Algebra Examples

$\begin{array}{c} x & y \\ 10 & 7 \\ 15 & 10.5 \\ 20 & 14 \end{array}$

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 $y = a x + b$ .

$7 = a (10) + b 10.5 = a (15) + b 14 = a (20) + b$

Step 1.3

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

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Step 1.3.1

Solve for $b$ in $7 = a (10) + b$ .

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Step 1.3.1.1

Rewrite the equation as $a (10) + b = 7$ .

$a (10) + b = 7$

$10.5 = a (15) + b$

$14 = a (20) + b$

Step 1.3.1.2

Move $10$ to the left of $a$ .

$10 a + b = 7$

$10.5 = a (15) + b$

$14 = a (20) + b$

Step 1.3.1.3

Subtract $10 a$ from both sides of the equation.

$b = 7 - 10 a$

$10.5 = a (15) + b$

$14 = a (20) + b$

$b = 7 - 10 a$

$10.5 = a (15) + b$

$14 = a (20) + b$

Step 1.3.2

Replace all occurrences of $b$ with $7 - 10 a$ in each equation.

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Step 1.3.2.1

Replace all occurrences of $b$ in $10.5 = a (15) + b$ with $7 - 10 a$ .

$10.5 = a (15) + 7 - 10 a$

$b = 7 - 10 a$

$14 = a (20) + b$

Step 1.3.2.2

Simplify $10.5 = a (15) + 7 - 10 a$ .

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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.

$10.5 = a (15) + 7 - 10 a$

$b = 7 - 10 a$

$14 = a (20) + b$

$10.5 = a (15) + 7 - 10 a$

$b = 7 - 10 a$

$14 = a (20) + b$

Step 1.3.2.2.2

Simplify the right side.

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Step 1.3.2.2.2.1

Simplify $a (15) + 7 - 10 a$ .

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Step 1.3.2.2.2.1.1

Move $15$ to the left of $a$ .

$10.5 = 15 a + 7 - 10 a$

$b = 7 - 10 a$

$14 = a (20) + b$

Step 1.3.2.2.2.1.2

Subtract $10 a$ from $15 a$ .

$10.5 = 5 a + 7$

$b = 7 - 10 a$

$14 = a (20) + b$

$10.5 = 5 a + 7$

$b = 7 - 10 a$

$14 = a (20) + b$

$10.5 = 5 a + 7$

$b = 7 - 10 a$

$14 = a (20) + b$

$10.5 = 5 a + 7$

$b = 7 - 10 a$

$14 = a (20) + b$

Step 1.3.2.3

Replace all occurrences of $b$ in $14 = a (20) + b$ with $7 - 10 a$ .

$14 = a (20) + 7 - 10 a$

$10.5 = 5 a + 7$

$b = 7 - 10 a$

Step 1.3.2.4

Simplify $14 = a (20) + 7 - 10 a$ .

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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.

$14 = a (20) + 7 - 10 a$

$10.5 = 5 a + 7$

$b = 7 - 10 a$

$14 = a (20) + 7 - 10 a$

$10.5 = 5 a + 7$

$b = 7 - 10 a$

Step 1.3.2.4.2

Simplify the right side.

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Step 1.3.2.4.2.1

Simplify $a (20) + 7 - 10 a$ .

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Step 1.3.2.4.2.1.1

Move $20$ to the left of $a$ .

$14 = 20 a + 7 - 10 a$

$10.5 = 5 a + 7$

$b = 7 - 10 a$

Step 1.3.2.4.2.1.2

Subtract $10 a$ from $20 a$ .

$14 = 10 a + 7$

$10.5 = 5 a + 7$

$b = 7 - 10 a$

$14 = 10 a + 7$

$10.5 = 5 a + 7$

$b = 7 - 10 a$

$14 = 10 a + 7$

$10.5 = 5 a + 7$

$b = 7 - 10 a$

$14 = 10 a + 7$

$10.5 = 5 a + 7$

$b = 7 - 10 a$

$14 = 10 a + 7$

$10.5 = 5 a + 7$

$b = 7 - 10 a$

Step 1.3.3

Solve for $a$ in $14 = 10 a + 7$ .

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Step 1.3.3.1

Rewrite the equation as $10 a + 7 = 14$ .

$10 a + 7 = 14$

$10.5 = 5 a + 7$

$b = 7 - 10 a$

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 $7$ from both sides of the equation.

$10 a = 14 - 7$

$10.5 = 5 a + 7$

$b = 7 - 10 a$

Step 1.3.3.2.2

Subtract $7$ from $14$ .

$10 a = 7$

$10.5 = 5 a + 7$

$b = 7 - 10 a$

$10 a = 7$

$10.5 = 5 a + 7$

$b = 7 - 10 a$

Step 1.3.3.3

Divide each term in $10 a = 7$ by $10$ and simplify.

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Step 1.3.3.3.1

Divide each term in $10 a = 7$ by $10$ .

$\frac{10 a}{10} = \frac{7}{10}$

$10.5 = 5 a + 7$

$b = 7 - 10 a$

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 $10$ .

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Step 1.3.3.3.2.1.1

Cancel the common factor.

$\frac{10 a}{10} = \frac{7}{10}$

$10.5 = 5 a + 7$

$b = 7 - 10 a$

Step 1.3.3.3.2.1.2

Divide $a$ by $1$ .

$a = \frac{7}{10}$

$10.5 = 5 a + 7$

$b = 7 - 10 a$

$a = \frac{7}{10}$

$10.5 = 5 a + 7$

$b = 7 - 10 a$

$a = \frac{7}{10}$

$10.5 = 5 a + 7$

$b = 7 - 10 a$

$a = \frac{7}{10}$

$10.5 = 5 a + 7$

$b = 7 - 10 a$

$a = \frac{7}{10}$

$10.5 = 5 a + 7$

$b = 7 - 10 a$

Step 1.3.4

Replace all occurrences of $a$ with $\frac{7}{10}$ in each equation.

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Step 1.3.4.1

Replace all occurrences of $a$ in $10.5 = 5 a + 7$ with $\frac{7}{10}$ .

$10.5 = 5 (\frac{7}{10}) + 7$

$a = \frac{7}{10}$

$b = 7 - 10 a$

Step 1.3.4.2

Simplify the right side.

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Step 1.3.4.2.1

Simplify $5 (\frac{7}{10}) + 7$ .

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Step 1.3.4.2.1.1

Cancel the common factor of $5$ .

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Step 1.3.4.2.1.1.1

Factor $5$ out of $10$ .

$10.5 = 5 (\frac{7}{5 (2)}) + 7$

$a = \frac{7}{10}$

$b = 7 - 10 a$

Step 1.3.4.2.1.1.2

Cancel the common factor.

$10.5 = 5 (\frac{7}{5 \cdot 2}) + 7$

$a = \frac{7}{10}$

$b = 7 - 10 a$

Step 1.3.4.2.1.1.3

Rewrite the expression.

