a + 2 = 4$ .

$600 a + 2 = 4$

$3 = 100 a + 2$

$b = 2 - 25 a$

Step 1.3.3.2

Move all terms not containing $a$ to the right side of the equation.

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Step 1.3.3.2.1

Subtract $2$ from both sides of the equation.

$600 a = 4 - 2$

$3 = 100 a + 2$

$b = 2 - 25 a$

Step 1.3.3.2.2

Subtract $2$ from $4$ .

$600 a = 2$

$3 = 100 a + 2$

$b = 2 - 25 a$

$600 a = 2$

$3 = 100 a + 2$

$b = 2 - 25 a$

Step 1.3.3.3

Divide each term in $600 a = 2$ by $600$ and simplify.

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Step 1.3.3.3.1

Divide each term in $600 a = 2$ by $600$ .

$\frac{600 a}{600} = \frac{2}{600}$

$3 = 100 a + 2$

$b = 2 - 25 a$

Step 1.3.3.3.2

Simplify the left side.

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Step 1.3.3.3.2.1

Cancel the common factor of $600$ .

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Step 1.3.3.3.2.1.1

Cancel the common factor.

$\frac{600 a}{600} = \frac{2}{600}$

$3 = 100 a + 2$

$b = 2 - 25 a$

Step 1.3.3.3.2.1.2

Divide $a$ by $1$ .

$a = \frac{2}{600}$

$3 = 100 a + 2$

$b = 2 - 25 a$

$a = \frac{2}{600}$

$3 = 100 a + 2$

$b = 2 - 25 a$

$a = \frac{2}{600}$

$3 = 100 a + 2$

$b = 2 - 25 a$

Step 1.3.3.3.3

Simplify the right side.

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Step 1.3.3.3.3.1

Cancel the common factor of $2$ and $600$ .

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Step 1.3.3.3.3.1.1

Factor $2$ out of $2$ .

$a = \frac{2 (1)}{600}$

$3 = 100 a + 2$

$b = 2 - 25 a$

Step 1.3.3.3.3.1.2

Cancel the common factors.

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Step 1.3.3.3.3.1.2.1

Factor $2$ out of $600$ .

$a = \frac{2 \cdot 1}{2 \cdot 300}$

$3 = 100 a + 2$

$b = 2 - 25 a$

Step 1.3.3.3.3.1.2.2

Cancel the common factor.

$a = \frac{2 \cdot 1}{2 \cdot 300}$

$3 = 100 a + 2$

$b = 2 - 25 a$

Step 1.3.3.3.3.1.2.3

Rewrite the expression.

$a = \frac{1}{300}$

$3 = 100 a + 2$

$b = 2 - 25 a$

$a = \frac{1}{300}$

$3 = 100 a + 2$

$b = 2 - 25 a$

$a = \frac{1}{300}$

$3 = 100 a + 2$

$b = 2 - 25 a$

$a = \frac{1}{300}$

$3 = 100 a + 2$

$b = 2 - 25 a$

$a = \frac{1}{300}$

$3 = 100 a + 2$

$b = 2 - 25 a$

$a = \frac{1}{300}$

$3 = 100 a + 2$

$b = 2 - 25 a$

Step 1.3.4

Replace all occurrences of $a$ with $\frac{1}{300}$ in each equation.

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Step 1.3.4.1

Replace all occurrences of $a$ in $3 = 100 a + 2$ with $\frac{1}{300}$ .

$3 = 100 (\frac{1}{300}) + 2$

$a = \frac{1}{300}$

$b = 2 - 25 a$

Step 1.3.4.2

Simplify the right side.

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Step 1.3.4.2.1

Simplify $100 (\frac{1}{300}) + 2$ .

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Step 1.3.4.2.1.1

Cancel the common factor of $100$ .

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Step 1.3.4.2.1.1.1

Factor $100$ out of $300$ .

$3 = 100 (\frac{1}{100 (3)}) + 2$

$a = \frac{1}{300}$

$b = 2 - 25 a$

Step 1.3.4.2.1.1.2

Cancel the common factor.

$3 = 100 (\frac{1}{100 \cdot 3}) + 2$

$a = \frac{1}{300}$

$b = 2 - 25 a$

Step 1.3.4.2.1.1.3

Rewrite the expression.

$3 = \frac{1}{3} + 2$

$a = \frac{1}{300}$

$b = 2 - 25 a$

$3 = \frac{1}{3} + 2$

$a = \frac{1}{300}$

$b = 2 - 25 a$

Step 1.3.4.2.1.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$3 = \frac{1}{3} + 2 \cdot \frac{3}{3}$

$a = \frac{1}{300}$

$b = 2 - 25 a$

Step 1.3.4.2.1.3

Combine $2$ and $\frac{3}{3}$ .

$3 = \frac{1}{3} + \frac{2 \cdot 3}{3}$

$a = \frac{1}{300}$

$b = 2 - 25 a$

Step 1.3.4.2.1.4

Combine the numerators over the common denominator.

$3 = \frac{1 + 2 \cdot 3}{3}$

$a = \frac{1}{300}$

$b = 2 - 25 a$

Step 1.3.4.2.1.5

Simplify the numerator.

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Step 1.3.4.2.1.5.1

Multiply $2$ by $3$ .

$3 = \frac{1 + 6}{3}$

$a = \frac{1}{300}$

$b = 2 - 25 a$

Step 1.3.4.2.1.5.2

Add $1$ and $6$ .

$3 = \frac{7}{3}$

$a = \frac{1}{300}$

$b = 2 - 25 a$

$3 = \frac{7}{3}$

$a = \frac{1}{300}$

$b = 2 - 25 a$

$3 = \frac{7}{3}$

$a = \frac{1}{300}$

$b = 2 - 25 a$

$3 = \frac{7}{3}$

$a = \frac{1}{300}$

$b = 2 - 25 a$

Step 1.3.4.3

Replace all occurrences of $a$ in $b = 2 - 25 a$ with $\frac{1}{300}$ .

$b = 2 - 25 (\frac{1}{300})$

$3 = \frac{7}{3}$

$a = \frac{1}{300}$

Step 1.3.4.4

Simplify the right side.

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Step 1.3.4.4.1

Simplify $2 - 25 (\frac{1}{300})$ .

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Step 1.3.4.4.1.1

Simplify each term.

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Step 1.3.4.4.1.1.1

Cancel the common factor of $25$ .

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Step 1.3.4.4.1.1.1.1

Factor $25$ out of $- 25$ .

$b = 2 + 25 (- 1) (\frac{1}{300})$

$3 = \frac{7}{3}$

$a = \frac{1}{300}$

Step 1.3.4.4.1.1.1.2

Factor $25$ out of $300$ .

$b = 2 + 25 \cdot (- 1 \frac{1}{25 \cdot 12})$

$3 = \frac{7}{3}$

$a = \frac{1}{300}$

Step 1.3.4.4.1.1.1.3

Cancel the common factor.

$b = 2 + 25 \cdot (- 1 \frac{1}{25 \cdot 12})$

$3 = \frac{7}{3}$

$a = \frac{1}{300}$

Step 1.3.4.4.1.1.1.4

Rewrite the expression.

$b = 2 - 1 (\frac{1}{12})$

$3 = \frac{7}{3}$

$a = \frac{1}{300}$

$b = 2 - 1 (\frac{1}{12})$

$3 = \frac{7}{3}$

$a = \frac{1}{300}$

Step 1.3.4.4.1.1.2

Rewrite $- 1 (\frac{1}{12})$ as $- (\frac{1}{12})$ .

$b = 2 - \frac{1}{12}$

$3 = \frac{7}{3}$

$a = \frac{1}{300}$

$b = 2 - \frac{1}{12}$

$3 = \frac{7}{3}$

$a = \frac{1}{300}$

Step 1.3.4.4.1.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$b = 2 \cdot \frac{12}{12} - \frac{1}{12}$

$3 = \frac{7}{3}$

$a = \frac{1}{300}$

Step 1.3.4.4.1.3

Combine $2$ and $\frac{12}{12}$ .

$b = \frac{2 \cdot 12}{12} - \frac{1}{12}$

$3 = \frac{7}{3}$

$a = \frac{1}{300}$

Step 1.3.4.4.1.4

Combine the numerators over the common denominator.

$b = \frac{2 \cdot 12 - 1}{12}$

$3 = \frac{7}{3}$

$a = \frac{1}{300}$

Step 1.3.4.4.1.5

Simplify the numerator.

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Step 1.3.4.4.1.5.1

Multiply $2$ by $12$ .

$b = \frac{24 - 1}{12}$

$3 = \frac{7}{3}$

$a = \frac{1}{300}$

Step 1.3.4.4.1.5.2

Subtract $1$ from $24$ .

$b = \frac{23}{12}$

$3 = \frac{7}{3}$

$a = \frac{1}{300}$

$b = \frac{23}{12}$

$3 = \frac{7}{3}$

$a = \frac{1}{300}$

$b = \frac{23}{12}$

$3 = \frac{7}{3}$

$a = \frac{1}{300}$

$b = \frac{23}{12}$

$3 = \frac{7}{3}$

$a = \frac{1}{300}$

$b = \frac{23}{12}$

$3 = \frac{7}{3}$

$a = \frac{1}{300}$

Step 1.3.5

Since $3 = \frac{7}{3}$ is not true, there is no solution.

No solution

Step 1.4

Since $y \neq f (x)$ for the corresponding $x$ values, the function is not linear.

The function is not linear

Step 2

Check if the function rule is quadratic.

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Step 2.1

To find if the table follows a function rule, check whether the function rule could follow the form $y = a x^{2} + b x + c$ .

$y = a x^{2} + b x + c$

Step 2.2

Build a set of $3$ equations from the table such that $f (x) = a x^{2} + b x + c$ .

