Algebra Examples

$\begin{array}{c} x & y \\ 2 & 11 \\ 4 & 23 \\ 5 & 30 \end{array}$

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 $y = a x + b$ .

$11 = a (2) + b 23 = a (4) + b 30 = a (5) + b$

Step 1.3

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

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Step 1.3.1

Solve for $b$ in $11 = a (2) + b$ .

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Step 1.3.1.1

Rewrite the equation as $a (2) + b = 11$ .

$a (2) + b = 11$

$23 = a (4) + b$

$30 = a (5) + b$

Step 1.3.1.2

Move $2$ to the left of $a$ .

$2 a + b = 11$

$23 = a (4) + b$

$30 = a (5) + b$

Step 1.3.1.3

Subtract $2 a$ from both sides of the equation.

$b = 11 - 2 a$

$23 = a (4) + b$

$30 = a (5) + b$

$b = 11 - 2 a$

$23 = a (4) + b$

$30 = a (5) + b$

Step 1.3.2

Replace all occurrences of $b$ with $11 - 2 a$ in each equation.

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Step 1.3.2.1

Replace all occurrences of $b$ in $23 = a (4) + b$ with $11 - 2 a$ .

$23 = a (4) + 11 - 2 a$

$b = 11 - 2 a$

$30 = a (5) + b$

Step 1.3.2.2

Simplify $23 = a (4) + 11 - 2 a$ .

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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.

$23 = a (4) + 11 - 2 a$

$b = 11 - 2 a$

$30 = a (5) + b$

$23 = a (4) + 11 - 2 a$

$b = 11 - 2 a$

$30 = a (5) + b$

Step 1.3.2.2.2

Simplify the right side.

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Step 1.3.2.2.2.1

Simplify $a (4) + 11 - 2 a$ .

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Step 1.3.2.2.2.1.1

Move $4$ to the left of $a$ .

$23 = 4 a + 11 - 2 a$

$b = 11 - 2 a$

$30 = a (5) + b$

Step 1.3.2.2.2.1.2

Subtract $2 a$ from $4 a$ .

$23 = 2 a + 11$

$b = 11 - 2 a$

$30 = a (5) + b$

$23 = 2 a + 11$

$b = 11 - 2 a$

$30 = a (5) + b$

$23 = 2 a + 11$

$b = 11 - 2 a$

$30 = a (5) + b$

$23 = 2 a + 11$

$b = 11 - 2 a$

$30 = a (5) + b$

Step 1.3.2.3

Replace all occurrences of $b$ in $30 = a (5) + b$ with $11 - 2 a$ .

$30 = a (5) + 11 - 2 a$

$23 = 2 a + 11$

$b = 11 - 2 a$

Step 1.3.2.4

Simplify $30 = a (5) + 11 - 2 a$ .

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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.

$30 = a (5) + 11 - 2 a$

$23 = 2 a + 11$

$b = 11 - 2 a$

$30 = a (5) + 11 - 2 a$

$23 = 2 a + 11$

$b = 11 - 2 a$

Step 1.3.2.4.2

Simplify the right side.

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Step 1.3.2.4.2.1

Simplify $a (5) + 11 - 2 a$ .

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Step 1.3.2.4.2.1.1

Move $5$ to the left of $a$ .

$30 = 5 a + 11 - 2 a$

$23 = 2 a + 11$

$b = 11 - 2 a$

Step 1.3.2.4.2.1.2

Subtract $2 a$ from $5 a$ .

$30 = 3 a + 11$

$23 = 2 a + 11$

$b = 11 - 2 a$

$30 = 3 a + 11$

$23 = 2 a + 11$

$b = 11 - 2 a$

$30 = 3 a + 11$

$23 = 2 a + 11$

$b = 11 - 2 a$

$30 = 3 a + 11$

$23 = 2 a + 11$

$b = 11 - 2 a$

$30 = 3 a + 11$

$23 = 2 a + 11$

$b = 11 - 2 a$

Step 1.3.3

Solve for $a$ in $30 = 3 a + 11$ .

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Step 1.3.3.1

Rewrite the equation as $3 a + 11 = 30$ .

$3 a + 11 = 30$

$23 = 2 a + 11$

$b = 11 - 2 a$

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 $11$ from both sides of the equation.

$3 a = 30 - 11$

$23 = 2 a + 11$

$b = 11 - 2 a$

Step 1.3.3.2.2

Subtract $11$ from $30$ .

$3 a = 19$

$23 = 2 a + 11$

$b = 11 - 2 a$

$3 a = 19$

$23 = 2 a + 11$

$b = 11 - 2 a$

Step 1.3.3.3

Divide each term in $3 a = 19$ by $3$ and simplify.

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Step 1.3.3.3.1

Divide each term in $3 a = 19$ by $3$ .

$\frac{3 a}{3} = \frac{19}{3}$

$23 = 2 a + 11$

$b = 11 - 2 a$

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 $3$ .