$10.5 = \frac{7}{2} + 7$

$a = \frac{7}{10}$

$b = 7 - 10 a$

$10.5 = \frac{7}{2} + 7$

$a = \frac{7}{10}$

$b = 7 - 10 a$

Step 1.3.4.2.1.2

To write $7$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$10.5 = \frac{7}{2} + 7 \cdot \frac{2}{2}$

$a = \frac{7}{10}$

$b = 7 - 10 a$

Step 1.3.4.2.1.3

Combine $7$ and $\frac{2}{2}$ .

$10.5 = \frac{7}{2} + \frac{7 \cdot 2}{2}$

$a = \frac{7}{10}$

$b = 7 - 10 a$

Step 1.3.4.2.1.4

Combine the numerators over the common denominator.

$10.5 = \frac{7 + 7 \cdot 2}{2}$

$a = \frac{7}{10}$

$b = 7 - 10 a$

Step 1.3.4.2.1.5

Simplify the numerator.

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Step 1.3.4.2.1.5.1

Multiply $7$ by $2$ .

$10.5 = \frac{7 + 14}{2}$

$a = \frac{7}{10}$

$b = 7 - 10 a$

Step 1.3.4.2.1.5.2

Add $7$ and $14$ .

$10.5 = \frac{21}{2}$

$a = \frac{7}{10}$

$b = 7 - 10 a$

$10.5 = \frac{21}{2}$

$a = \frac{7}{10}$

$b = 7 - 10 a$

$10.5 = \frac{21}{2}$

$a = \frac{7}{10}$

$b = 7 - 10 a$

$10.5 = \frac{21}{2}$

$a = \frac{7}{10}$

$b = 7 - 10 a$

Step 1.3.4.3

Replace all occurrences of $a$ in $b = 7 - 10 a$ with $\frac{7}{10}$ .

$b = 7 - 10 (\frac{7}{10})$

$10.5 = \frac{21}{2}$

$a = \frac{7}{10}$

Step 1.3.4.4

Simplify the right side.

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Step 1.3.4.4.1

Simplify $7 - 10 (\frac{7}{10})$ .

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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 $10$ .

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Step 1.3.4.4.1.1.1.1

Factor $10$ out of $- 10$ .

$b = 7 + 10 (- 1) (\frac{7}{10})$

$10.5 = \frac{21}{2}$

$a = \frac{7}{10}$

Step 1.3.4.4.1.1.1.2

Cancel the common factor.

$b = 7 + 10 \cdot (- 1 (\frac{7}{10}))$

$10.5 = \frac{21}{2}$

$a = \frac{7}{10}$

Step 1.3.4.4.1.1.1.3

Rewrite the expression.

$b = 7 - 1 \cdot 7$

$10.5 = \frac{21}{2}$

$a = \frac{7}{10}$

$b = 7 - 1 \cdot 7$

$10.5 = \frac{21}{2}$

$a = \frac{7}{10}$

Step 1.3.4.4.1.1.2

Multiply $- 1$ by $7$ .

$b = 7 - 7$

$10.5 = \frac{21}{2}$

$a = \frac{7}{10}$

$b = 7 - 7$

$10.5 = \frac{21}{2}$

$a = \frac{7}{10}$

Step 1.3.4.4.1.2

Subtract $7$ from $7$ .

$b = 0$

$10.5 = \frac{21}{2}$

$a = \frac{7}{10}$

$b = 0$

$10.5 = \frac{21}{2}$

$a = \frac{7}{10}$

$b = 0$

$10.5 = \frac{21}{2}$

$a = \frac{7}{10}$

$b = 0$

$10.5 = \frac{21}{2}$

$a = \frac{7}{10}$

Step 1.3.5

Since $10.5 = \frac{21}{2}$ is not true, there is no solution.

No solution

Step 1.4

Since $y \neq y$ 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 $y = a x^{2} + b x + c$ .

Step 2.3

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

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Step 2.3.1

Solve for $c$ in $7 = a \cdot 10^{2} + b (10) + c$ .

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Step 2.3.1.1

Rewrite the equation as $a \cdot 10^{2} + b (10) + c = 7$ .

$a \cdot 10^{2} + b (10) + c = 7$

$10.5 = a \cdot 15^{2} + b (15) + c$

$14 = a \cdot 20^{2} + b (20) + c$

Step 2.3.1.2

Simplify each term.

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Step 2.3.1.2.1

Raise $10$ to the power of $2$ .

$a \cdot 100 + b (10) + c = 7$

$10.5 = a \cdot 15^{2} + b (15) + c$

$14 = a \cdot 20^{2} + b (20) + c$

Step 2.3.1.2.2

Move $100$ to the left of $a$ .

$100 \cdot a + b (10) + c = 7$

$10.5 = a \cdot 15^{2} + b (15) + c$

$14 = a \cdot 20^{2} + b (20) + c$

Step 2.3.1.2.3

Move $10$ to the left of $b$ .

$100 a + 10 b + c = 7$

$10.5 = a \cdot 15^{2} + b (15) + c$

$14 = a \cdot 20^{2} + b (20) + c$

$100 a + 10 b + c = 7$

$10.5 = a \cdot 15^{2} + b (15) + c$

$14 = a \cdot 20^{2} + b (20) + c$

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 $100 a$ from both sides of the equation.

$10 b + c = 7 - 100 a$

$10.5 = a \cdot 15^{2} + b (15) + c$

$14 = a \cdot 20^{2} + b (20) + c$

Step 2.3.1.3.2

Subtract $10 b$ from both sides of the equation.

$c = 7 - 100 a - 10 b$

$10.5 = a \cdot 15^{2} + b (15) + c$

$14 = a \cdot 20^{2} + b (20) + c$

$c = 7 - 100 a - 10 b$

$10.5 = a \cdot 15^{2} + b (15) + c$

$14 = a \cdot 20^{2} + b (20) + c$

$c = 7 - 100 a - 10 b$

$10.5 = a \cdot 15^{2} + b (15) + c$

$14 = a \cdot 20^{2} + b (20) + c$

Step 2.3.2

Replace all occurrences of $c$ with $7 - 100 a - 10 b$ in each equation.

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Step 2.3.2.1

Replace all occurrences of $c$ in $10.5 = a \cdot 15^{2} + b (15) + c$ with $7 - 100 a - 10 b$ .

$10.5 = a \cdot 15^{2} + b (15) + 7 - 100 a - 10 b$

$c = 7 - 100 a - 10 b$

$14 = a \cdot 20^{2} + b (20) + c$

Step 2.3.2.2

Simplify $10.5 = a \cdot 15^{2} + b (15) + 7 - 100 a - 10 b$ .

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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.

$10.5 = a \cdot 15^{2} + b (15) + 7 - 100 a - 10 b$

$c = 7 - 100 a - 10 b$

$14 = a \cdot 20^{2} + b (20) + c$

$10.5 = a \cdot 15^{2} + b (15) + 7 - 100 a - 10 b$

$c = 7 - 100 a - 10 b$

$14 = a \cdot 20^{2} + b (20) + c$

Step 2.3.2.2.2

Simplify the right side.

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Step 2.3.2.2.2.1

Simplify $a \cdot 15^{2} + b (15) + 7 - 100 a - 10 b$ .

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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 $15$ to the power of $2$ .