Step 2.3

Calculate the values of $a$ , $b$ , and $c$ .

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Step 2.3.1

Solve for $c$ in $2 = a \cdot 25^{2} + b (25) + c$ .

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Step 2.3.1.1

Rewrite the equation as $a \cdot 25^{2} + b (25) + c = 2$ .

$a \cdot 25^{2} + b (25) + c = 2$

$3 = a \cdot 125^{2} + b (125) + c$

$4 = a \cdot 625^{2} + b (625) + c$

Step 2.3.1.2

Simplify each term.

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Step 2.3.1.2.1

Raise $25$ to the power of $2$ .

$a \cdot 625 + b (25) + c = 2$

$3 = a \cdot 125^{2} + b (125) + c$

$4 = a \cdot 625^{2} + b (625) + c$

Step 2.3.1.2.2

Move $625$ to the left of $a$ .

$625 \cdot a + b (25) + c = 2$

$3 = a \cdot 125^{2} + b (125) + c$

$4 = a \cdot 625^{2} + b (625) + c$

Step 2.3.1.2.3

Move $25$ to the left of $b$ .

$625 a + 25 b + c = 2$

$3 = a \cdot 125^{2} + b (125) + c$

$4 = a \cdot 625^{2} + b (625) + c$

$625 a + 25 b + c = 2$

$3 = a \cdot 125^{2} + b (125) + c$

$4 = a \cdot 625^{2} + b (625) + c$

Step 2.3.1.3

Move all terms not containing $c$ to the right side of the equation.

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Step 2.3.1.3.1

Subtract $625 a$ from both sides of the equation.

$25 b + c = 2 - 625 a$

$3 = a \cdot 125^{2} + b (125) + c$

$4 = a \cdot 625^{2} + b (625) + c$

Step 2.3.1.3.2

Subtract $25 b$ from both sides of the equation.

$c = 2 - 625 a - 25 b$

$3 = a \cdot 125^{2} + b (125) + c$

$4 = a \cdot 625^{2} + b (625) + c$

$c = 2 - 625 a - 25 b$

$3 = a \cdot 125^{2} + b (125) + c$

$4 = a \cdot 625^{2} + b (625) + c$

$c = 2 - 625 a - 25 b$

$3 = a \cdot 125^{2} + b (125) + c$

$4 = a \cdot 625^{2} + b (625) + c$

Step 2.3.2

Replace all occurrences of $c$ with $2 - 625 a - 25 b$ in each equation.

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Step 2.3.2.1

Replace all occurrences of $c$ in $3 = a \cdot 125^{2} + b (125) + c$ with $2 - 625 a - 25 b$ .

$3 = a \cdot 125^{2} + b (125) + 2 - 625 a - 25 b$

$c = 2 - 625 a - 25 b$

$4 = a \cdot 625^{2} + b (625) + c$

Step 2.3.2.2

Simplify $3 = a \cdot 125^{2} + b (125) + 2 - 625 a - 25 b$ .

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Step 2.3.2.2.1

Simplify the left side.

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Step 2.3.2.2.1.1

Remove parentheses.

$3 = a \cdot 125^{2} + b (125) + 2 - 625 a - 25 b$

$c = 2 - 625 a - 25 b$

$4 = a \cdot 625^{2} + b (625) + c$

$3 = a \cdot 125^{2} + b (125) + 2 - 625 a - 25 b$

$c = 2 - 625 a - 25 b$

$4 = a \cdot 625^{2} + b (625) + c$

Step 2.3.2.2.2

Simplify the right side.

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Step 2.3.2.2.2.1

Simplify $a \cdot 125^{2} + b (125) + 2 - 625 a - 25 b$ .

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Step 2.3.2.2.2.1.1

Simplify each term.

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Step 2.3.2.2.2.1.1.1

Raise $125$ to the power of $2$ .

$3 = a \cdot 15625 + b (125) + 2 - 625 a - 25 b$

$c = 2 - 625 a - 25 b$

$4 = a \cdot 625^{2} + b (625) + c$

Step 2.3.2.2.2.1.1.2

Move $15625$ to the left of $a$ .

$3 = 15625 \cdot a + b (125) + 2 - 625 a - 25 b$

$c = 2 - 625 a - 25 b$

$4 = a \cdot 625^{2} + b (625) + c$

Step 2.3.2.2.2.1.1.3

Move $125$ to the left of $b$ .

$3 = 15625 a + 125 b + 2 - 625 a - 25 b$

$c = 2 - 625 a - 25 b$

$4 = a \cdot 625^{2} + b (625) + c$

$3 = 15625 a + 125 b + 2 - 625 a - 25 b$

$c = 2 - 625 a - 25 b$

$4 = a \cdot 625^{2} + b (625) + c$

Step 2.3.2.2.2.1.2

Simplify by adding terms.

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Step 2.3.2.2.2.1.2.1

Subtract $625 a$ from $15625 a$ .

$3 = 15000 a + 125 b + 2 - 25 b$

$c = 2 - 625 a - 25 b$

$4 = a \cdot 625^{2} + b (625) + c$

Step 2.3.2.2.2.1.2.2

Subtract $25 b$ from $125 b$ .

$3 = 15000 a + 100 b + 2$

$c = 2 - 625 a - 25 b$

$4 = a \cdot 625^{2} + b (625) + c$

$3 = 15000 a + 100 b + 2$

$c = 2 - 625 a - 25 b$

$4 = a \cdot 625^{2} + b (625) + c$

$3 = 15000 a + 100 b + 2$

$c = 2 - 625 a - 25 b$

$4 = a \cdot 625^{2} + b (625) + c$

$3 = 15000 a + 100 b + 2$

$c = 2 - 625 a - 25 b$

$4 = a \cdot 625^{2} + b (625) + c$

$3 = 15000 a + 100 b + 2$

$c = 2 - 625 a - 25 b$

$4 = a \cdot 625^{2} + b (625) + c$

Step 2.3.2.3

Replace all occurrences of $c$ in $4 = a \cdot 625^{2} + b (625) + c$ with $2 - 625 a - 25 b$ .

$4 = a \cdot 625^{2} + b (625) + 2 - 625 a - 25 b$

$3 = 15000 a + 100 b + 2$

$c = 2 - 625 a - 25 b$

Step 2.3.2.4

Simplify $4 = a \cdot 625^{2} + b (625) + 2 - 625 a - 25 b$ .

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Step 2.3.2.4.1

Simplify the left side.

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Step 2.3.2.4.1.1

Remove parentheses.

$4 = a \cdot 625^{2} + b (625) + 2 - 625 a - 25 b$

$3 = 15000 a + 100 b + 2$

$c = 2 - 625 a - 25 b$

$4 = a \cdot 625^{2} + b (625) + 2 - 625 a - 25 b$

$3 = 15000 a + 100 b + 2$

$c = 2 - 625 a - 25 b$

Step 2.3.2.4.2

Simplify the right side.

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Step 2.3.2.4.2.1

Simplify $a \cdot 625^{2} + b (625) + 2 - 625 a - 25 b$ .

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Step 2.3.2.4.2.1.1

Simplify each term.

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Step 2.3.2.4.2.1.1.1

Raise $625$ to the power of $2$ .

$4 = a \cdot 390625 + b (625) + 2 - 625 a - 25 b$

$3 = 15000 a + 100 b + 2$

$c = 2 - 625 a - 25 b$

Step 2.3.2.4.2.1.1.2

Move $390625$ to the left of $a$ .

$4 = 390625 \cdot a + b (625) + 2 - 625 a - 25 b$

$3 = 15000 a + 100 b + 2$

$c = 2 - 625 a - 25 b$

Step 2.3.2.4.2.1.1.3

Move $625$ to the left of $b$ .

$4 = 390625 a + 625 b + 2 - 625 a - 25 b$

$3 = 15000 a + 100 b + 2$

$c = 2 - 625 a - 25 b$

$4 = 390625 a + 625 b + 2 - 625 a - 25 b$

$3 = 15000 a + 100 b + 2$

$c = 2 - 625 a - 25 b$

Step 2.3.2.4.2.1.2

Simplify by adding terms.

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Step 2.3.2.4.2.1.2.1

Subtract $625 a$ from $390625 a$ .

$4 = 390000 a + 625 b + 2 - 25 b$

$3 = 15000 a + 100 b + 2$

$c = 2 - 625 a - 25 b$

Step 2.3.2.4.2.1.2.2

Subtract $25 b$ from $625 b$ .

$4 = 390000 a + 600 b + 2$

$3 = 15000 a + 100 b + 2$

$c = 2 - 625 a - 25 b$

$4 = 390000 a + 600 b + 2$

$3 = 15000 a + 100 b + 2$

$c = 2 - 625 a - 25 b$

$4 = 390000 a + 600 b + 2$

$3 = 15000 a + 100 b + 2$

$c = 2 - 625 a - 25 b$

$4 = 390000 a + 600 b + 2$

$3 = 15000 a + 100 b + 2$

$c = 2 - 625 a - 25 b$

$4 = 390000 a + 600 b + 2$

$3 = 15000 a + 100 b + 2$

$c = 2 - 625 a - 25 b$

$4 = 390000 a + 600 b + 2$

$3 = 15000 a + 100 b + 2$

$c = 2 - 625 a - 25 b$

Step 2.3.3

Solve for $a$ in $4 = 390000 a + 600 b + 2$ .

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Step 2.3.3.1

Rewrite the equation as $390000 a + 600 b + 2 = 4$ .

$390000 a + 600 b + 2 = 4$

$3 = 15000 a + 100 b + 2$

$c = 2 - 625 a - 25 b$

Step 2.3.3.2

Move all terms not containing $a$ to the right side of the equation.