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Step 1.3.3.3.2.1.1

Cancel the common factor.

$\frac{3 a}{3} = \frac{19}{3}$

$23 = 2 a + 11$

$b = 11 - 2 a$

Step 1.3.3.3.2.1.2

Divide $a$ by $1$ .

$a = \frac{19}{3}$

$23 = 2 a + 11$

$b = 11 - 2 a$

$a = \frac{19}{3}$

$23 = 2 a + 11$

$b = 11 - 2 a$

$a = \frac{19}{3}$

$23 = 2 a + 11$

$b = 11 - 2 a$

$a = \frac{19}{3}$

$23 = 2 a + 11$

$b = 11 - 2 a$

$a = \frac{19}{3}$

$23 = 2 a + 11$

$b = 11 - 2 a$

Step 1.3.4

Replace all occurrences of $a$ with $\frac{19}{3}$ in each equation.

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Step 1.3.4.1

Replace all occurrences of $a$ in $23 = 2 a + 11$ with $\frac{19}{3}$ .

$23 = 2 (\frac{19}{3}) + 11$

$a = \frac{19}{3}$

$b = 11 - 2 a$

Step 1.3.4.2

Simplify the right side.

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Step 1.3.4.2.1

Simplify $2 (\frac{19}{3}) + 11$ .

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Step 1.3.4.2.1.1

Multiply $2 (\frac{19}{3})$ .

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Step 1.3.4.2.1.1.1

Combine $2$ and $\frac{19}{3}$ .

$23 = \frac{2 \cdot 19}{3} + 11$

$a = \frac{19}{3}$

$b = 11 - 2 a$

Step 1.3.4.2.1.1.2

Multiply $2$ by $19$ .

$23 = \frac{38}{3} + 11$

$a = \frac{19}{3}$

$b = 11 - 2 a$

$23 = \frac{38}{3} + 11$

$a = \frac{19}{3}$

$b = 11 - 2 a$

Step 1.3.4.2.1.2

To write $11$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$23 = \frac{38}{3} + 11 \cdot \frac{3}{3}$

$a = \frac{19}{3}$

$b = 11 - 2 a$

Step 1.3.4.2.1.3

Combine $11$ and $\frac{3}{3}$ .

$23 = \frac{38}{3} + \frac{11 \cdot 3}{3}$

$a = \frac{19}{3}$

$b = 11 - 2 a$

Step 1.3.4.2.1.4

Combine the numerators over the common denominator.

$23 = \frac{38 + 11 \cdot 3}{3}$

$a = \frac{19}{3}$

$b = 11 - 2 a$

Step 1.3.4.2.1.5

Simplify the numerator.

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Step 1.3.4.2.1.5.1

Multiply $11$ by $3$ .

$23 = \frac{38 + 33}{3}$

$a = \frac{19}{3}$

$b = 11 - 2 a$

Step 1.3.4.2.1.5.2

Add $38$ and $33$ .

$23 = \frac{71}{3}$

$a = \frac{19}{3}$

$b = 11 - 2 a$

$23 = \frac{71}{3}$

$a = \frac{19}{3}$

$b = 11 - 2 a$

$23 = \frac{71}{3}$

$a = \frac{19}{3}$

$b = 11 - 2 a$

$23 = \frac{71}{3}$

$a = \frac{19}{3}$

$b = 11 - 2 a$

Step 1.3.4.3

Replace all occurrences of $a$ in $b = 11 - 2 a$ with $\frac{19}{3}$ .

$b = 11 - 2 (\frac{19}{3})$

$23 = \frac{71}{3}$

$a = \frac{19}{3}$

Step 1.3.4.4

Simplify the right side.

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Step 1.3.4.4.1

Simplify $11 - 2 (\frac{19}{3})$ .

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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

Multiply $- 2 (\frac{19}{3})$ .

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Step 1.3.4.4.1.1.1.1

Combine $- 2$ and $\frac{19}{3}$ .

$b = 11 + \frac{- 2 \cdot 19}{3}$

$23 = \frac{71}{3}$

$a = \frac{19}{3}$

Step 1.3.4.4.1.1.1.2

Multiply $- 2$ by $19$ .

$b = 11 + \frac{- 38}{3}$

$23 = \frac{71}{3}$

$a = \frac{19}{3}$

$b = 11 + \frac{- 38}{3}$

$23 = \frac{71}{3}$

$a = \frac{19}{3}$

Step 1.3.4.4.1.1.2

Move the negative in front of the fraction.

$b = 11 - \frac{38}{3}$

$23 = \frac{71}{3}$

$a = \frac{19}{3}$

$b = 11 - \frac{38}{3}$

$23 = \frac{71}{3}$

$a = \frac{19}{3}$

Step 1.3.4.4.1.2

To write $11$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$b = 11 \cdot \frac{3}{3} - \frac{38}{3}$

$23 = \frac{71}{3}$

$a = \frac{19}{3}$

Step 1.3.4.4.1.3

Combine $11$ and $\frac{3}{3}$ .