$10.5 = a \cdot 225 + b (15) + 7 - 100 a - 10 b$

$c = 7 - 100 a - 10 b$

$14 = a \cdot 20^{2} + b (20) + c$

Step 2.3.2.2.2.1.1.2

Move $225$ to the left of $a$ .

$10.5 = 225 \cdot a + b (15) + 7 - 100 a - 10 b$

$c = 7 - 100 a - 10 b$

$14 = a \cdot 20^{2} + b (20) + c$

Step 2.3.2.2.2.1.1.3

Move $15$ to the left of $b$ .

$10.5 = 225 a + 15 b + 7 - 100 a - 10 b$

$c = 7 - 100 a - 10 b$

$14 = a \cdot 20^{2} + b (20) + c$

$10.5 = 225 a + 15 b + 7 - 100 a - 10 b$

$c = 7 - 100 a - 10 b$

$14 = a \cdot 20^{2} + b (20) + c$

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 $100 a$ from $225 a$ .

$10.5 = 125 a + 15 b + 7 - 10 b$

$c = 7 - 100 a - 10 b$

$14 = a \cdot 20^{2} + b (20) + c$

Step 2.3.2.2.2.1.2.2

Subtract $10 b$ from $15 b$ .

$10.5 = 125 a + 5 b + 7$

$c = 7 - 100 a - 10 b$

$14 = a \cdot 20^{2} + b (20) + c$

$10.5 = 125 a + 5 b + 7$

$c = 7 - 100 a - 10 b$

$14 = a \cdot 20^{2} + b (20) + c$

$10.5 = 125 a + 5 b + 7$

$c = 7 - 100 a - 10 b$

$14 = a \cdot 20^{2} + b (20) + c$

$10.5 = 125 a + 5 b + 7$

$c = 7 - 100 a - 10 b$

$14 = a \cdot 20^{2} + b (20) + c$

$10.5 = 125 a + 5 b + 7$

$c = 7 - 100 a - 10 b$

$14 = a \cdot 20^{2} + b (20) + c$

Step 2.3.2.3

Replace all occurrences of $c$ in $14 = a \cdot 20^{2} + b (20) + c$ with $7 - 100 a - 10 b$ .

$14 = a \cdot 20^{2} + b (20) + 7 - 100 a - 10 b$

$10.5 = 125 a + 5 b + 7$

$c = 7 - 100 a - 10 b$

Step 2.3.2.4

Simplify $14 = a \cdot 20^{2} + b (20) + 7 - 100 a - 10 b$ .

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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.

$14 = a \cdot 20^{2} + b (20) + 7 - 100 a - 10 b$

$10.5 = 125 a + 5 b + 7$

$c = 7 - 100 a - 10 b$

$14 = a \cdot 20^{2} + b (20) + 7 - 100 a - 10 b$

$10.5 = 125 a + 5 b + 7$

$c = 7 - 100 a - 10 b$

Step 2.3.2.4.2

Simplify the right side.

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Step 2.3.2.4.2.1

Simplify $a \cdot 20^{2} + b (20) + 7 - 100 a - 10 b$ .

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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 $20$ to the power of $2$ .

$14 = a \cdot 400 + b (20) + 7 - 100 a - 10 b$

$10.5 = 125 a + 5 b + 7$

$c = 7 - 100 a - 10 b$

Step 2.3.2.4.2.1.1.2

Move $400$ to the left of $a$ .

$14 = 400 \cdot a + b (20) + 7 - 100 a - 10 b$

$10.5 = 125 a + 5 b + 7$

$c = 7 - 100 a - 10 b$

Step 2.3.2.4.2.1.1.3

Move $20$ to the left of $b$ .

$14 = 400 a + 20 b + 7 - 100 a - 10 b$

$10.5 = 125 a + 5 b + 7$

$c = 7 - 100 a - 10 b$

$14 = 400 a + 20 b + 7 - 100 a - 10 b$

$10.5 = 125 a + 5 b + 7$

$c = 7 - 100 a - 10 b$

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 $100 a$ from $400 a$ .

$14 = 300 a + 20 b + 7 - 10 b$

$10.5 = 125 a + 5 b + 7$

$c = 7 - 100 a - 10 b$

Step 2.3.2.4.2.1.2.2

Subtract $10 b$ from $20 b$ .

$14 = 300 a + 10 b + 7$

$10.5 = 125 a + 5 b + 7$

$c = 7 - 100 a - 10 b$

$14 = 300 a + 10 b + 7$

$10.5 = 125 a + 5 b + 7$

$c = 7 - 100 a - 10 b$

$14 = 300 a + 10 b + 7$

$10.5 = 125 a + 5 b + 7$

$c = 7 - 100 a - 10 b$

$14 = 300 a + 10 b + 7$

$10.5 = 125 a + 5 b + 7$

$c = 7 - 100 a - 10 b$

$14 = 300 a + 10 b + 7$

$10.5 = 125 a + 5 b + 7$

$c = 7 - 100 a - 10 b$

$14 = 300 a + 10 b + 7$

$10.5 = 125 a + 5 b + 7$

$c = 7 - 100 a - 10 b$

Step 2.3.3

Solve for $a$ in $14 = 300 a + 10 b + 7$ .

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Step 2.3.3.1

Rewrite the equation as $300 a + 10 b + 7 = 14$ .

$300 a + 10 b + 7 = 14$

$10.5 = 125 a + 5 b + 7$

$c = 7 - 100 a - 10 b$

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 $10 b$ from both sides of the equation.

$300 a + 7 = 14 - 10 b$

$10.5 = 125 a + 5 b + 7$

$c = 7 - 100 a - 10 b$

Step 2.3.3.2.2

Subtract $7$ from both sides of the equation.

$300 a = 14 - 10 b - 7$

$10.5 = 125 a + 5 b + 7$

$c = 7 - 100 a - 10 b$

Step 2.3.3.2.3

Subtract $7$ from $14$ .

$300 a = - 10 b + 7$

$10.5 = 125 a + 5 b + 7$

$c = 7 - 100 a - 10 b$

$300 a = - 10 b + 7$

$10.5 = 125 a + 5 b + 7$

$c = 7 - 100 a - 10 b$

Step 2.3.3.3

Divide each term in $300 a = - 10 b + 7$ by $300$ and simplify.

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Step 2.3.3.3.1

Divide each term in $300 a = - 10 b + 7$ by $300$ .

$\frac{300 a}{300} = \frac{- 10 b}{300} + \frac{7}{300}$

$10.5 = 125 a + 5 b + 7$

$c = 7 - 100 a - 10 b$

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 $300$ .

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Step 2.3.3.3.2.1.1

Cancel the common factor.

$\frac{300 a}{300} = \frac{- 10 b}{300} + \frac{7}{300}$

$10.5 = 125 a + 5 b + 7$

$c = 7 - 100 a - 10 b$

Step 2.3.3.3.2.1.2

Divide $a$ by $1$ .

$a = \frac{- 10 b}{300} + \frac{7}{300}$

$10.5 = 125 a + 5 b + 7$

$c = 7 - 100 a - 10 b$

$a = \frac{- 10 b}{300} + \frac{7}{300}$

$10.5 = 125 a + 5 b + 7$

$c = 7 - 100 a - 10 b$

$a = \frac{- 10 b}{300} + \frac{7}{300}$

$10.5 = 125 a + 5 b + 7$

$c = 7 - 100 a - 10 b$

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 $- 10$ and $300$ .