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Step 2.3.3.2.1

Subtract $600 b$ from both sides of the equation.

$390000 a + 2 = 4 - 600 b$

$3 = 15000 a + 100 b + 2$

$c = 2 - 625 a - 25 b$

Step 2.3.3.2.2

Subtract $2$ from both sides of the equation.

$390000 a = 4 - 600 b - 2$

$3 = 15000 a + 100 b + 2$

$c = 2 - 625 a - 25 b$

Step 2.3.3.2.3

Subtract $2$ from $4$ .

$390000 a = - 600 b + 2$

$3 = 15000 a + 100 b + 2$

$c = 2 - 625 a - 25 b$

$390000 a = - 600 b + 2$

$3 = 15000 a + 100 b + 2$

$c = 2 - 625 a - 25 b$

Step 2.3.3.3

Divide each term in $390000 a = - 600 b + 2$ by $390000$ and simplify.

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Step 2.3.3.3.1

Divide each term in $390000 a = - 600 b + 2$ by $390000$ .

$\frac{390000 a}{390000} = \frac{- 600 b}{390000} + \frac{2}{390000}$

$3 = 15000 a + 100 b + 2$

$c = 2 - 625 a - 25 b$

Step 2.3.3.3.2

Simplify the left side.

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Step 2.3.3.3.2.1

Cancel the common factor of $390000$ .

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Step 2.3.3.3.2.1.1

Cancel the common factor.

$\frac{390000 a}{390000} = \frac{- 600 b}{390000} + \frac{2}{390000}$

$3 = 15000 a + 100 b + 2$

$c = 2 - 625 a - 25 b$

Step 2.3.3.3.2.1.2

Divide $a$ by $1$ .

$a = \frac{- 600 b}{390000} + \frac{2}{390000}$

$3 = 15000 a + 100 b + 2$

$c = 2 - 625 a - 25 b$

$a = \frac{- 600 b}{390000} + \frac{2}{390000}$

$3 = 15000 a + 100 b + 2$

$c = 2 - 625 a - 25 b$

$a = \frac{- 600 b}{390000} + \frac{2}{390000}$

$3 = 15000 a + 100 b + 2$

$c = 2 - 625 a - 25 b$

Step 2.3.3.3.3

Simplify the right side.

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Step 2.3.3.3.3.1

Simplify each term.

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Step 2.3.3.3.3.1.1

Cancel the common factor of $- 600$ and $390000$ .

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Step 2.3.3.3.3.1.1.1

Factor $600$ out of $- 600 b$ .

$a = \frac{600 (- b)}{390000} + \frac{2}{390000}$

$3 = 15000 a + 100 b + 2$

$c = 2 - 625 a - 25 b$

Step 2.3.3.3.3.1.1.2

Cancel the common factors.

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Step 2.3.3.3.3.1.1.2.1

Factor $600$ out of $390000$ .

$a = \frac{600 (- b)}{600 (650)} + \frac{2}{390000}$

$3 = 15000 a + 100 b + 2$

$c = 2 - 625 a - 25 b$

Step 2.3.3.3.3.1.1.2.2

Cancel the common factor.

$a = \frac{600 (- b)}{600 \cdot 650} + \frac{2}{390000}$

$3 = 15000 a + 100 b + 2$

$c = 2 - 625 a - 25 b$

Step 2.3.3.3.3.1.1.2.3

Rewrite the expression.

$a = \frac{- b}{650} + \frac{2}{390000}$

$3 = 15000 a + 100 b + 2$

$c = 2 - 625 a - 25 b$

$a = \frac{- b}{650} + \frac{2}{390000}$

$3 = 15000 a + 100 b + 2$

$c = 2 - 625 a - 25 b$

$a = \frac{- b}{650} + \frac{2}{390000}$

$3 = 15000 a + 100 b + 2$

$c = 2 - 625 a - 25 b$

Step 2.3.3.3.3.1.2

Move the negative in front of the fraction.

$a = - \frac{b}{650} + \frac{2}{390000}$

$3 = 15000 a + 100 b + 2$

$c = 2 - 625 a - 25 b$

Step 2.3.3.3.3.1.3

Cancel the common factor of $2$ and $390000$ .

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Step 2.3.3.3.3.1.3.1

Factor $2$ out of $2$ .

$a = - \frac{b}{650} + \frac{2 (1)}{390000}$

$3 = 15000 a + 100 b + 2$

$c = 2 - 625 a - 25 b$

Step 2.3.3.3.3.1.3.2

Cancel the common factors.

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Step 2.3.3.3.3.1.3.2.1

Factor $2$ out of $390000$ .

$a = - \frac{b}{650} + \frac{2 \cdot 1}{2 \cdot 195000}$

$3 = 15000 a + 100 b + 2$

$c = 2 - 625 a - 25 b$

Step 2.3.3.3.3.1.3.2.2

Cancel the common factor.

$a = - \frac{b}{650} + \frac{2 \cdot 1}{2 \cdot 195000}$

$3 = 15000 a + 100 b + 2$

$c = 2 - 625 a - 25 b$

Step 2.3.3.3.3.1.3.2.3

Rewrite the expression.

$a = - \frac{b}{650} + \frac{1}{195000}$

$3 = 15000 a + 100 b + 2$

$c = 2 - 625 a - 25 b$

$a = - \frac{b}{650} + \frac{1}{195000}$

$3 = 15000 a + 100 b + 2$

$c = 2 - 625 a - 25 b$

$a = - \frac{b}{650} + \frac{1}{195000}$

$3 = 15000 a + 100 b + 2$

$c = 2 - 625 a - 25 b$

$a = - \frac{b}{650} + \frac{1}{195000}$

$3 = 15000 a + 100 b + 2$

$c = 2 - 625 a - 25 b$

$a = - \frac{b}{650} + \frac{1}{195000}$

$3 = 15000 a + 100 b + 2$

$c = 2 - 625 a - 25 b$

$a = - \frac{b}{650} + \frac{1}{195000}$

$3 = 15000 a + 100 b + 2$

$c = 2 - 625 a - 25 b$

$a = - \frac{b}{650} + \frac{1}{195000}$

$3 = 15000 a + 100 b + 2$

$c = 2 - 625 a - 25 b$

Step 2.3.4

Replace all occurrences of $a$ with $- \frac{b}{650} + \frac{1}{195000}$ in each equation.

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Step 2.3.4.1

Replace all occurrences of $a$ in $3 = 15000 a + 100 b + 2$ with $- \frac{b}{650} + \frac{1}{195000}$ .

$3 = 15000 (- \frac{b}{650} + \frac{1}{195000}) + 100 b + 2$

$a = - \frac{b}{650} + \frac{1}{195000}$

$c = 2 - 625 a - 25 b$

Step 2.3.4.2

Simplify the right side.

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Step 2.3.4.2.1

Simplify $15000 (- \frac{b}{650} + \frac{1}{195000}) + 100 b + 2$ .

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Step 2.3.4.2.1.1

Simplify each term.

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Step 2.3.4.2.1.1.1

Apply the distributive property.

$3 = 15000 (- \frac{b}{650}) + 15000 (\frac{1}{195000}) + 100 b + 2$

$a = - \frac{b}{650} + \frac{1}{195000}$

$c = 2 - 625 a - 25 b$

Step 2.3.4.2.1.1.2

Cancel the common factor of $50$ .

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Step 2.3.4.2.1.1.2.1

Move the leading negative in $- \frac{b}{650}$ into the numerator.

$3 = 15000 (\frac{- b}{650}) + 15000 (\frac{1}{195000}) + 100 b + 2$

$a = - \frac{b}{650} + \frac{1}{195000}$

$c = 2 - 625 a - 25 b$

Step 2.3.4.2.1.1.2.2

Factor $50$ out of $15000$ .

$3 = 50 (300) (\frac{- b}{650}) + 15000 (\frac{1}{195000}) + 100 b + 2$

$a = - \frac{b}{650} + \frac{1}{195000}$

$c = 2 - 625 a - 25 b$

Step 2.3.4.2.1.1.2.3

Factor $50$ out of $650$ .

$3 = 50 \cdot (300 (\frac{- b}{50 \cdot 13})) + 15000 (\frac{1}{195000}) + 100 b + 2$

$a = - \frac{b}{650} + \frac{1}{195000}$

$c = 2 - 625 a - 25 b$

Step 2.3.4.2.1.1.2.4

Cancel the common factor.

$3 = 50 \cdot (300 (\frac{- b}{50 \cdot 13})) + 15000 (\frac{1}{195000}) + 100 b + 2$

$a = - \frac{b}{650} + \frac{1}{195000}$

$c = 2 - 625 a - 25 b$

Step 2.3.4.2.1.1.2.5

Rewrite the expression.

$3 = 300 (\frac{- b}{13}) + 15000 (\frac{1}{195000}) + 100 b + 2$

$a = - \frac{b}{650} + \frac{1}{195000}$

$c = 2 - 625 a - 25 b$

$3 = 300 (\frac{- b}{13}) + 15000 (\frac{1}{195000}) + 100 b + 2$

$a = - \frac{b}{650} + \frac{1}{195000}$

$c = 2 - 625 a - 25 b$

Step 2.3.4.2.1.1.3

Combine $300$ and $\frac{- b}{13}$ .

$3 = \frac{300 (- b)}{13} + 15000 (\frac{1}{195000}) + 100 b + 2$

$a = - \frac{b}{650} + \frac{1}{195000}$

$c = 2 - 625 a - 25 b$

Step 2.3.4.2.1.1.4

Multiply $- 1$ by $300$ .

$3 = \frac{- 300 b}{13} + 15000 (\frac{1}{195000}) + 100 b + 2$

$a = - \frac{b}{650} + \frac{1}{195000}$

$c = 2 - 625 a - 25 b$

Step 2.3.4.2.1.1.5

Cancel the common factor of $15000$ .