$b = \frac{11 \cdot 3}{3} - \frac{38}{3}$

$23 = \frac{71}{3}$

$a = \frac{19}{3}$

Step 1.3.4.4.1.4

Combine the numerators over the common denominator.

$b = \frac{11 \cdot 3 - 38}{3}$

$23 = \frac{71}{3}$

$a = \frac{19}{3}$

Step 1.3.4.4.1.5

Simplify the numerator.

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Step 1.3.4.4.1.5.1

Multiply $11$ by $3$ .

$b = \frac{33 - 38}{3}$

$23 = \frac{71}{3}$

$a = \frac{19}{3}$

Step 1.3.4.4.1.5.2

Subtract $38$ from $33$ .

$b = \frac{- 5}{3}$

$23 = \frac{71}{3}$

$a = \frac{19}{3}$

$b = \frac{- 5}{3}$

$23 = \frac{71}{3}$

$a = \frac{19}{3}$

Step 1.3.4.4.1.6

Move the negative in front of the fraction.

$b = - \frac{5}{3}$

$23 = \frac{71}{3}$

$a = \frac{19}{3}$

$b = - \frac{5}{3}$

$23 = \frac{71}{3}$

$a = \frac{19}{3}$

$b = - \frac{5}{3}$

$23 = \frac{71}{3}$

$a = \frac{19}{3}$

$b = - \frac{5}{3}$

$23 = \frac{71}{3}$

$a = \frac{19}{3}$

Step 1.3.5

Since $23 = \frac{71}{3}$ 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 $11 = a \cdot 2^{2} + b (2) + c$ .

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Step 2.3.1.1

Rewrite the equation as $a \cdot 2^{2} + b (2) + c = 11$ .

$a \cdot 2^{2} + b (2) + c = 11$

$23 = a \cdot 4^{2} + b (4) + c$

$30 = a \cdot 5^{2} + b (5) + c$

Step 2.3.1.2

Simplify each term.

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Step 2.3.1.2.1

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

$a \cdot 4 + b (2) + c = 11$

$23 = a \cdot 4^{2} + b (4) + c$

$30 = a \cdot 5^{2} + b (5) + c$

Step 2.3.1.2.2

Move $4$ to the left of $a$ .

$4 \cdot a + b (2) + c = 11$

$23 = a \cdot 4^{2} + b (4) + c$

$30 = a \cdot 5^{2} + b (5) + c$

Step 2.3.1.2.3

Move $2$ to the left of $b$ .

$4 a + 2 b + c = 11$

$23 = a \cdot 4^{2} + b (4) + c$

$30 = a \cdot 5^{2} + b (5) + c$

$4 a + 2 b + c = 11$

$23 = a \cdot 4^{2} + b (4) + c$

$30 = a \cdot 5^{2} + b (5) + c$

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 $4 a$ from both sides of the equation.

$2 b + c = 11 - 4 a$

$23 = a \cdot 4^{2} + b (4) + c$

$30 = a \cdot 5^{2} + b (5) + c$

Step 2.3.1.3.2

Subtract $2 b$ from both sides of the equation.

$c = 11 - 4 a - 2 b$

$23 = a \cdot 4^{2} + b (4) + c$

$30 = a \cdot 5^{2} + b (5) + c$

$c = 11 - 4 a - 2 b$

$23 = a \cdot 4^{2} + b (4) + c$

$30 = a \cdot 5^{2} + b (5) + c$

$c = 11 - 4 a - 2 b$

$23 = a \cdot 4^{2} + b (4) + c$

$30 = a \cdot 5^{2} + b (5) + c$

Step 2.3.2

Replace all occurrences of $c$ with $11 - 4 a - 2 b$ in each equation.

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Step 2.3.2.1

Replace all occurrences of $c$ in $23 = a \cdot 4^{2} + b (4) + c$ with $11 - 4 a - 2 b$ .

$23 = a \cdot 4^{2} + b (4) + 11 - 4 a - 2 b$

$c = 11 - 4 a - 2 b$

$30 = a \cdot 5^{2} + b (5) + c$

Step 2.3.2.2

Simplify $23 = a \cdot 4^{2} + b (4) + 11 - 4 a - 2 b$ .

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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.

$23 = a \cdot 4^{2} + b (4) + 11 - 4 a - 2 b$

$c = 11 - 4 a - 2 b$

$30 = a \cdot 5^{2} + b (5) + c$

$23 = a \cdot 4^{2} + b (4) + 11 - 4 a - 2 b$

$c = 11 - 4 a - 2 b$

$30 = a \cdot 5^{2} + b (5) + c$

Step 2.3.2.2.2

Simplify the right side.

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Step 2.3.2.2.2.1

Simplify $a \cdot 4^{2} + b (4) + 11 - 4 a - 2 b$ .

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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 $4$ to the power of $2$ .