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Step 2.3.3.3.3.1.1.1

Factor $10$ out of $- 10 b$ .

$a = \frac{10 (- b)}{300} + \frac{7}{300}$

$10.5 = 125 a + 5 b + 7$

$c = 7 - 100 a - 10 b$

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 $10$ out of $300$ .

$a = \frac{10 (- b)}{10 (30)} + \frac{7}{300}$

$10.5 = 125 a + 5 b + 7$

$c = 7 - 100 a - 10 b$

Step 2.3.3.3.3.1.1.2.2

Cancel the common factor.

$a = \frac{10 (- b)}{10 \cdot 30} + \frac{7}{300}$

$10.5 = 125 a + 5 b + 7$

$c = 7 - 100 a - 10 b$

Step 2.3.3.3.3.1.1.2.3

Rewrite the expression.

$a = \frac{- b}{30} + \frac{7}{300}$

$10.5 = 125 a + 5 b + 7$

$c = 7 - 100 a - 10 b$

$a = \frac{- b}{30} + \frac{7}{300}$

$10.5 = 125 a + 5 b + 7$

$c = 7 - 100 a - 10 b$

$a = \frac{- b}{30} + \frac{7}{300}$

$10.5 = 125 a + 5 b + 7$

$c = 7 - 100 a - 10 b$

Step 2.3.3.3.3.1.2

Move the negative in front of the fraction.

$a = - \frac{b}{30} + \frac{7}{300}$

$10.5 = 125 a + 5 b + 7$

$c = 7 - 100 a - 10 b$

$a = - \frac{b}{30} + \frac{7}{300}$

$10.5 = 125 a + 5 b + 7$

$c = 7 - 100 a - 10 b$

$a = - \frac{b}{30} + \frac{7}{300}$

$10.5 = 125 a + 5 b + 7$

$c = 7 - 100 a - 10 b$

$a = - \frac{b}{30} + \frac{7}{300}$

$10.5 = 125 a + 5 b + 7$

$c = 7 - 100 a - 10 b$

$a = - \frac{b}{30} + \frac{7}{300}$

$10.5 = 125 a + 5 b + 7$

$c = 7 - 100 a - 10 b$

Step 2.3.4

Replace all occurrences of $a$ with $- \frac{b}{30} + \frac{7}{300}$ in each equation.

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Step 2.3.4.1

Replace all occurrences of $a$ in $10.5 = 125 a + 5 b + 7$ with $- \frac{b}{30} + \frac{7}{300}$ .

$10.5 = 125 (- \frac{b}{30} + \frac{7}{300}) + 5 b + 7$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = 7 - 100 a - 10 b$

Step 2.3.4.2

Simplify the right side.

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Step 2.3.4.2.1

Simplify $125 (- \frac{b}{30} + \frac{7}{300}) + 5 b + 7$ .

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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.

$10.5 = 125 (- \frac{b}{30}) + 125 (\frac{7}{300}) + 5 b + 7$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = 7 - 100 a - 10 b$

Step 2.3.4.2.1.1.2

Cancel the common factor of $5$ .

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Step 2.3.4.2.1.1.2.1

Move the leading negative in $- \frac{b}{30}$ into the numerator.

$10.5 = 125 (\frac{- b}{30}) + 125 (\frac{7}{300}) + 5 b + 7$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = 7 - 100 a - 10 b$

Step 2.3.4.2.1.1.2.2

Factor $5$ out of $125$ .

$10.5 = 5 (25) (\frac{- b}{30}) + 125 (\frac{7}{300}) + 5 b + 7$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = 7 - 100 a - 10 b$

Step 2.3.4.2.1.1.2.3

Factor $5$ out of $30$ .

$10.5 = 5 \cdot (25 (\frac{- b}{5 \cdot 6})) + 125 (\frac{7}{300}) + 5 b + 7$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = 7 - 100 a - 10 b$

Step 2.3.4.2.1.1.2.4

Cancel the common factor.

$10.5 = 5 \cdot (25 (\frac{- b}{5 \cdot 6})) + 125 (\frac{7}{300}) + 5 b + 7$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = 7 - 100 a - 10 b$

Step 2.3.4.2.1.1.2.5

Rewrite the expression.

$10.5 = 25 (\frac{- b}{6}) + 125 (\frac{7}{300}) + 5 b + 7$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = 7 - 100 a - 10 b$

$10.5 = 25 (\frac{- b}{6}) + 125 (\frac{7}{300}) + 5 b + 7$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = 7 - 100 a - 10 b$

Step 2.3.4.2.1.1.3

Combine $25$ and $\frac{- b}{6}$ .

$10.5 = \frac{25 (- b)}{6} + 125 (\frac{7}{300}) + 5 b + 7$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = 7 - 100 a - 10 b$

Step 2.3.4.2.1.1.4

Multiply $- 1$ by $25$ .

$10.5 = \frac{- 25 b}{6} + 125 (\frac{7}{300}) + 5 b + 7$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = 7 - 100 a - 10 b$

Step 2.3.4.2.1.1.5

Cancel the common factor of $25$ .

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Step 2.3.4.2.1.1.5.1

Factor $25$ out of $125$ .

$10.5 = \frac{- 25 b}{6} + 25 (5) (\frac{7}{300}) + 5 b + 7$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = 7 - 100 a - 10 b$

Step 2.3.4.2.1.1.5.2

Factor $25$ out of $300$ .

$10.5 = \frac{- 25 b}{6} + 25 \cdot (5 (\frac{7}{25 \cdot 12})) + 5 b + 7$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = 7 - 100 a - 10 b$

Step 2.3.4.2.1.1.5.3

Cancel the common factor.

$10.5 = \frac{- 25 b}{6} + 25 \cdot (5 (\frac{7}{25 \cdot 12})) + 5 b + 7$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = 7 - 100 a - 10 b$

Step 2.3.4.2.1.1.5.4

Rewrite the expression.

$10.5 = \frac{- 25 b}{6} + 5 (\frac{7}{12}) + 5 b + 7$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = 7 - 100 a - 10 b$

$10.5 = \frac{- 25 b}{6} + 5 (\frac{7}{12}) + 5 b + 7$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = 7 - 100 a - 10 b$

Step 2.3.4.2.1.1.6

Combine $5$ and $\frac{7}{12}$ .

$10.5 = \frac{- 25 b}{6} + \frac{5 \cdot 7}{12} + 5 b + 7$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = 7 - 100 a - 10 b$

Step 2.3.4.2.1.1.7

Multiply $5$ by $7$ .

$10.5 = \frac{- 25 b}{6} + \frac{35}{12} + 5 b + 7$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = 7 - 100 a - 10 b$

Step 2.3.4.2.1.1.8

Move the negative in front of the fraction.