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Step 2.3.4.2.1.1.5.1

Factor $15000$ out of $195000$ .

$3 = \frac{- 300 b}{13} + 15000 (\frac{1}{15000 (13)}) + 100 b + 2$

$a = - \frac{b}{650} + \frac{1}{195000}$

$c = 2 - 625 a - 25 b$

Step 2.3.4.2.1.1.5.2

Cancel the common factor.

$3 = \frac{- 300 b}{13} + 15000 (\frac{1}{15000 \cdot 13}) + 100 b + 2$

$a = - \frac{b}{650} + \frac{1}{195000}$

$c = 2 - 625 a - 25 b$

Step 2.3.4.2.1.1.5.3

Rewrite the expression.

$3 = \frac{- 300 b}{13} + \frac{1}{13} + 100 b + 2$

$a = - \frac{b}{650} + \frac{1}{195000}$

$c = 2 - 625 a - 25 b$

$3 = \frac{- 300 b}{13} + \frac{1}{13} + 100 b + 2$

$a = - \frac{b}{650} + \frac{1}{195000}$

$c = 2 - 625 a - 25 b$

Step 2.3.4.2.1.1.6

Move the negative in front of the fraction.

$3 = - \frac{300 b}{13} + \frac{1}{13} + 100 b + 2$

$a = - \frac{b}{650} + \frac{1}{195000}$

$c = 2 - 625 a - 25 b$

$3 = - \frac{300 b}{13} + \frac{1}{13} + 100 b + 2$

$a = - \frac{b}{650} + \frac{1}{195000}$

$c = 2 - 625 a - 25 b$

Step 2.3.4.2.1.2

To write $100 b$ as a fraction with a common denominator, multiply by $\frac{13}{13}$ .

$3 = - \frac{300 b}{13} + 100 b \cdot \frac{13}{13} + \frac{1}{13} + 2$

$a = - \frac{b}{650} + \frac{1}{195000}$

$c = 2 - 625 a - 25 b$

Step 2.3.4.2.1.3

Combine $100 b$ and $\frac{13}{13}$ .

$3 = - \frac{300 b}{13} + \frac{100 b \cdot 13}{13} + \frac{1}{13} + 2$

$a = - \frac{b}{650} + \frac{1}{195000}$

$c = 2 - 625 a - 25 b$

Step 2.3.4.2.1.4

Combine the numerators over the common denominator.

$3 = \frac{- 300 b + 100 b \cdot 13}{13} + \frac{1}{13} + 2$

$a = - \frac{b}{650} + \frac{1}{195000}$

$c = 2 - 625 a - 25 b$

Step 2.3.4.2.1.5

Combine the numerators over the common denominator.

$3 = 2 + \frac{- 300 b + 100 b \cdot 13 + 1}{13}$

$a = - \frac{b}{650} + \frac{1}{195000}$

$c = 2 - 625 a - 25 b$

Step 2.3.4.2.1.6

Multiply $13$ by $100$ .

$3 = 2 + \frac{- 300 b + 1300 b + 1}{13}$

$a = - \frac{b}{650} + \frac{1}{195000}$

$c = 2 - 625 a - 25 b$

Step 2.3.4.2.1.7

Add $- 300 b$ and $1300 b$ .

$3 = 2 + \frac{1000 b + 1}{13}$

$a = - \frac{b}{650} + \frac{1}{195000}$

$c = 2 - 625 a - 25 b$

Step 2.3.4.2.1.8

To write $2$ as a fraction with a common denominator, multiply by $\frac{13}{13}$ .

$3 = 2 \cdot \frac{13}{13} + \frac{1000 b + 1}{13}$

$a = - \frac{b}{650} + \frac{1}{195000}$

$c = 2 - 625 a - 25 b$

Step 2.3.4.2.1.9

Simplify terms.

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Step 2.3.4.2.1.9.1

Combine $2$ and $\frac{13}{13}$ .

$3 = \frac{2 \cdot 13}{13} + \frac{1000 b + 1}{13}$

$a = - \frac{b}{650} + \frac{1}{195000}$

$c = 2 - 625 a - 25 b$

Step 2.3.4.2.1.9.2

Combine the numerators over the common denominator.

$3 = \frac{2 \cdot 13 + 1000 b + 1}{13}$

$a = - \frac{b}{650} + \frac{1}{195000}$

$c = 2 - 625 a - 25 b$

$3 = \frac{2 \cdot 13 + 1000 b + 1}{13}$

$a = - \frac{b}{650} + \frac{1}{195000}$

$c = 2 - 625 a - 25 b$

Step 2.3.4.2.1.10

Simplify the numerator.

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Step 2.3.4.2.1.10.1

Multiply $2$ by $13$ .

$3 = \frac{26 + 1000 b + 1}{13}$

$a = - \frac{b}{650} + \frac{1}{195000}$

$c = 2 - 625 a - 25 b$

Step 2.3.4.2.1.10.2

Add $26$ and $1$ .

$3 = \frac{1000 b + 27}{13}$

$a = - \frac{b}{650} + \frac{1}{195000}$

$c = 2 - 625 a - 25 b$

$3 = \frac{1000 b + 27}{13}$

$a = - \frac{b}{650} + \frac{1}{195000}$

$c = 2 - 625 a - 25 b$

$3 = \frac{1000 b + 27}{13}$

$a = - \frac{b}{650} + \frac{1}{195000}$

$c = 2 - 625 a - 25 b$

$3 = \frac{1000 b + 27}{13}$

$a = - \frac{b}{650} + \frac{1}{195000}$

$c = 2 - 625 a - 25 b$

Step 2.3.4.3

Replace all occurrences of $a$ in $c = 2 - 625 a - 25 b$ with $- \frac{b}{650} + \frac{1}{195000}$ .

$c = 2 - 625 (- \frac{b}{650} + \frac{1}{195000}) - 25 b$

$3 = \frac{1000 b + 27}{13}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.4.4

Simplify the right side.

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Step 2.3.4.4.1

Simplify $2 - 625 (- \frac{b}{650} + \frac{1}{195000}) - 25 b$ .

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Step 2.3.4.4.1.1

Simplify each term.

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Step 2.3.4.4.1.1.1

Apply the distributive property.

$c = 2 - 625 (- \frac{b}{650}) - 625 (\frac{1}{195000}) - 25 b$

$3 = \frac{1000 b + 27}{13}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.4.4.1.1.2

Cancel the common factor of $25$ .

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Step 2.3.4.4.1.1.2.1

Move the leading negative in $- \frac{b}{650}$ into the numerator.

$c = 2 - 625 \frac{- b}{650} - 625 (\frac{1}{195000}) - 25 b$

$3 = \frac{1000 b + 27}{13}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.4.4.1.1.2.2

Factor $25$ out of $- 625$ .

$c = 2 + 25 (- 25) (\frac{- b}{650}) - 625 (\frac{1}{195000}) - 25 b$

$3 = \frac{1000 b + 27}{13}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.4.4.1.1.2.3

Factor $25$ out of $650$ .

$c = 2 + 25 \cdot (- 25 \frac{- b}{25 \cdot 26}) - 625 (\frac{1}{195000}) - 25 b$

$3 = \frac{1000 b + 27}{13}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.4.4.1.1.2.4

Cancel the common factor.

$c = 2 + 25 \cdot (- 25 \frac{- b}{25 \cdot 26}) - 625 (\frac{1}{195000}) - 25 b$

$3 = \frac{1000 b + 27}{13}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.4.4.1.1.2.5

Rewrite the expression.

$c = 2 - 25 \frac{- b}{26} - 625 (\frac{1}{195000}) - 25 b$

$3 = \frac{1000 b + 27}{13}$

$a = - \frac{b}{650} + \frac{1}{195000}$

$c = 2 - 25 \frac{- b}{26} - 625 (\frac{1}{195000}) - 25 b$

$3 = \frac{1000 b + 27}{13}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.4.4.1.1.3

Combine $- 25$ and $\frac{- b}{26}$ .

$c = 2 + \frac{- 25 (- b)}{26} - 625 (\frac{1}{195000}) - 25 b$

$3 = \frac{1000 b + 27}{13}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.4.4.1.1.4

Multiply $- 1$ by $- 25$ .

$c = 2 + \frac{25 b}{26} - 625 (\frac{1}{195000}) - 25 b$

$3 = \frac{1000 b + 27}{13}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.4.4.1.1.5

Cancel the common factor of $625$ .

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Step 2.3.4.4.1.1.5.1

Factor $625$ out of $- 625$ .

$c = 2 + \frac{25 b}{26} + 625 (- 1) (\frac{1}{195000}) - 25 b$

$3 = \frac{1000 b + 27}{13}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.4.4.1.1.5.2

Factor $625$ out of $195000$ .

$c = 2 + \frac{25 b}{26} + 625 \cdot (- 1 \frac{1}{625 \cdot 312}) - 25 b$

$3 = \frac{1000 b + 27}{13}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.4.4.1.1.5.3

Cancel the common factor.

$c = 2 + \frac{25 b}{26} + 625 \cdot (- 1 \frac{1}{625 \cdot 312}) - 25 b$

$3 = \frac{1000 b + 27}{13}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.4.4.1.1.5.4

Rewrite the expression.

$c = 2 + \frac{25 b}{26} - 1 (\frac{1}{312}) - 25 b$

$3 = \frac{1000 b + 27}{13}$

$a = - \frac{b}{650} + \frac{1}{195000}$

$c = 2 + \frac{25 b}{26} - 1 (\frac{1}{312}) - 25 b$

$3 = \frac{1000 b + 27}{13}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.4.4.1.1.6

Rewrite $- 1 (\frac{1}{312})$ as $- (\frac{1}{312})$ .