$23 = a \cdot 16 + b (4) + 11 - 4 a - 2 b$

$c = 11 - 4 a - 2 b$

$30 = a \cdot 5^{2} + b (5) + c$

Step 2.3.2.2.2.1.1.2

Move $16$ to the left of $a$ .

$23 = 16 \cdot a + b (4) + 11 - 4 a - 2 b$

$c = 11 - 4 a - 2 b$

$30 = a \cdot 5^{2} + b (5) + c$

Step 2.3.2.2.2.1.1.3

Move $4$ to the left of $b$ .

$23 = 16 a + 4 b + 11 - 4 a - 2 b$

$c = 11 - 4 a - 2 b$

$30 = a \cdot 5^{2} + b (5) + c$

$23 = 16 a + 4 b + 11 - 4 a - 2 b$

$c = 11 - 4 a - 2 b$

$30 = a \cdot 5^{2} + b (5) + c$

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 $4 a$ from $16 a$ .

$23 = 12 a + 4 b + 11 - 2 b$

$c = 11 - 4 a - 2 b$

$30 = a \cdot 5^{2} + b (5) + c$

Step 2.3.2.2.2.1.2.2

Subtract $2 b$ from $4 b$ .

$23 = 12 a + 2 b + 11$

$c = 11 - 4 a - 2 b$

$30 = a \cdot 5^{2} + b (5) + c$

$23 = 12 a + 2 b + 11$

$c = 11 - 4 a - 2 b$

$30 = a \cdot 5^{2} + b (5) + c$

$23 = 12 a + 2 b + 11$

$c = 11 - 4 a - 2 b$

$30 = a \cdot 5^{2} + b (5) + c$

$23 = 12 a + 2 b + 11$

$c = 11 - 4 a - 2 b$

$30 = a \cdot 5^{2} + b (5) + c$

$23 = 12 a + 2 b + 11$

$c = 11 - 4 a - 2 b$

$30 = a \cdot 5^{2} + b (5) + c$

Step 2.3.2.3

Replace all occurrences of $c$ in $30 = a \cdot 5^{2} + b (5) + c$ with $11 - 4 a - 2 b$ .

$30 = a \cdot 5^{2} + b (5) + 11 - 4 a - 2 b$

$23 = 12 a + 2 b + 11$

$c = 11 - 4 a - 2 b$

Step 2.3.2.4

Simplify $30 = a \cdot 5^{2} + b (5) + 11 - 4 a - 2 b$ .

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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.

$30 = a \cdot 5^{2} + b (5) + 11 - 4 a - 2 b$

$23 = 12 a + 2 b + 11$

$c = 11 - 4 a - 2 b$

$30 = a \cdot 5^{2} + b (5) + 11 - 4 a - 2 b$

$23 = 12 a + 2 b + 11$

$c = 11 - 4 a - 2 b$

Step 2.3.2.4.2

Simplify the right side.

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Step 2.3.2.4.2.1

Simplify $a \cdot 5^{2} + b (5) + 11 - 4 a - 2 b$ .

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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 $5$ to the power of $2$ .

$30 = a \cdot 25 + b (5) + 11 - 4 a - 2 b$

$23 = 12 a + 2 b + 11$

$c = 11 - 4 a - 2 b$

Step 2.3.2.4.2.1.1.2

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

$30 = 25 \cdot a + b (5) + 11 - 4 a - 2 b$

$23 = 12 a + 2 b + 11$

$c = 11 - 4 a - 2 b$

Step 2.3.2.4.2.1.1.3

Move $5$ to the left of $b$ .

$30 = 25 a + 5 b + 11 - 4 a - 2 b$

$23 = 12 a + 2 b + 11$

$c = 11 - 4 a - 2 b$

$30 = 25 a + 5 b + 11 - 4 a - 2 b$

$23 = 12 a + 2 b + 11$

$c = 11 - 4 a - 2 b$

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 $4 a$ from $25 a$ .

$30 = 21 a + 5 b + 11 - 2 b$

$23 = 12 a + 2 b + 11$

$c = 11 - 4 a - 2 b$

Step 2.3.2.4.2.1.2.2

Subtract $2 b$ from $5 b$ .

$30 = 21 a + 3 b + 11$

$23 = 12 a + 2 b + 11$

$c = 11 - 4 a - 2 b$

$30 = 21 a + 3 b + 11$

$23 = 12 a + 2 b + 11$

$c = 11 - 4 a - 2 b$

$30 = 21 a + 3 b + 11$

$23 = 12 a + 2 b + 11$

$c = 11 - 4 a - 2 b$

$30 = 21 a + 3 b + 11$

$23 = 12 a + 2 b + 11$

$c = 11 - 4 a - 2 b$

$30 = 21 a + 3 b + 11$

$23 = 12 a + 2 b + 11$

$c = 11 - 4 a - 2 b$

$30 = 21 a + 3 b + 11$

$23 = 12 a + 2 b + 11$

$c = 11 - 4 a - 2 b$

Step 2.3.3

Solve for $a$ in $30 = 21 a + 3 b + 11$ .