$10.5 = - \frac{25 b}{6} + \frac{35}{12} + 5 b + 7$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = 7 - 100 a - 10 b$

$10.5 = - \frac{25 b}{6} + \frac{35}{12} + 5 b + 7$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = 7 - 100 a - 10 b$

Step 2.3.4.2.1.2

To write $5 b$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$10.5 = - \frac{25 b}{6} + 5 b \cdot \frac{6}{6} + \frac{35}{12} + 7$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = 7 - 100 a - 10 b$

Step 2.3.4.2.1.3

Combine $5 b$ and $\frac{6}{6}$ .

$10.5 = - \frac{25 b}{6} + \frac{5 b \cdot 6}{6} + \frac{35}{12} + 7$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = 7 - 100 a - 10 b$

Step 2.3.4.2.1.4

Combine the numerators over the common denominator.

$10.5 = \frac{- 25 b + 5 b \cdot 6}{6} + \frac{35}{12} + 7$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = 7 - 100 a - 10 b$

Step 2.3.4.2.1.5

Find the common denominator.

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Step 2.3.4.2.1.5.1

Multiply $\frac{- 25 b + 5 b \cdot 6}{6}$ by $\frac{2}{2}$ .

$10.5 = \frac{- 25 b + 5 b \cdot 6}{6} \cdot \frac{2}{2} + \frac{35}{12} + 7$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = 7 - 100 a - 10 b$

Step 2.3.4.2.1.5.2

Multiply $\frac{- 25 b + 5 b \cdot 6}{6}$ by $\frac{2}{2}$ .

$10.5 = \frac{(- 25 b + 5 b \cdot 6) \cdot 2}{6 \cdot 2} + \frac{35}{12} + 7$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = 7 - 100 a - 10 b$

Step 2.3.4.2.1.5.3

Write $7$ as a fraction with denominator $1$ .

$10.5 = \frac{(- 25 b + 5 b \cdot 6) \cdot 2}{6 \cdot 2} + \frac{35}{12} + \frac{7}{1}$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = 7 - 100 a - 10 b$

Step 2.3.4.2.1.5.4

Multiply $\frac{7}{1}$ by $\frac{12}{12}$ .

$10.5 = \frac{(- 25 b + 5 b \cdot 6) \cdot 2}{6 \cdot 2} + \frac{35}{12} + \frac{7}{1} \cdot \frac{12}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = 7 - 100 a - 10 b$

Step 2.3.4.2.1.5.5

Multiply $\frac{7}{1}$ by $\frac{12}{12}$ .

$10.5 = \frac{(- 25 b + 5 b \cdot 6) \cdot 2}{6 \cdot 2} + \frac{35}{12} + \frac{7 \cdot 12}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = 7 - 100 a - 10 b$

Step 2.3.4.2.1.5.6

Reorder the factors of $6 \cdot 2$ .

$10.5 = \frac{(- 25 b + 5 b \cdot 6) \cdot 2}{2 \cdot 6} + \frac{35}{12} + \frac{7 \cdot 12}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = 7 - 100 a - 10 b$

Step 2.3.4.2.1.5.7

Multiply $2$ by $6$ .

$10.5 = \frac{(- 25 b + 5 b \cdot 6) \cdot 2}{12} + \frac{35}{12} + \frac{7 \cdot 12}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = 7 - 100 a - 10 b$

$10.5 = \frac{(- 25 b + 5 b \cdot 6) \cdot 2}{12} + \frac{35}{12} + \frac{7 \cdot 12}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = 7 - 100 a - 10 b$

Step 2.3.4.2.1.6

Combine the numerators over the common denominator.

$10.5 = \frac{(- 25 b + 5 b \cdot 6) \cdot 2 + 35 + 7 \cdot 12}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = 7 - 100 a - 10 b$

Step 2.3.4.2.1.7

Simplify each term.

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Step 2.3.4.2.1.7.1

Multiply $6$ by $5$ .

$10.5 = \frac{(- 25 b + 30 b) \cdot 2 + 35 + 7 \cdot 12}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = 7 - 100 a - 10 b$

Step 2.3.4.2.1.7.2

Add $- 25 b$ and $30 b$ .

$10.5 = \frac{5 b \cdot 2 + 35 + 7 \cdot 12}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = 7 - 100 a - 10 b$

Step 2.3.4.2.1.7.3

Multiply $2$ by $5$ .

$10.5 = \frac{10 b + 35 + 7 \cdot 12}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = 7 - 100 a - 10 b$

Step 2.3.4.2.1.7.4

Multiply $7$ by $12$ .

$10.5 = \frac{10 b + 35 + 84}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = 7 - 100 a - 10 b$

$10.5 = \frac{10 b + 35 + 84}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = 7 - 100 a - 10 b$

Step 2.3.4.2.1.8

Add $35$ and $84$ .

$10.5 = \frac{10 b + 119}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = 7 - 100 a - 10 b$

$10.5 = \frac{10 b + 119}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = 7 - 100 a - 10 b$

$10.5 = \frac{10 b + 119}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = 7 - 100 a - 10 b$

Step 2.3.4.3

Replace all occurrences of $a$ in $c = 7 - 100 a - 10 b$ with $- \frac{b}{30} + \frac{7}{300}$ .

$c = 7 - 100 (- \frac{b}{30} + \frac{7}{300}) - 10 b$

$10.5 = \frac{10 b + 119}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

Step 2.3.4.4

Simplify the right side.

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Step 2.3.4.4.1

Simplify $7 - 100 (- \frac{b}{30} + \frac{7}{300}) - 10 b$ .

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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.

$c = 7 - 100 (- \frac{b}{30}) - 100 (\frac{7}{300}) - 10 b$

$10.5 = \frac{10 b + 119}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

Step 2.3.4.4.1.1.2

Cancel the common factor of $10$ .

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Step 2.3.4.4.1.1.2.1

Move the leading negative in $- \frac{b}{30}$ into the numerator.

$c = 7 - 100 \frac{- b}{30} - 100 (\frac{7}{300}) - 10 b$

$10.5 = \frac{10 b + 119}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

Step 2.3.4.4.1.1.2.2

Factor $10$ out of $- 100$ .

$c = 7 + 10 (- 10) (\frac{- b}{30}) - 100 (\frac{7}{300}) - 10 b$

$10.5 = \frac{10 b + 119}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

Step 2.3.4.4.1.1.2.3

Factor $10$ out of $30$ .

$c = 7 + 10 \cdot (- 10 \frac{- b}{10 \cdot 3}) - 100 (\frac{7}{300}) - 10 b$

$10.5 = \frac{10 b + 119}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

Step 2.3.4.4.1.1.2.4

Cancel the common factor.

$c = 7 + 10 \cdot (- 10 \frac{- b}{10 \cdot 3}) - 100 (\frac{7}{300}) - 10 b$

$10.5 = \frac{10 b + 119}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

Step 2.3.4.4.1.1.2.5

Rewrite the expression.

$c = 7 - 10 \frac{- b}{3} - 100 (\frac{7}{300}) - 10 b$

$10.5 = \frac{10 b + 119}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = 7 - 10 \frac{- b}{3} - 100 (\frac{7}{300}) - 10 b$

$10.5 = \frac{10 b + 119}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

Step 2.3.4.4.1.1.3

Combine $- 10$ and $\frac{- b}{3}$ .