$c = 2 + \frac{25 b}{26} - \frac{1}{312} - 25 b$

$3 = \frac{1000 b + 27}{13}$

$a = - \frac{b}{650} + \frac{1}{195000}$

$c = 2 + \frac{25 b}{26} - \frac{1}{312} - 25 b$

$3 = \frac{1000 b + 27}{13}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.4.4.1.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{312}{312}$ .

$c = \frac{25 b}{26} + 2 \cdot \frac{312}{312} - \frac{1}{312} - 25 b$

$3 = \frac{1000 b + 27}{13}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.4.4.1.3

Combine $2$ and $\frac{312}{312}$ .

$c = \frac{25 b}{26} + \frac{2 \cdot 312}{312} - \frac{1}{312} - 25 b$

$3 = \frac{1000 b + 27}{13}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.4.4.1.4

Combine the numerators over the common denominator.

$c = \frac{25 b}{26} + \frac{2 \cdot 312 - 1}{312} - 25 b$

$3 = \frac{1000 b + 27}{13}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.4.4.1.5

Simplify the numerator.

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Step 2.3.4.4.1.5.1

Multiply $2$ by $312$ .

$c = \frac{25 b}{26} + \frac{624 - 1}{312} - 25 b$

$3 = \frac{1000 b + 27}{13}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.4.4.1.5.2

Subtract $1$ from $624$ .

$c = \frac{25 b}{26} + \frac{623}{312} - 25 b$

$3 = \frac{1000 b + 27}{13}$

$a = - \frac{b}{650} + \frac{1}{195000}$

$c = \frac{25 b}{26} + \frac{623}{312} - 25 b$

$3 = \frac{1000 b + 27}{13}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.4.4.1.6

To write $- 25 b$ as a fraction with a common denominator, multiply by $\frac{26}{26}$ .

$c = \frac{25 b}{26} - 25 b \cdot \frac{26}{26} + \frac{623}{312}$

$3 = \frac{1000 b + 27}{13}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.4.4.1.7

Simplify terms.

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Step 2.3.4.4.1.7.1

Combine $- 25 b$ and $\frac{26}{26}$ .

$c = \frac{25 b}{26} + \frac{- 25 b \cdot 26}{26} + \frac{623}{312}$

$3 = \frac{1000 b + 27}{13}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.4.4.1.7.2

Combine the numerators over the common denominator.

$c = \frac{25 b - 25 b \cdot 26}{26} + \frac{623}{312}$

$3 = \frac{1000 b + 27}{13}$

$a = - \frac{b}{650} + \frac{1}{195000}$

$c = \frac{25 b - 25 b \cdot 26}{26} + \frac{623}{312}$

$3 = \frac{1000 b + 27}{13}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.4.4.1.8

Simplify each term.

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Step 2.3.4.4.1.8.1

Simplify the numerator.

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Step 2.3.4.4.1.8.1.1

Factor $25 b$ out of $25 b - 25 b \cdot 26$ .

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Step 2.3.4.4.1.8.1.1.1

Factor $25 b$ out of $25 b$ .

$c = \frac{25 b (1) - 25 b \cdot 26}{26} + \frac{623}{312}$

$3 = \frac{1000 b + 27}{13}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.4.4.1.8.1.1.2

Factor $25 b$ out of $- 25 b \cdot 26$ .

$c = \frac{25 b (1) + 25 b (- 1 \cdot 26)}{26} + \frac{623}{312}$

$3 = \frac{1000 b + 27}{13}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.4.4.1.8.1.1.3

Factor $25 b$ out of $25 b (1) + 25 b (- 1 \cdot 26)$ .

$c = \frac{25 b (1 - 1 \cdot 26)}{26} + \frac{623}{312}$

$3 = \frac{1000 b + 27}{13}$

$a = - \frac{b}{650} + \frac{1}{195000}$

$c = \frac{25 b (1 - 1 \cdot 26)}{26} + \frac{623}{312}$

$3 = \frac{1000 b + 27}{13}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.4.4.1.8.1.2

Multiply $- 1$ by $26$ .

$c = \frac{25 b (1 - 26)}{26} + \frac{623}{312}$

$3 = \frac{1000 b + 27}{13}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.4.4.1.8.1.3

Subtract $26$ from $1$ .

$c = \frac{25 b \cdot - 25}{26} + \frac{623}{312}$

$3 = \frac{1000 b + 27}{13}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.4.4.1.8.1.4

Multiply $- 25$ by $25$ .

$c = \frac{- 625 b}{26} + \frac{623}{312}$

$3 = \frac{1000 b + 27}{13}$

$a = - \frac{b}{650} + \frac{1}{195000}$

$c = \frac{- 625 b}{26} + \frac{623}{312}$

$3 = \frac{1000 b + 27}{13}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.4.4.1.8.2

Move the negative in front of the fraction.

$c = - \frac{625 b}{26} + \frac{623}{312}$

$3 = \frac{1000 b + 27}{13}$

$a = - \frac{b}{650} + \frac{1}{195000}$

$c = - \frac{625 b}{26} + \frac{623}{312}$

$3 = \frac{1000 b + 27}{13}$

$a = - \frac{b}{650} + \frac{1}{195000}$

$c = - \frac{625 b}{26} + \frac{623}{312}$

$3 = \frac{1000 b + 27}{13}$

$a = - \frac{b}{650} + \frac{1}{195000}$

$c = - \frac{625 b}{26} + \frac{623}{312}$

$3 = \frac{1000 b + 27}{13}$

$a = - \frac{b}{650} + \frac{1}{195000}$

$c = - \frac{625 b}{26} + \frac{623}{312}$

$3 = \frac{1000 b + 27}{13}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.5

Solve for $b$ in $3 = \frac{1000 b + 27}{13}$ .

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Step 2.3.5.1

Rewrite the equation as $\frac{1000 b + 27}{13} = 3$ .

$\frac{1000 b + 27}{13} = 3$

$c = - \frac{625 b}{26} + \frac{623}{312}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.5.2

Multiply both sides by $13$ .

$\frac{1000 b + 27}{13} \cdot 13 = 3 \cdot 13$

$c = - \frac{625 b}{26} + \frac{623}{312}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.5.3

Simplify.

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Step 2.3.5.3.1

Simplify the left side.

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Step 2.3.5.3.1.1

Cancel the common factor of $13$ .

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Step 2.3.5.3.1.1.1

Cancel the common factor.

$\frac{1000 b + 27}{13} \cdot 13 = 3 \cdot 13$

$c = - \frac{625 b}{26} + \frac{623}{312}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.5.3.1.1.2

Rewrite the expression.

$1000 b + 27 = 3 \cdot 13$

$c = - \frac{625 b}{26} + \frac{623}{312}$

$a = - \frac{b}{650} + \frac{1}{195000}$

$1000 b + 27 = 3 \cdot 13$

$c = - \frac{625 b}{26} + \frac{623}{312}$

$a = - \frac{b}{650} + \frac{1}{195000}$

$1000 b + 27 = 3 \cdot 13$

$c = - \frac{625 b}{26} + \frac{623}{312}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.5.3.2

Simplify the right side.

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Step 2.3.5.3.2.1

Multiply $3$ by $13$ .

$1000 b + 27 = 39$

$c = - \frac{625 b}{26} + \frac{623}{312}$

$a = - \frac{b}{650} + \frac{1}{195000}$

$1000 b + 27 = 39$

$c = - \frac{625 b}{26} + \frac{623}{312}$

$a = - \frac{b}{650} + \frac{1}{195000}$

$1000 b + 27 = 39$

$c = - \frac{625 b}{26} + \frac{623}{312}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.5.4

Solve for $b$ .

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Step 2.3.5.4.1

Move all terms not containing $b$ to the right side of the equation.

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Step 2.3.5.4.1.1

Subtract $27$ from both sides of the equation.

$1000 b = 39 - 27$

$c = - \frac{625 b}{26} + \frac{623}{312}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.5.4.1.2

Subtract $27$ from $39$ .

$1000 b = 12$

$c = - \frac{625 b}{26} + \frac{623}{312}$

$a = - \frac{b}{650} + \frac{1}{195000}$

$1000 b = 12$

$c = - \frac{625 b}{26} + \frac{623}{312}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.5.4.2

Divide each term in $1000 b = 12$ by $1000$ and simplify.

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Step 2.3.5.4.2.1

Divide each term in $1000 b = 12$ by $1000$ .

$\frac{1000 b}{1000} = \frac{12}{1000}$

$c = - \frac{625 b}{26} + \frac{623}{312}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.5.4.2.2

Simplify the left side.

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Step 2.3.5.4.2.2.1

Cancel the common factor of $1000$ .

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Step 2.3.5.4.2.2.1.1

Cancel the common factor.

$\frac{1000 b}{1000} = \frac{12}{1000}$

$c = - \frac{625 b}{26} + \frac{623}{312}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.5.4.2.2.1.2

Divide $b$ by $1$ .

$b = \frac{12}{1000}$

$c = - \frac{625 b}{26} + \frac{623}{312}$

$a = - \frac{b}{650} + \frac{1}{195000}$

$b = \frac{12}{1000}$

$c = - \frac{625 b}{26} + \frac{623}{312}$

$a = - \frac{b}{650} + \frac{1}{195000}$

$b = \frac{12}{1000}$

$c = - \frac{625 b}{26} + \frac{623}{312}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.5.4.2.3

Simplify the right side.

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Step 2.3.5.4.2.3.1

Cancel the common factor of $12$ and $1000$ .

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Step 2.3.5.4.2.3.1.1

Factor $4$ out of $12$ .