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Step 2.3.3.1

Rewrite the equation as $21 a + 3 b + 11 = 30$ .

$21 a + 3 b + 11 = 30$

$23 = 12 a + 2 b + 11$

$c = 11 - 4 a - 2 b$

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 $3 b$ from both sides of the equation.

$21 a + 11 = 30 - 3 b$

$23 = 12 a + 2 b + 11$

$c = 11 - 4 a - 2 b$

Step 2.3.3.2.2

Subtract $11$ from both sides of the equation.

$21 a = 30 - 3 b - 11$

$23 = 12 a + 2 b + 11$

$c = 11 - 4 a - 2 b$

Step 2.3.3.2.3

Subtract $11$ from $30$ .

$21 a = - 3 b + 19$

$23 = 12 a + 2 b + 11$

$c = 11 - 4 a - 2 b$

$21 a = - 3 b + 19$

$23 = 12 a + 2 b + 11$

$c = 11 - 4 a - 2 b$

Step 2.3.3.3

Divide each term in $21 a = - 3 b + 19$ by $21$ and simplify.

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Step 2.3.3.3.1

Divide each term in $21 a = - 3 b + 19$ by $21$ .

$\frac{21 a}{21} = \frac{- 3 b}{21} + \frac{19}{21}$

$23 = 12 a + 2 b + 11$

$c = 11 - 4 a - 2 b$

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 $21$ .

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Step 2.3.3.3.2.1.1

Cancel the common factor.

$\frac{21 a}{21} = \frac{- 3 b}{21} + \frac{19}{21}$

$23 = 12 a + 2 b + 11$

$c = 11 - 4 a - 2 b$

Step 2.3.3.3.2.1.2

Divide $a$ by $1$ .

$a = \frac{- 3 b}{21} + \frac{19}{21}$

$23 = 12 a + 2 b + 11$

$c = 11 - 4 a - 2 b$

$a = \frac{- 3 b}{21} + \frac{19}{21}$

$23 = 12 a + 2 b + 11$

$c = 11 - 4 a - 2 b$

$a = \frac{- 3 b}{21} + \frac{19}{21}$

$23 = 12 a + 2 b + 11$

$c = 11 - 4 a - 2 b$

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 $- 3$ and $21$ .

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Step 2.3.3.3.3.1.1.1

Factor $3$ out of $- 3 b$ .

$a = \frac{3 (- b)}{21} + \frac{19}{21}$

$23 = 12 a + 2 b + 11$

$c = 11 - 4 a - 2 b$

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 $3$ out of $21$ .

$a = \frac{3 (- b)}{3 (7)} + \frac{19}{21}$

$23 = 12 a + 2 b + 11$

$c = 11 - 4 a - 2 b$

Step 2.3.3.3.3.1.1.2.2

Cancel the common factor.

$a = \frac{3 (- b)}{3 \cdot 7} + \frac{19}{21}$

$23 = 12 a + 2 b + 11$

$c = 11 - 4 a - 2 b$

Step 2.3.3.3.3.1.1.2.3

Rewrite the expression.

$a = \frac{- b}{7} + \frac{19}{21}$

$23 = 12 a + 2 b + 11$

$c = 11 - 4 a - 2 b$

$a = \frac{- b}{7} + \frac{19}{21}$

$23 = 12 a + 2 b + 11$

$c = 11 - 4 a - 2 b$

$a = \frac{- b}{7} + \frac{19}{21}$

$23 = 12 a + 2 b + 11$

$c = 11 - 4 a - 2 b$

Step 2.3.3.3.3.1.2

Move the negative in front of the fraction.

$a = - \frac{b}{7} + \frac{19}{21}$

$23 = 12 a + 2 b + 11$

$c = 11 - 4 a - 2 b$

$a = - \frac{b}{7} + \frac{19}{21}$

$23 = 12 a + 2 b + 11$

$c = 11 - 4 a - 2 b$

$a = - \frac{b}{7} + \frac{19}{21}$

$23 = 12 a + 2 b + 11$

$c = 11 - 4 a - 2 b$

$a = - \frac{b}{7} + \frac{19}{21}$

$23 = 12 a + 2 b + 11$

$c = 11 - 4 a - 2 b$

$a = - \frac{b}{7} + \frac{19}{21}$

$23 = 12 a + 2 b + 11$

$c = 11 - 4 a - 2 b$

Step 2.3.4

Replace all occurrences of $a$ with $- \frac{b}{7} + \frac{19}{21}$ in each equation.

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Step 2.3.4.1

Replace all occurrences of $a$ in $23 = 12 a + 2 b + 11$ with $- \frac{b}{7} + \frac{19}{21}$ .

$23 = 12 (- \frac{b}{7} + \frac{19}{21}) + 2 b + 11$

$a = - \frac{b}{7} + \frac{19}{21}$

$c = 11 - 4 a - 2 b$

Step 2.3.4.2

Simplify the right side.