$c = 7 + \frac{- 10 (- b)}{3} - 100 (\frac{7}{300}) - 10 b$

$10.5 = \frac{10 b + 119}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

Step 2.3.4.4.1.1.4

Multiply $- 1$ by $- 10$ .

$c = 7 + \frac{10 b}{3} - 100 (\frac{7}{300}) - 10 b$

$10.5 = \frac{10 b + 119}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

Step 2.3.4.4.1.1.5

Cancel the common factor of $100$ .

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Step 2.3.4.4.1.1.5.1

Factor $100$ out of $- 100$ .

$c = 7 + \frac{10 b}{3} + 100 (- 1) (\frac{7}{300}) - 10 b$

$10.5 = \frac{10 b + 119}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

Step 2.3.4.4.1.1.5.2

Factor $100$ out of $300$ .

$c = 7 + \frac{10 b}{3} + 100 \cdot (- 1 \frac{7}{100 \cdot 3}) - 10 b$

$10.5 = \frac{10 b + 119}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

Step 2.3.4.4.1.1.5.3

Cancel the common factor.

$c = 7 + \frac{10 b}{3} + 100 \cdot (- 1 \frac{7}{100 \cdot 3}) - 10 b$

$10.5 = \frac{10 b + 119}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

Step 2.3.4.4.1.1.5.4

Rewrite the expression.

$c = 7 + \frac{10 b}{3} - 1 (\frac{7}{3}) - 10 b$

$10.5 = \frac{10 b + 119}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = 7 + \frac{10 b}{3} - 1 (\frac{7}{3}) - 10 b$

$10.5 = \frac{10 b + 119}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

Step 2.3.4.4.1.1.6

Rewrite $- 1 (\frac{7}{3})$ as $- (\frac{7}{3})$ .

$c = 7 + \frac{10 b}{3} - \frac{7}{3} - 10 b$

$10.5 = \frac{10 b + 119}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = 7 + \frac{10 b}{3} - \frac{7}{3} - 10 b$

$10.5 = \frac{10 b + 119}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

Step 2.3.4.4.1.2

To write $7$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$c = \frac{10 b}{3} + 7 \cdot \frac{3}{3} - \frac{7}{3} - 10 b$

$10.5 = \frac{10 b + 119}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

Step 2.3.4.4.1.3

Combine $7$ and $\frac{3}{3}$ .

$c = \frac{10 b}{3} + \frac{7 \cdot 3}{3} - \frac{7}{3} - 10 b$

$10.5 = \frac{10 b + 119}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

Step 2.3.4.4.1.4

Combine the numerators over the common denominator.

$c = \frac{10 b}{3} + \frac{7 \cdot 3 - 7}{3} - 10 b$

$10.5 = \frac{10 b + 119}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

Step 2.3.4.4.1.5

Simplify the numerator.

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Step 2.3.4.4.1.5.1

Multiply $7$ by $3$ .

$c = \frac{10 b}{3} + \frac{21 - 7}{3} - 10 b$

$10.5 = \frac{10 b + 119}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

Step 2.3.4.4.1.5.2

Subtract $7$ from $21$ .

$c = \frac{10 b}{3} + \frac{14}{3} - 10 b$

$10.5 = \frac{10 b + 119}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = \frac{10 b}{3} + \frac{14}{3} - 10 b$

$10.5 = \frac{10 b + 119}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

Step 2.3.4.4.1.6

To write $- 10 b$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$c = \frac{10 b}{3} - 10 b \cdot \frac{3}{3} + \frac{14}{3}$

$10.5 = \frac{10 b + 119}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

Step 2.3.4.4.1.7

Combine $- 10 b$ and $\frac{3}{3}$ .

$c = \frac{10 b}{3} + \frac{- 10 b \cdot 3}{3} + \frac{14}{3}$

$10.5 = \frac{10 b + 119}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

Step 2.3.4.4.1.8

Combine the numerators over the common denominator.

$c = \frac{10 b - 10 b \cdot 3}{3} + \frac{14}{3}$

$10.5 = \frac{10 b + 119}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

Step 2.3.4.4.1.9

Combine the numerators over the common denominator.

$c = \frac{10 b - 10 b \cdot 3 + 14}{3}$

$10.5 = \frac{10 b + 119}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

Step 2.3.4.4.1.10

Multiply $3$ by $- 10$ .

$c = \frac{10 b - 30 b + 14}{3}$

$10.5 = \frac{10 b + 119}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

Step 2.3.4.4.1.11

Subtract $30 b$ from $10 b$ .

$c = \frac{- 20 b + 14}{3}$

$10.5 = \frac{10 b + 119}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

Step 2.3.4.4.1.12

Factor $2$ out of $- 20 b + 14$ .

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Step 2.3.4.4.1.12.1

Factor $2$ out of $- 20 b$ .

$c = \frac{2 (- 10 b) + 14}{3}$

$10.5 = \frac{10 b + 119}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

Step 2.3.4.4.1.12.2

Factor $2$ out of $14$ .

$c = \frac{2 (- 10 b) + 2 (7)}{3}$

$10.5 = \frac{10 b + 119}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

Step 2.3.4.4.1.12.3

Factor $2$ out of $2 (- 10 b) + 2 (7)$ .

$c = \frac{2 (- 10 b + 7)}{3}$

$10.5 = \frac{10 b + 119}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = \frac{2 (- 10 b + 7)}{3}$

$10.5 = \frac{10 b + 119}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

Step 2.3.4.4.1.13

Factor $- 1$ out of $- 10 b$ .

$c = \frac{2 (- (10 b) + 7)}{3}$

$10.5 = \frac{10 b + 119}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

Step 2.3.4.4.1.14

Rewrite $7$ as $- 1 (- 7)$ .

$c = \frac{2 (- (10 b) - 1 \cdot - 7)}{3}$

$10.5 = \frac{10 b + 119}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

Step 2.3.4.4.1.15

Factor $- 1$ out of $- (10 b) - 1 (- 7)$ .

$c = \frac{2 (- (10 b - 7))}{3}$

$10.5 = \frac{10 b + 119}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

Step 2.3.4.4.1.16

Simplify the expression.

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Step 2.3.4.4.1.16.1

Rewrite $- (10 b - 7)$ as $- 1 (10 b - 7)$ .

$c = \frac{2 (- 1 (10 b - 7))}{3}$

$10.5 = \frac{10 b + 119}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

Step 2.3.4.4.1.16.2

Move the negative in front of the fraction.

$c = - \frac{2 (10 b - 7)}{3}$

$10.5 = \frac{10 b + 119}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = - \frac{2 (10 b - 7)}{3}$

$10.5 = \frac{10 b + 119}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = - \frac{2 (10 b - 7)}{3}$

$10.5 = \frac{10 b + 119}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = - \frac{2 (10 b - 7)}{3}$

$10.5 = \frac{10 b + 119}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = - \frac{2 (10 b - 7)}{3}$

$10.5 = \frac{10 b + 119}{12}$

$a = - \frac{b}{30} + \frac{7}{300}$

Step 2.3.5

Solve for $b$ in $10.5 = \frac{10 b + 119}{12}$ .