$b = \frac{4 (3)}{1000}$

$c = - \frac{625 b}{26} + \frac{623}{312}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.5.4.2.3.1.2

Cancel the common factors.

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Step 2.3.5.4.2.3.1.2.1

Factor $4$ out of $1000$ .

$b = \frac{4 \cdot 3}{4 \cdot 250}$

$c = - \frac{625 b}{26} + \frac{623}{312}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.5.4.2.3.1.2.2

Cancel the common factor.

$b = \frac{4 \cdot 3}{4 \cdot 250}$

$c = - \frac{625 b}{26} + \frac{623}{312}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.5.4.2.3.1.2.3

Rewrite the expression.

$b = \frac{3}{250}$

$c = - \frac{625 b}{26} + \frac{623}{312}$

$a = - \frac{b}{650} + \frac{1}{195000}$

$b = \frac{3}{250}$

$c = - \frac{625 b}{26} + \frac{623}{312}$

$a = - \frac{b}{650} + \frac{1}{195000}$

$b = \frac{3}{250}$

$c = - \frac{625 b}{26} + \frac{623}{312}$

$a = - \frac{b}{650} + \frac{1}{195000}$

$b = \frac{3}{250}$

$c = - \frac{625 b}{26} + \frac{623}{312}$

$a = - \frac{b}{650} + \frac{1}{195000}$

$b = \frac{3}{250}$

$c = - \frac{625 b}{26} + \frac{623}{312}$

$a = - \frac{b}{650} + \frac{1}{195000}$

$b = \frac{3}{250}$

$c = - \frac{625 b}{26} + \frac{623}{312}$

$a = - \frac{b}{650} + \frac{1}{195000}$

$b = \frac{3}{250}$

$c = - \frac{625 b}{26} + \frac{623}{312}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.6

Replace all occurrences of $b$ with $\frac{3}{250}$ in each equation.

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Step 2.3.6.1

Replace all occurrences of $b$ in $c = - \frac{625 b}{26} + \frac{623}{312}$ with $\frac{3}{250}$ .

$c = - \frac{625 (\frac{3}{250})}{26} + \frac{623}{312}$

$b = \frac{3}{250}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.6.2

Simplify the right side.

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Step 2.3.6.2.1

Simplify $- \frac{625 (\frac{3}{250})}{26} + \frac{623}{312}$ .

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Step 2.3.6.2.1.1

Simplify each term.

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Step 2.3.6.2.1.1.1

Combine $625$ and $\frac{3}{250}$ .

$c = - \frac{\frac{625 \cdot 3}{250}}{26} + \frac{623}{312}$

$b = \frac{3}{250}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.6.2.1.1.2

Multiply $625$ by $3$ .

$c = - \frac{\frac{1875}{250}}{26} + \frac{623}{312}$

$b = \frac{3}{250}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.6.2.1.1.3

Cancel the common factor of $1875$ and $250$ .

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Step 2.3.6.2.1.1.3.1

Factor $125$ out of $1875$ .

$c = - \frac{\frac{125 (15)}{250}}{26} + \frac{623}{312}$

$b = \frac{3}{250}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.6.2.1.1.3.2

Cancel the common factors.

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Step 2.3.6.2.1.1.3.2.1

Factor $125$ out of $250$ .

$c = - \frac{\frac{125 \cdot 15}{125 \cdot 2}}{26} + \frac{623}{312}$

$b = \frac{3}{250}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.6.2.1.1.3.2.2

Cancel the common factor.

$c = - \frac{\frac{125 \cdot 15}{125 \cdot 2}}{26} + \frac{623}{312}$

$b = \frac{3}{250}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.6.2.1.1.3.2.3

Rewrite the expression.

$c = - \frac{\frac{15}{2}}{26} + \frac{623}{312}$

$b = \frac{3}{250}$

$a = - \frac{b}{650} + \frac{1}{195000}$

$c = - \frac{\frac{15}{2}}{26} + \frac{623}{312}$

$b = \frac{3}{250}$

$a = - \frac{b}{650} + \frac{1}{195000}$

$c = - \frac{\frac{15}{2}}{26} + \frac{623}{312}$

$b = \frac{3}{250}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.6.2.1.1.4

Multiply the numerator by the reciprocal of the denominator.

$c = - (\frac{15}{2} \cdot \frac{1}{26}) + \frac{623}{312}$

$b = \frac{3}{250}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.6.2.1.1.5

Multiply $\frac{15}{2} \cdot \frac{1}{26}$ .

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Step 2.3.6.2.1.1.5.1

Multiply $\frac{15}{2}$ by $\frac{1}{26}$ .

$c = - \frac{15}{2 \cdot 26} + \frac{623}{312}$

$b = \frac{3}{250}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.6.2.1.1.5.2

Multiply $2$ by $26$ .

$c = - \frac{15}{52} + \frac{623}{312}$

$b = \frac{3}{250}$

$a = - \frac{b}{650} + \frac{1}{195000}$

$c = - \frac{15}{52} + \frac{623}{312}$

$b = \frac{3}{250}$

$a = - \frac{b}{650} + \frac{1}{195000}$

$c = - \frac{15}{52} + \frac{623}{312}$

$b = \frac{3}{250}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.6.2.1.2

To write $- \frac{15}{52}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$c = - \frac{15}{52} \cdot \frac{6}{6} + \frac{623}{312}$

$b = \frac{3}{250}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.6.2.1.3

Write each expression with a common denominator of $312$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.6.2.1.3.1

Multiply $\frac{15}{52}$ by $\frac{6}{6}$ .

$c = - \frac{15 \cdot 6}{52 \cdot 6} + \frac{623}{312}$

$b = \frac{3}{250}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.6.2.1.3.2

Multiply $52$ by $6$ .

$c = - \frac{15 \cdot 6}{312} + \frac{623}{312}$

$b = \frac{3}{250}$

$a = - \frac{b}{650} + \frac{1}{195000}$

$c = - \frac{15 \cdot 6}{312} + \frac{623}{312}$

$b = \frac{3}{250}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.6.2.1.4

Combine the numerators over the common denominator.

$c = \frac{- 15 \cdot 6 + 623}{312}$

$b = \frac{3}{250}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.6.2.1.5

Simplify the numerator.

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Step 2.3.6.2.1.5.1

Multiply $- 15$ by $6$ .

$c = \frac{- 90 + 623}{312}$

$b = \frac{3}{250}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.6.2.1.5.2

Add $- 90$ and $623$ .

$c = \frac{533}{312}$

$b = \frac{3}{250}$

$a = - \frac{b}{650} + \frac{1}{195000}$

$c = \frac{533}{312}$

$b = \frac{3}{250}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.6.2.1.6

Cancel the common factor of $533$ and $312$ .

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Step 2.3.6.2.1.6.1

Factor $13$ out of $533$ .

$c = \frac{13 (41)}{312}$

$b = \frac{3}{250}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.6.2.1.6.2

Cancel the common factors.

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Step 2.3.6.2.1.6.2.1

Factor $13$ out of $312$ .

$c = \frac{13 \cdot 41}{13 \cdot 24}$

$b = \frac{3}{250}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.6.2.1.6.2.2

Cancel the common factor.

$c = \frac{13 \cdot 41}{13 \cdot 24}$

$b = \frac{3}{250}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.6.2.1.6.2.3

Rewrite the expression.

$c = \frac{41}{24}$

$b = \frac{3}{250}$

$a = - \frac{b}{650} + \frac{1}{195000}$

$c = \frac{41}{24}$

$b = \frac{3}{250}$

$a = - \frac{b}{650} + \frac{1}{195000}$

$c = \frac{41}{24}$

$b = \frac{3}{250}$

$a = - \frac{b}{650} + \frac{1}{195000}$

$c = \frac{41}{24}$

$b = \frac{3}{250}$

$a = - \frac{b}{650} + \frac{1}{195000}$

$c = \frac{41}{24}$

$b = \frac{3}{250}$

$a = - \frac{b}{650} + \frac{1}{195000}$

Step 2.3.6.3

Replace all occurrences of $b$ in $a = - \frac{b}{650} + \frac{1}{195000}$ with $\frac{3}{250}$ .

$a = - \frac{\frac{3}{250}}{650} + \frac{1}{195000}$

$c = \frac{41}{24}$

$b = \frac{3}{250}$

Step 2.3.6.4

Simplify the right side.

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Step 2.3.6.4.1

Simplify $- \frac{\frac{3}{250}}{650} + \frac{1}{195000}$ .

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Step 2.3.6.4.1.1

Simplify each term.

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Step 2.3.6.4.1.1.1

Multiply the numerator by the reciprocal of the denominator.

$a = - (\frac{3}{250} \cdot \frac{1}{650}) + \frac{1}{195000}$

$c = \frac{41}{24}$

$b = \frac{3}{250}$

Step 2.3.6.4.1.1.2

Multiply $\frac{3}{250} \cdot \frac{1}{650}$ .

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Step 2.3.6.4.1.1.2.1

Multiply $\frac{3}{250}$ by $\frac{1}{650}$ .

$a = - \frac{3}{250 \cdot 650} + \frac{1}{195000}$

$c = \frac{41}{24}$

$b = \frac{3}{250}$

Step 2.3.6.4.1.1.2.2

Multiply $250$ by $650$ .