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Step 2.3.4.2.1

Simplify $12 (- \frac{b}{7} + \frac{19}{21}) + 2 b + 11$ .

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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.

$23 = 12 (- \frac{b}{7}) + 12 (\frac{19}{21}) + 2 b + 11$

$a = - \frac{b}{7} + \frac{19}{21}$

$c = 11 - 4 a - 2 b$

Step 2.3.4.2.1.1.2

Multiply $12 (- \frac{b}{7})$ .

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Step 2.3.4.2.1.1.2.1

Multiply $- 1$ by $12$ .

$23 = - 12 \frac{b}{7} + 12 (\frac{19}{21}) + 2 b + 11$

$a = - \frac{b}{7} + \frac{19}{21}$

$c = 11 - 4 a - 2 b$

Step 2.3.4.2.1.1.2.2

Combine $- 12$ and $\frac{b}{7}$ .

$23 = \frac{- 12 b}{7} + 12 (\frac{19}{21}) + 2 b + 11$

$a = - \frac{b}{7} + \frac{19}{21}$

$c = 11 - 4 a - 2 b$

$23 = \frac{- 12 b}{7} + 12 (\frac{19}{21}) + 2 b + 11$

$a = - \frac{b}{7} + \frac{19}{21}$

$c = 11 - 4 a - 2 b$

Step 2.3.4.2.1.1.3

Cancel the common factor of $3$ .

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Step 2.3.4.2.1.1.3.1

Factor $3$ out of $12$ .

$23 = \frac{- 12 b}{7} + 3 (4) (\frac{19}{21}) + 2 b + 11$

$a = - \frac{b}{7} + \frac{19}{21}$

$c = 11 - 4 a - 2 b$

Step 2.3.4.2.1.1.3.2

Factor $3$ out of $21$ .

$23 = \frac{- 12 b}{7} + 3 \cdot (4 (\frac{19}{3 \cdot 7})) + 2 b + 11$

$a = - \frac{b}{7} + \frac{19}{21}$

$c = 11 - 4 a - 2 b$

Step 2.3.4.2.1.1.3.3

Cancel the common factor.

$23 = \frac{- 12 b}{7} + 3 \cdot (4 (\frac{19}{3 \cdot 7})) + 2 b + 11$

$a = - \frac{b}{7} + \frac{19}{21}$

$c = 11 - 4 a - 2 b$

Step 2.3.4.2.1.1.3.4

Rewrite the expression.

$23 = \frac{- 12 b}{7} + 4 (\frac{19}{7}) + 2 b + 11$

$a = - \frac{b}{7} + \frac{19}{21}$

$c = 11 - 4 a - 2 b$

$23 = \frac{- 12 b}{7} + 4 (\frac{19}{7}) + 2 b + 11$

$a = - \frac{b}{7} + \frac{19}{21}$

$c = 11 - 4 a - 2 b$

Step 2.3.4.2.1.1.4

Combine $4$ and $\frac{19}{7}$ .

$23 = \frac{- 12 b}{7} + \frac{4 \cdot 19}{7} + 2 b + 11$

$a = - \frac{b}{7} + \frac{19}{21}$

$c = 11 - 4 a - 2 b$

Step 2.3.4.2.1.1.5

Multiply $4$ by $19$ .

$23 = \frac{- 12 b}{7} + \frac{76}{7} + 2 b + 11$

$a = - \frac{b}{7} + \frac{19}{21}$

$c = 11 - 4 a - 2 b$

Step 2.3.4.2.1.1.6

Move the negative in front of the fraction.

$23 = - \frac{12 b}{7} + \frac{76}{7} + 2 b + 11$

$a = - \frac{b}{7} + \frac{19}{21}$

$c = 11 - 4 a - 2 b$

$23 = - \frac{12 b}{7} + \frac{76}{7} + 2 b + 11$

$a = - \frac{b}{7} + \frac{19}{21}$

$c = 11 - 4 a - 2 b$

Step 2.3.4.2.1.2

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

$23 = - \frac{12 b}{7} + 2 b \cdot \frac{7}{7} + \frac{76}{7} + 11$

$a = - \frac{b}{7} + \frac{19}{21}$

$c = 11 - 4 a - 2 b$

Step 2.3.4.2.1.3

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

$23 = - \frac{12 b}{7} + \frac{2 b \cdot 7}{7} + \frac{76}{7} + 11$

$a = - \frac{b}{7} + \frac{19}{21}$

$c = 11 - 4 a - 2 b$

Step 2.3.4.2.1.4

Combine the numerators over the common denominator.

$23 = \frac{- 12 b + 2 b \cdot 7}{7} + \frac{76}{7} + 11$

$a = - \frac{b}{7} + \frac{19}{21}$

$c = 11 - 4 a - 2 b$

Step 2.3.4.2.1.5

Combine the numerators over the common denominator.