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Step 2.3.5.1

Rewrite the equation as $\frac{10 b + 119}{12} = 10.5$ .

$\frac{10 b + 119}{12} = 10.5$

$c = - \frac{2 (10 b - 7)}{3}$

$a = - \frac{b}{30} + \frac{7}{300}$

Step 2.3.5.2

Multiply both sides by $12$ .

$\frac{10 b + 119}{12} \cdot 12 = 10.5 \cdot 12$

$c = - \frac{2 (10 b - 7)}{3}$

$a = - \frac{b}{30} + \frac{7}{300}$

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 $12$ .

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Step 2.3.5.3.1.1.1

Cancel the common factor.

$\frac{10 b + 119}{12} \cdot 12 = 10.5 \cdot 12$

$c = - \frac{2 (10 b - 7)}{3}$

$a = - \frac{b}{30} + \frac{7}{300}$

Step 2.3.5.3.1.1.2

Rewrite the expression.

$10 b + 119 = 10.5 \cdot 12$

$c = - \frac{2 (10 b - 7)}{3}$

$a = - \frac{b}{30} + \frac{7}{300}$

$10 b + 119 = 10.5 \cdot 12$

$c = - \frac{2 (10 b - 7)}{3}$

$a = - \frac{b}{30} + \frac{7}{300}$

$10 b + 119 = 10.5 \cdot 12$

$c = - \frac{2 (10 b - 7)}{3}$

$a = - \frac{b}{30} + \frac{7}{300}$

Step 2.3.5.3.2

Simplify the right side.

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Step 2.3.5.3.2.1

Multiply $10.5$ by $12$ .

$10 b + 119 = 126$

$c = - \frac{2 (10 b - 7)}{3}$

$a = - \frac{b}{30} + \frac{7}{300}$

$10 b + 119 = 126$

$c = - \frac{2 (10 b - 7)}{3}$

$a = - \frac{b}{30} + \frac{7}{300}$

$10 b + 119 = 126$

$c = - \frac{2 (10 b - 7)}{3}$

$a = - \frac{b}{30} + \frac{7}{300}$

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 $119$ from both sides of the equation.

$10 b = 126 - 119$

$c = - \frac{2 (10 b - 7)}{3}$

$a = - \frac{b}{30} + \frac{7}{300}$

Step 2.3.5.4.1.2

Subtract $119$ from $126$ .

$10 b = 7$

$c = - \frac{2 (10 b - 7)}{3}$

$a = - \frac{b}{30} + \frac{7}{300}$

$10 b = 7$

$c = - \frac{2 (10 b - 7)}{3}$

$a = - \frac{b}{30} + \frac{7}{300}$

Step 2.3.5.4.2

Divide each term in $10 b = 7$ by $10$ and simplify.

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Step 2.3.5.4.2.1

Divide each term in $10 b = 7$ by $10$ .

$\frac{10 b}{10} = \frac{7}{10}$

$c = - \frac{2 (10 b - 7)}{3}$

$a = - \frac{b}{30} + \frac{7}{300}$

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 $10$ .

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Step 2.3.5.4.2.2.1.1

Cancel the common factor.

$\frac{10 b}{10} = \frac{7}{10}$

$c = - \frac{2 (10 b - 7)}{3}$

$a = - \frac{b}{30} + \frac{7}{300}$

Step 2.3.5.4.2.2.1.2

Divide $b$ by $1$ .

$b = \frac{7}{10}$

$c = - \frac{2 (10 b - 7)}{3}$

$a = - \frac{b}{30} + \frac{7}{300}$

$b = \frac{7}{10}$

$c = - \frac{2 (10 b - 7)}{3}$

$a = - \frac{b}{30} + \frac{7}{300}$

$b = \frac{7}{10}$

$c = - \frac{2 (10 b - 7)}{3}$

$a = - \frac{b}{30} + \frac{7}{300}$

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 $7$ and $10$ .

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Step 2.3.5.4.2.3.1.1

Rewrite $7$ as $1 (7)$ .

$b = \frac{1 (7)}{10}$

$c = - \frac{2 (10 b - 7)}{3}$

$a = - \frac{b}{30} + \frac{7}{300}$

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

Rewrite $10$ as $1 (10)$ .

$b = \frac{1 \cdot 7}{1 \cdot 10}$

$c = - \frac{2 (10 b - 7)}{3}$

$a = - \frac{b}{30} + \frac{7}{300}$

Step 2.3.5.4.2.3.1.2.2

Cancel the common factor.

$b = \frac{1 \cdot 7}{1 \cdot 10}$

$c = - \frac{2 (10 b - 7)}{3}$

$a = - \frac{b}{30} + \frac{7}{300}$

Step 2.3.5.4.2.3.1.2.3

Rewrite the expression.

$b = \frac{7}{10}$

$c = - \frac{2 (10 b - 7)}{3}$

$a = - \frac{b}{30} + \frac{7}{300}$

$b = \frac{7}{10}$

$c = - \frac{2 (10 b - 7)}{3}$

$a = - \frac{b}{30} + \frac{7}{300}$

$b = \frac{7}{10}$

$c = - \frac{2 (10 b - 7)}{3}$

$a = - \frac{b}{30} + \frac{7}{300}$

$b = \frac{7}{10}$

$c = - \frac{2 (10 b - 7)}{3}$

$a = - \frac{b}{30} + \frac{7}{300}$

$b = \frac{7}{10}$

$c = - \frac{2 (10 b - 7)}{3}$

$a = - \frac{b}{30} + \frac{7}{300}$

$b = \frac{7}{10}$

$c = - \frac{2 (10 b - 7)}{3}$

$a = - \frac{b}{30} + \frac{7}{300}$

$b = \frac{7}{10}$

$c = - \frac{2 (10 b - 7)}{3}$

$a = - \frac{b}{30} + \frac{7}{300}$

Step 2.3.6

Replace all occurrences of $b$ with $\frac{7}{10}$ in each equation.

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Step 2.3.6.1

Replace all occurrences of $b$ in $c = - \frac{2 (10 b - 7)}{3}$ with $\frac{7}{10}$ .

$c = - \frac{2 (10 (\frac{7}{10}) - 7)}{3}$

$b = \frac{7}{10}$

$a = - \frac{b}{30} + \frac{7}{300}$

Step 2.3.6.2

Simplify the right side.

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Step 2.3.6.2.1

Simplify $- \frac{2 (10 (\frac{7}{10}) - 7)}{3}$ .

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Step 2.3.6.2.1.1

Simplify the numerator.

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Step 2.3.6.2.1.1.1

Cancel the common factor of $10$ .

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Step 2.3.6.2.1.1.1.1

Cancel the common factor.

$c = - \frac{2 (10 (\frac{7}{10}) - 7)}{3}$

$b = \frac{7}{10}$

$a = - \frac{b}{30} + \frac{7}{300}$

Step 2.3.6.2.1.1.1.2

Rewrite the expression.

$c = - \frac{2 (7 - 7)}{3}$

$b = \frac{7}{10}$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = - \frac{2 (7 - 7)}{3}$

$b = \frac{7}{10}$

$a = - \frac{b}{30} + \frac{7}{300}$

Step 2.3.6.2.1.1.2

Subtract $7$ from $7$ .