$a = - \frac{3}{162500} + \frac{1}{195000}$

$c = \frac{41}{24}$

$b = \frac{3}{250}$

$a = - \frac{3}{162500} + \frac{1}{195000}$

$c = \frac{41}{24}$

$b = \frac{3}{250}$

$a = - \frac{3}{162500} + \frac{1}{195000}$

$c = \frac{41}{24}$

$b = \frac{3}{250}$

Step 2.3.6.4.1.2

To write $- \frac{3}{162500}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$a = - \frac{3}{162500} \cdot \frac{6}{6} + \frac{1}{195000}$

$c = \frac{41}{24}$

$b = \frac{3}{250}$

Step 2.3.6.4.1.3

To write $\frac{1}{195000}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$a = - \frac{3}{162500} \cdot \frac{6}{6} + \frac{1}{195000} \cdot \frac{5}{5}$

$c = \frac{41}{24}$

$b = \frac{3}{250}$

Step 2.3.6.4.1.4

Write each expression with a common denominator of $975000$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.6.4.1.4.1

Multiply $\frac{3}{162500}$ by $\frac{6}{6}$ .

$a = - \frac{3 \cdot 6}{162500 \cdot 6} + \frac{1}{195000} \cdot \frac{5}{5}$

$c = \frac{41}{24}$

$b = \frac{3}{250}$

Step 2.3.6.4.1.4.2

Multiply $162500$ by $6$ .

$a = - \frac{3 \cdot 6}{975000} + \frac{1}{195000} \cdot \frac{5}{5}$

$c = \frac{41}{24}$

$b = \frac{3}{250}$

Step 2.3.6.4.1.4.3

Multiply $\frac{1}{195000}$ by $\frac{5}{5}$ .

$a = - \frac{3 \cdot 6}{975000} + \frac{5}{195000 \cdot 5}$

$c = \frac{41}{24}$

$b = \frac{3}{250}$

Step 2.3.6.4.1.4.4

Multiply $195000$ by $5$ .

$a = - \frac{3 \cdot 6}{975000} + \frac{5}{975000}$

$c = \frac{41}{24}$

$b = \frac{3}{250}$

$a = - \frac{3 \cdot 6}{975000} + \frac{5}{975000}$

$c = \frac{41}{24}$

$b = \frac{3}{250}$

Step 2.3.6.4.1.5

Combine the numerators over the common denominator.

$a = \frac{- 3 \cdot 6 + 5}{975000}$

$c = \frac{41}{24}$

$b = \frac{3}{250}$

Step 2.3.6.4.1.6

Simplify the numerator.

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Step 2.3.6.4.1.6.1

Multiply $- 3$ by $6$ .

$a = \frac{- 18 + 5}{975000}$

$c = \frac{41}{24}$

$b = \frac{3}{250}$

Step 2.3.6.4.1.6.2

Add $- 18$ and $5$ .

$a = \frac{- 13}{975000}$

$c = \frac{41}{24}$

$b = \frac{3}{250}$

$a = \frac{- 13}{975000}$

$c = \frac{41}{24}$

$b = \frac{3}{250}$

Step 2.3.6.4.1.7

Cancel the common factor of $- 13$ and $975000$ .

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Step 2.3.6.4.1.7.1

Factor $13$ out of $- 13$ .

$a = \frac{13 (- 1)}{975000}$

$c = \frac{41}{24}$

$b = \frac{3}{250}$

Step 2.3.6.4.1.7.2

Cancel the common factors.

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Step 2.3.6.4.1.7.2.1

Factor $13$ out of $975000$ .

$a = \frac{13 \cdot - 1}{13 \cdot 75000}$

$c = \frac{41}{24}$

$b = \frac{3}{250}$

Step 2.3.6.4.1.7.2.2

Cancel the common factor.

$a = \frac{13 \cdot - 1}{13 \cdot 75000}$

$c = \frac{41}{24}$

$b = \frac{3}{250}$

Step 2.3.6.4.1.7.2.3

Rewrite the expression.

$a = \frac{- 1}{75000}$

$c = \frac{41}{24}$

$b = \frac{3}{250}$

$a = \frac{- 1}{75000}$

$c = \frac{41}{24}$

$b = \frac{3}{250}$

$a = \frac{- 1}{75000}$

$c = \frac{41}{24}$

$b = \frac{3}{250}$

Step 2.3.6.4.1.8

Move the negative in front of the fraction.

$a = - \frac{1}{75000}$

$c = \frac{41}{24}$

$b = \frac{3}{250}$

$a = - \frac{1}{75000}$

$c = \frac{41}{24}$

$b = \frac{3}{250}$

$a = - \frac{1}{75000}$

$c = \frac{41}{24}$

$b = \frac{3}{250}$

$a = - \frac{1}{75000}$

$c = \frac{41}{24}$

$b = \frac{3}{250}$

Step 2.3.7

List all of the solutions.

$a = - \frac{1}{75000}, c = \frac{41}{24}, b = \frac{3}{250}$

Step 2.4

Calculate the value of $y$ using each $x$ value in the table and compare this value to the given $f (x)$ value in the table.

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Step 2.4.1

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = - \frac{1}{75000}$ , $b = \frac{3}{250}$ , $c = \frac{41}{24}$ , and $x = 25$ .

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Step 2.4.1.1

Simplify each term.

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Step 2.4.1.1.1

Raise $25$ to the power of $2$ .

$y = - \frac{1}{75000} \cdot 625 + (\frac{3}{250}) \cdot (25) + \frac{41}{24}$

Step 2.4.1.1.2

Cancel the common factor of $625$ .

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Step 2.4.1.1.2.1

Move the leading negative in $- \frac{1}{75000}$ into the numerator.

$y = \frac{- 1}{75000} \cdot 625 + (\frac{3}{250}) \cdot (25) + \frac{41}{24}$

Step 2.4.1.1.2.2

Factor $625$ out of $75000$ .

$y = \frac{- 1}{625 (120)} \cdot 625 + (\frac{3}{250}) \cdot (25) + \frac{41}{24}$

Step 2.4.1.1.2.3

Cancel the common factor.

$y = \frac{- 1}{625 \cdot 120} \cdot 625 + (\frac{3}{250}) \cdot (25) + \frac{41}{24}$

Step 2.4.1.1.2.4

Rewrite the expression.

$y = \frac{- 1}{120} + (\frac{3}{250}) \cdot (25) + \frac{41}{24}$

Step 2.4.1.1.3

Move the negative in front of the fraction.

$y = - \frac{1}{120} + (\frac{3}{250}) \cdot (25) + \frac{41}{24}$

Step 2.4.1.1.4

Cancel the common factor of $25$ .

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Step 2.4.1.1.4.1

Factor $25$ out of $250$ .

$y = - \frac{1}{120} + \frac{3}{25 (10)} \cdot 25 + \frac{41}{24}$

Step 2.4.1.1.4.2

Cancel the common factor.

$y = - \frac{1}{120} + \frac{3}{25 \cdot 10} \cdot 25 + \frac{41}{24}$

Step 2.4.1.1.4.3

Rewrite the expression.

$y = - \frac{1}{120} + \frac{3}{10} + \frac{41}{24}$

Step 2.4.1.2

Find the common denominator.

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Step 2.4.1.2.1

Multiply $\frac{3}{10}$ by $\frac{12}{12}$ .

$y = - \frac{1}{120} + \frac{3}{10} \cdot \frac{12}{12} + \frac{41}{24}$

Step 2.4.1.2.2

Multiply $\frac{3}{10}$ by $\frac{12}{12}$ .

$y = - \frac{1}{120} + \frac{3 \cdot 12}{10 \cdot 12} + \frac{41}{24}$

Step 2.4.1.2.3

Multiply $\frac{41}{24}$ by $\frac{5}{5}$ .

$y = - \frac{1}{120} + \frac{3 \cdot 12}{10 \cdot 12} + \frac{41}{24} \cdot \frac{5}{5}$

Step 2.4.1.2.4

Multiply $\frac{41}{24}$ by $\frac{5}{5}$ .

$y = - \frac{1}{120} + \frac{3 \cdot 12}{10 \cdot 12} + \frac{41 \cdot 5}{24 \cdot 5}$

Step 2.4.1.2.5

Multiply $10$ by $12$ .

$y = - \frac{1}{120} + \frac{3 \cdot 12}{120} + \frac{41 \cdot 5}{24 \cdot 5}$

Step 2.4.1.2.6

Reorder the factors of $24 \cdot 5$ .

$y = - \frac{1}{120} + \frac{3 \cdot 12}{120} + \frac{41 \cdot 5}{5 \cdot 24}$

Step 2.4.1.2.7

Multiply $5$ by $24$ .

$y = - \frac{1}{120} + \frac{3 \cdot 12}{120} + \frac{41 \cdot 5}{120}$

Step 2.4.1.3

Combine the numerators over the common denominator.

$y = \frac{- 1 + 3 \cdot 12 + 41 \cdot 5}{120}$

Step 2.4.1.4

Simplify each term.

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Step 2.4.1.4.1

Multiply $3$ by $12$ .

$y = \frac{- 1 + 36 + 41 \cdot 5}{120}$

Step 2.4.1.4.2

Multiply $41$ by $5$ .

$y = \frac{- 1 + 36 + 205}{120}$

Step 2.4.1.5

Simplify the expression.

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Step 2.4.1.5.1

Add $- 1$ and $36$ .

$y = \frac{35 + 205}{120}$

Step 2.4.1.5.2

Add $35$ and $205$ .

$y = \frac{240}{120}$

Step 2.4.1.5.3

Divide $240$ by $120$ .

$y = 2$

Step 2.4.2

If the table has a quadratic function rule, $y = f (x)$ for the corresponding $x$ value, $x = 25$ . This check passes since $y = 2$ and $f (x) = 2$ .

$2 = 2$

Step 2.4.3

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = - \frac{1}{75000}$ , $b = \frac{3}{250}$ , $c = \frac{41}{24}$ , and $x = 125$ .

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Step 2.4.3.1

Simplify each term.

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Step 2.4.3.1.1

Raise $125$ to the power of $2$ .

$y = - \frac{1}{75000} \cdot 15625 + (\frac{3}{250}) \cdot (125) + \frac{41}{24}$

Step 2.4.3.1.2

Cancel the common factor of $3125$ .