$23 = 11 + \frac{- 12 b + 2 b \cdot 7 + 76}{7}$

$a = - \frac{b}{7} + \frac{19}{21}$

$c = 11 - 4 a - 2 b$

Step 2.3.4.2.1.6

Multiply $7$ by $2$ .

$23 = 11 + \frac{- 12 b + 14 b + 76}{7}$

$a = - \frac{b}{7} + \frac{19}{21}$

$c = 11 - 4 a - 2 b$

Step 2.3.4.2.1.7

Add $- 12 b$ and $14 b$ .

$23 = 11 + \frac{2 b + 76}{7}$

$a = - \frac{b}{7} + \frac{19}{21}$

$c = 11 - 4 a - 2 b$

Step 2.3.4.2.1.8

Factor $2$ out of $2 b + 76$ .

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Step 2.3.4.2.1.8.1

Factor $2$ out of $2 b$ .

$23 = 11 + \frac{2 (b) + 76}{7}$

$a = - \frac{b}{7} + \frac{19}{21}$

$c = 11 - 4 a - 2 b$

Step 2.3.4.2.1.8.2

Factor $2$ out of $76$ .

$23 = 11 + \frac{2 b + 2 \cdot 38}{7}$

$a = - \frac{b}{7} + \frac{19}{21}$

$c = 11 - 4 a - 2 b$

Step 2.3.4.2.1.8.3

Factor $2$ out of $2 b + 2 \cdot 38$ .

$23 = 11 + \frac{2 (b + 38)}{7}$

$a = - \frac{b}{7} + \frac{19}{21}$

$c = 11 - 4 a - 2 b$

$23 = 11 + \frac{2 (b + 38)}{7}$

$a = - \frac{b}{7} + \frac{19}{21}$

$c = 11 - 4 a - 2 b$

Step 2.3.4.2.1.9

To write $11$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$23 = 11 \cdot \frac{7}{7} + \frac{2 (b + 38)}{7}$

$a = - \frac{b}{7} + \frac{19}{21}$

$c = 11 - 4 a - 2 b$

Step 2.3.4.2.1.10

Simplify terms.

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Step 2.3.4.2.1.10.1

Combine $11$ and $\frac{7}{7}$ .

$23 = \frac{11 \cdot 7}{7} + \frac{2 (b + 38)}{7}$

$a = - \frac{b}{7} + \frac{19}{21}$

$c = 11 - 4 a - 2 b$

Step 2.3.4.2.1.10.2

Combine the numerators over the common denominator.

$23 = \frac{11 \cdot 7 + 2 (b + 38)}{7}$

$a = - \frac{b}{7} + \frac{19}{21}$

$c = 11 - 4 a - 2 b$

$23 = \frac{11 \cdot 7 + 2 (b + 38)}{7}$

$a = - \frac{b}{7} + \frac{19}{21}$

$c = 11 - 4 a - 2 b$

Step 2.3.4.2.1.11

Simplify the numerator.

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Step 2.3.4.2.1.11.1

Multiply $11$ by $7$ .

$23 = \frac{77 + 2 (b + 38)}{7}$

$a = - \frac{b}{7} + \frac{19}{21}$

$c = 11 - 4 a - 2 b$

Step 2.3.4.2.1.11.2

Apply the distributive property.

$23 = \frac{77 + 2 b + 2 \cdot 38}{7}$

$a = - \frac{b}{7} + \frac{19}{21}$

$c = 11 - 4 a - 2 b$

Step 2.3.4.2.1.11.3

Multiply $2$ by $38$ .

$23 = \frac{77 + 2 b + 76}{7}$

$a = - \frac{b}{7} + \frac{19}{21}$

$c = 11 - 4 a - 2 b$

Step 2.3.4.2.1.11.4

Add $77$ and $76$ .

$23 = \frac{2 b + 153}{7}$

$a = - \frac{b}{7} + \frac{19}{21}$

$c = 11 - 4 a - 2 b$

$23 = \frac{2 b + 153}{7}$

$a = - \frac{b}{7} + \frac{19}{21}$

$c = 11 - 4 a - 2 b$

$23 = \frac{2 b + 153}{7}$

$a = - \frac{b}{7} + \frac{19}{21}$

$c = 11 - 4 a - 2 b$

$23 = \frac{2 b + 153}{7}$

$a = - \frac{b}{7} + \frac{19}{21}$

$c = 11 - 4 a - 2 b$

Step 2.3.4.3

Replace all occurrences of $a$ in $c = 11 - 4 a - 2 b$ with $- \frac{b}{7} + \frac{19}{21}$ .

$c = 11 - 4 (- \frac{b}{7} + \frac{19}{21}) - 2 b$

$23 = \frac{2 b + 153}{7}$

$a = - \frac{b}{7} + \frac{19}{21}$

Step 2.3.4.4

Simplify the right side.

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Step 2.3.4.4.1

Simplify $11 - 4 (- \frac{b}{7} + \frac{19}{21}) - 2 b$ .

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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.