$c = - \frac{2 \cdot 0}{3}$

$b = \frac{7}{10}$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = - \frac{2 \cdot 0}{3}$

$b = \frac{7}{10}$

$a = - \frac{b}{30} + \frac{7}{300}$

Step 2.3.6.2.1.2

Simplify the expression.

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Step 2.3.6.2.1.2.1

Multiply $2$ by $0$ .

$c = - \frac{0}{3}$

$b = \frac{7}{10}$

$a = - \frac{b}{30} + \frac{7}{300}$

Step 2.3.6.2.1.2.2

Divide $0$ by $3$ .

$c = - 0$

$b = \frac{7}{10}$

$a = - \frac{b}{30} + \frac{7}{300}$

Step 2.3.6.2.1.2.3

Multiply $- 1$ by $0$ .

$c = 0$

$b = \frac{7}{10}$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = 0$

$b = \frac{7}{10}$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = 0$

$b = \frac{7}{10}$

$a = - \frac{b}{30} + \frac{7}{300}$

$c = 0$

$b = \frac{7}{10}$

$a = - \frac{b}{30} + \frac{7}{300}$

Step 2.3.6.3

Replace all occurrences of $b$ in $a = - \frac{b}{30} + \frac{7}{300}$ with $\frac{7}{10}$ .

$a = - \frac{\frac{7}{10}}{30} + \frac{7}{300}$

$c = 0$

$b = \frac{7}{10}$

Step 2.3.6.4

Simplify the right side.

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Step 2.3.6.4.1

Simplify $- \frac{\frac{7}{10}}{30} + \frac{7}{300}$ .

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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.

$a = - (\frac{7}{10} \cdot \frac{1}{30}) + \frac{7}{300}$

$c = 0$

$b = \frac{7}{10}$

Step 2.3.6.4.1.1.2

Multiply $\frac{7}{10} \cdot \frac{1}{30}$ .

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Step 2.3.6.4.1.1.2.1

Multiply $\frac{7}{10}$ by $\frac{1}{30}$ .

$a = - \frac{7}{10 \cdot 30} + \frac{7}{300}$

$c = 0$

$b = \frac{7}{10}$

Step 2.3.6.4.1.1.2.2

Multiply $10$ by $30$ .

$a = - \frac{7}{300} + \frac{7}{300}$

$c = 0$

$b = \frac{7}{10}$

$a = - \frac{7}{300} + \frac{7}{300}$

$c = 0$

$b = \frac{7}{10}$

$a = - \frac{7}{300} + \frac{7}{300}$

$c = 0$

$b = \frac{7}{10}$

Step 2.3.6.4.1.2

Combine fractions.

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Step 2.3.6.4.1.2.1

Combine the numerators over the common denominator.

$a = \frac{- 7 + 7}{300}$

$c = 0$

$b = \frac{7}{10}$

Step 2.3.6.4.1.2.2

Simplify the expression.

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Step 2.3.6.4.1.2.2.1

Add $- 7$ and $7$ .

$a = \frac{0}{300}$

$c = 0$

$b = \frac{7}{10}$

Step 2.3.6.4.1.2.2.2

Divide $0$ by $300$ .

$a = 0$

$c = 0$

$b = \frac{7}{10}$

$a = 0$

$c = 0$

$b = \frac{7}{10}$

$a = 0$

$c = 0$

$b = \frac{7}{10}$

$a = 0$

$c = 0$

$b = \frac{7}{10}$

$a = 0$

$c = 0$

$b = \frac{7}{10}$

$a = 0$

$c = 0$

$b = \frac{7}{10}$

Step 2.3.7

List all of the solutions.

$a = 0, c = 0, b = \frac{7}{10}$

Step 2.4

Calculate the value of $y$ using each $x$ value in the table and compare this value to the given $y$ 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 $a = 0$ , $b = \frac{7}{10}$ , $c = 0$ , and $x = 10$ .

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Step 2.4.1.1

Simplify each term.

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Step 2.4.1.1.1

Convert $(0) \cdot {(10)}^{2}$ from scientific notation.

$y = 0 + (\frac{7}{10}) \cdot (10) + 0$

Step 2.4.1.1.2

Cancel the common factor of $10$ .

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Step 2.4.1.1.2.1

Cancel the common factor.

$y = 0 + \frac{7}{10} \cdot 10 + 0$

Step 2.4.1.1.2.2

Rewrite the expression.

$y = 0 + 7 + 0$

Step 2.4.1.2

Simplify by adding numbers.

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Step 2.4.1.2.1

Add $0$ and $7$ .

$y = 7 + 0$

Step 2.4.1.2.2

Add $7$ and $0$ .

$y = 7$

Step 2.4.2

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = 10$ . This check passes since $y = 7$ and $y = 7$ .

$7 = 7$

Step 2.4.3

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = 0$ , $b = \frac{7}{10}$ , $c = 0$ , and $x = 15$ .

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Step 2.4.3.1

Simplify each term.

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Step 2.4.3.1.1

Raise $15$ to the power of $2$ .

$y = 0 \cdot 225 + (\frac{7}{10}) \cdot (15) + 0$

Step 2.4.3.1.2

Multiply $0$ by $225$ .

$y = 0 + (\frac{7}{10}) \cdot (15) + 0$

Step 2.4.3.1.3

Cancel the common factor of $5$ .

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Step 2.4.3.1.3.1

Factor $5$ out of $10$ .

$y = 0 + \frac{7}{5 (2)} \cdot 15 + 0$

Step 2.4.3.1.3.2

Factor $5$ out of $15$ .

$y = 0 + \frac{7}{5 \cdot 2} \cdot (5 \cdot 3) + 0$

Step 2.4.3.1.3.3

Cancel the common factor.

$y = 0 + \frac{7}{5 \cdot 2} \cdot (5 \cdot 3) + 0$

Step 2.4.3.1.3.4

Rewrite the expression.

$y = 0 + \frac{7}{2} \cdot 3 + 0$

Step 2.4.3.1.4

Combine $\frac{7}{2}$ and $3$ .

$y = 0 + \frac{7 \cdot 3}{2} + 0$

Step 2.4.3.1.5

Multiply $7$ by $3$ .

$y = 0 + \frac{21}{2} + 0$

Step 2.4.3.2

Simplify by adding zeros.

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Step 2.4.3.2.1

Add $0$ and $\frac{21}{2}$ .

$y = \frac{21}{2} + 0$

Step 2.4.3.2.2

Add $\frac{21}{2}$ and $0$ .

$y = \frac{21}{2}$

Step 2.4.4

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = 15$ . This check does not pass, since $y = \frac{21}{2}$ and $y = 10.5$ . The function rule can't be quadratic.

$\frac{21}{2} \neq 10.5$

Step 2.4.5

Since $y \neq y$ for the corresponding $x$ values, the function is not quadratic.

The function is not quadratic

Step 3

There are no values of $a$ , $b$ , or $c$ in the equations $y = a x + b$ or $y = a x^{2} + b x + c$ that work for every pair of $x$ and $y$ .

The table does not have a function rule that is linear or quadratic.