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Step 2.4.3.1.2.1

Move the leading negative in $- \frac{1}{75000}$ into the numerator.

$y = \frac{- 1}{75000} \cdot 15625 + (\frac{3}{250}) \cdot (125) + \frac{41}{24}$

Step 2.4.3.1.2.2

Factor $3125$ out of $75000$ .

$y = \frac{- 1}{3125 (24)} \cdot 15625 + (\frac{3}{250}) \cdot (125) + \frac{41}{24}$

Step 2.4.3.1.2.3

Factor $3125$ out of $15625$ .

$y = \frac{- 1}{3125 \cdot 24} \cdot (3125 \cdot 5) + (\frac{3}{250}) \cdot (125) + \frac{41}{24}$

Step 2.4.3.1.2.4

Cancel the common factor.

$y = \frac{- 1}{3125 \cdot 24} \cdot (3125 \cdot 5) + (\frac{3}{250}) \cdot (125) + \frac{41}{24}$

Step 2.4.3.1.2.5

Rewrite the expression.

$y = \frac{- 1}{24} \cdot 5 + (\frac{3}{250}) \cdot (125) + \frac{41}{24}$

Step 2.4.3.1.3

Combine $\frac{- 1}{24}$ and $5$ .

$y = \frac{- 1 \cdot 5}{24} + (\frac{3}{250}) \cdot (125) + \frac{41}{24}$

Step 2.4.3.1.4

Multiply $- 1$ by $5$ .

$y = \frac{- 5}{24} + (\frac{3}{250}) \cdot (125) + \frac{41}{24}$

Step 2.4.3.1.5

Move the negative in front of the fraction.

$y = - \frac{5}{24} + (\frac{3}{250}) \cdot (125) + \frac{41}{24}$

Step 2.4.3.1.6

Cancel the common factor of $125$ .

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Step 2.4.3.1.6.1

Factor $125$ out of $250$ .

$y = - \frac{5}{24} + \frac{3}{125 (2)} \cdot 125 + \frac{41}{24}$

Step 2.4.3.1.6.2

Cancel the common factor.

$y = - \frac{5}{24} + \frac{3}{125 \cdot 2} \cdot 125 + \frac{41}{24}$

Step 2.4.3.1.6.3

Rewrite the expression.

$y = - \frac{5}{24} + \frac{3}{2} + \frac{41}{24}$

Step 2.4.3.2

Simplify terms.

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Step 2.4.3.2.1

Combine the numerators over the common denominator.

$y = \frac{- 5 + 41}{24} + \frac{3}{2}$

Step 2.4.3.2.2

Add $- 5$ and $41$ .

$y = \frac{36}{24} + \frac{3}{2}$

Step 2.4.3.2.3

Cancel the common factor of $36$ and $24$ .

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Step 2.4.3.2.3.1

Factor $12$ out of $36$ .

$y = \frac{12 (3)}{24} + \frac{3}{2}$

Step 2.4.3.2.3.2

Cancel the common factors.

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Step 2.4.3.2.3.2.1

Factor $12$ out of $24$ .

$y = \frac{12 \cdot 3}{12 \cdot 2} + \frac{3}{2}$

Step 2.4.3.2.3.2.2

Cancel the common factor.

$y = \frac{12 \cdot 3}{12 \cdot 2} + \frac{3}{2}$

Step 2.4.3.2.3.2.3

Rewrite the expression.

$y = \frac{3}{2} + \frac{3}{2}$

Step 2.4.3.2.4

Combine the numerators over the common denominator.

$y = \frac{3 + 3}{2}$

Step 2.4.3.2.5

Simplify the expression.

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Step 2.4.3.2.5.1

Add $3$ and $3$ .

$y = \frac{6}{2}$

Step 2.4.3.2.5.2

Divide $6$ by $2$ .

$y = 3$

Step 2.4.4

If the table has a quadratic function rule, $y = f (x)$ for the corresponding $x$ value, $x = 125$ . This check passes since $y = 3$ and $f (x) = 3$ .

$3 = 3$

Step 2.4.5

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = - \frac{1}{75000}$ , $b = \frac{3}{250}$ , $c = \frac{41}{24}$ , and $x = 625$ .

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Step 2.4.5.1

Simplify each term.

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Step 2.4.5.1.1

Raise $625$ to the power of $2$ .

$y = - \frac{1}{75000} \cdot 390625 + (\frac{3}{250}) \cdot (625) + \frac{41}{24}$

Step 2.4.5.1.2

Cancel the common factor of $3125$ .

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Step 2.4.5.1.2.1

Move the leading negative in $- \frac{1}{75000}$ into the numerator.

$y = \frac{- 1}{75000} \cdot 390625 + (\frac{3}{250}) \cdot (625) + \frac{41}{24}$

Step 2.4.5.1.2.2

Factor $3125$ out of $75000$ .

$y = \frac{- 1}{3125 (24)} \cdot 390625 + (\frac{3}{250}) \cdot (625) + \frac{41}{24}$

Step 2.4.5.1.2.3

Factor $3125$ out of $390625$ .

$y = \frac{- 1}{3125 \cdot 24} \cdot (3125 \cdot 125) + (\frac{3}{250}) \cdot (625) + \frac{41}{24}$

Step 2.4.5.1.2.4

Cancel the common factor.

$y = \frac{- 1}{3125 \cdot 24} \cdot (3125 \cdot 125) + (\frac{3}{250}) \cdot (625) + \frac{41}{24}$

Step 2.4.5.1.2.5

Rewrite the expression.

$y = \frac{- 1}{24} \cdot 125 + (\frac{3}{250}) \cdot (625) + \frac{41}{24}$

Step 2.4.5.1.3

Combine $\frac{- 1}{24}$ and $125$ .

$y = \frac{- 1 \cdot 125}{24} + (\frac{3}{250}) \cdot (625) + \frac{41}{24}$

Step 2.4.5.1.4

Multiply $- 1$ by $125$ .

$y = \frac{- 125}{24} + (\frac{3}{250}) \cdot (625) + \frac{41}{24}$

Step 2.4.5.1.5

Move the negative in front of the fraction.

$y = - \frac{125}{24} + (\frac{3}{250}) \cdot (625) + \frac{41}{24}$

Step 2.4.5.1.6

Cancel the common factor of $125$ .

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Step 2.4.5.1.6.1

Factor $125$ out of $250$ .

$y = - \frac{125}{24} + \frac{3}{125 (2)} \cdot 625 + \frac{41}{24}$

Step 2.4.5.1.6.2

Factor $125$ out of $625$ .

$y = - \frac{125}{24} + \frac{3}{125 \cdot 2} \cdot (125 \cdot 5) + \frac{41}{24}$

Step 2.4.5.1.6.3

Cancel the common factor.

$y = - \frac{125}{24} + \frac{3}{125 \cdot 2} \cdot (125 \cdot 5) + \frac{41}{24}$

Step 2.4.5.1.6.4

Rewrite the expression.

$y = - \frac{125}{24} + \frac{3}{2} \cdot 5 + \frac{41}{24}$

Step 2.4.5.1.7

Combine $\frac{3}{2}$ and $5$ .

$y = - \frac{125}{24} + \frac{3 \cdot 5}{2} + \frac{41}{24}$

Step 2.4.5.1.8

Multiply $3$ by $5$ .

$y = - \frac{125}{24} + \frac{15}{2} + \frac{41}{24}$

Step 2.4.5.2

Combine fractions.

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Step 2.4.5.2.1

Combine the numerators over the common denominator.

$y = \frac{- 125 + 41}{24} + \frac{15}{2}$

Step 2.4.5.2.2

Add $- 125$ and $41$ .

$y = \frac{- 84}{24} + \frac{15}{2}$

Step 2.4.5.3

Simplify each term.

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Step 2.4.5.3.1

Cancel the common factor of $- 84$ and $24$ .

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Step 2.4.5.3.1.1

Factor $12$ out of $- 84$ .

$y = \frac{12 (- 7)}{24} + \frac{15}{2}$

Step 2.4.5.3.1.2

Cancel the common factors.

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Step 2.4.5.3.1.2.1

Factor $12$ out of $24$ .

$y = \frac{12 \cdot - 7}{12 \cdot 2} + \frac{15}{2}$

Step 2.4.5.3.1.2.2

Cancel the common factor.

$y = \frac{12 \cdot - 7}{12 \cdot 2} + \frac{15}{2}$

Step 2.4.5.3.1.2.3

Rewrite the expression.

$y = \frac{- 7}{2} + \frac{15}{2}$

Step 2.4.5.3.2

Move the negative in front of the fraction.

$y = - \frac{7}{2} + \frac{15}{2}$

Step 2.4.5.4

Combine fractions.

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Step 2.4.5.4.1

Combine the numerators over the common denominator.

$y = \frac{- 7 + 15}{2}$

Step 2.4.5.4.2

Simplify the expression.

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Step 2.4.5.4.2.1

Add $- 7$ and $15$ .

$y = \frac{8}{2}$

Step 2.4.5.4.2.2

Divide $8$ by $2$ .

$y = 4$

Step 2.4.6

If the table has a quadratic function rule, $y = f (x)$ for the corresponding $x$ value, $x = 625$ . This check passes since $y = 4$ and $f (x) = 4$ .

$4 = 4$

Step 2.4.7

Since $y = f (x)$ for the corresponding $x$ values, the function is quadratic.

The function is quadratic

Step 3

Since all $y = f (x)$ , the function is quadratic and follows the form $y = - \frac{x^{2}}{75000} + \frac{3 x}{250} + \frac{41}{24}$ .

$y = - \frac{x^{2}}{75000} + \frac{3 x}{250} + \frac{41}{24}$