$c = 11 - 4 (- \frac{b}{7}) - 4 (\frac{19}{21}) - 2 b$

$23 = \frac{2 b + 153}{7}$

$a = - \frac{b}{7} + \frac{19}{21}$

Step 2.3.4.4.1.1.2

Multiply $- 4 (- \frac{b}{7})$ .

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Step 2.3.4.4.1.1.2.1

Multiply $- 1$ by $- 4$ .

$c = 11 + 4 (\frac{b}{7}) - 4 (\frac{19}{21}) - 2 b$

$23 = \frac{2 b + 153}{7}$

$a = - \frac{b}{7} + \frac{19}{21}$

Step 2.3.4.4.1.1.2.2

Combine $4$ and $\frac{b}{7}$ .

$c = 11 + \frac{4 b}{7} - 4 (\frac{19}{21}) - 2 b$

$23 = \frac{2 b + 153}{7}$

$a = - \frac{b}{7} + \frac{19}{21}$

$c = 11 + \frac{4 b}{7} - 4 (\frac{19}{21}) - 2 b$

$23 = \frac{2 b + 153}{7}$

$a = - \frac{b}{7} + \frac{19}{21}$

Step 2.3.4.4.1.1.3

Multiply $- 4 (\frac{19}{21})$ .

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Step 2.3.4.4.1.1.3.1

Combine $- 4$ and $\frac{19}{21}$ .

$c = 11 + \frac{4 b}{7} + \frac{- 4 \cdot 19}{21} - 2 b$

$23 = \frac{2 b + 153}{7}$

$a = - \frac{b}{7} + \frac{19}{21}$

Step 2.3.4.4.1.1.3.2

Multiply $- 4$ by $19$ .

$c = 11 + \frac{4 b}{7} + \frac{- 76}{21} - 2 b$

$23 = \frac{2 b + 153}{7}$

$a = - \frac{b}{7} + \frac{19}{21}$

$c = 11 + \frac{4 b}{7} + \frac{- 76}{21} - 2 b$

$23 = \frac{2 b + 153}{7}$

$a = - \frac{b}{7} + \frac{19}{21}$

Step 2.3.4.4.1.1.4

Move the negative in front of the fraction.

$c = 11 + \frac{4 b}{7} - \frac{76}{21} - 2 b$

$23 = \frac{2 b + 153}{7}$

$a = - \frac{b}{7} + \frac{19}{21}$

$c = 11 + \frac{4 b}{7} - \frac{76}{21} - 2 b$

$23 = \frac{2 b + 153}{7}$

$a = - \frac{b}{7} + \frac{19}{21}$

Step 2.3.4.4.1.2

To write $11$ as a fraction with a common denominator, multiply by $\frac{21}{21}$ .

$c = \frac{4 b}{7} + 11 \cdot \frac{21}{21} - \frac{76}{21} - 2 b$

$23 = \frac{2 b + 153}{7}$

$a = - \frac{b}{7} + \frac{19}{21}$

Step 2.3.4.4.1.3

Combine $11$ and $\frac{21}{21}$ .

$c = \frac{4 b}{7} + \frac{11 \cdot 21}{21} - \frac{76}{21} - 2 b$

$23 = \frac{2 b + 153}{7}$

$a = - \frac{b}{7} + \frac{19}{21}$

Step 2.3.4.4.1.4

Combine the numerators over the common denominator.

$c = \frac{4 b}{7} + \frac{11 \cdot 21 - 76}{21} - 2 b$

$23 = \frac{2 b + 153}{7}$

$a = - \frac{b}{7} + \frac{19}{21}$

Step 2.3.4.4.1.5

Simplify the numerator.

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Step 2.3.4.4.1.5.1

Multiply $11$ by $21$ .

$c = \frac{4 b}{7} + \frac{231 - 76}{21} - 2 b$

$23 = \frac{2 b + 153}{7}$

$a = - \frac{b}{7} + \frac{19}{21}$

Step 2.3.4.4.1.5.2

Subtract $76$ from $231$ .

$c = \frac{4 b}{7} + \frac{155}{21} - 2 b$

$23 = \frac{2 b + 153}{7}$

$a = - \frac{b}{7} + \frac{19}{21}$

$c = \frac{4 b}{7} + \frac{155}{21} - 2 b$

$23 = \frac{2 b + 153}{7}$

$a = - \frac{b}{7} + \frac{19}{21}$

Step 2.3.4.4.1.6

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

$c = \frac{4 b}{7} - 2 b \cdot \frac{7}{7} + \frac{155}{21}$

$23 = \frac{2 b + 153}{7}$

$a = - \frac{b}{7} + \frac{19}{21}$

Step 2.3.4.4.1.7

Simplify terms.

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Step 2.3.4.4.1.7.1

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

$c = \frac{4 b}{7} + \frac{- 2 b \cdot 7}{7} + \frac{155}{21}$

$23 = \frac{2 b + 153}{7}$

$a = - \frac{b}{7} + \frac{19}{21}$

Step 2.3.4.4.1.7.2

Combine the numerators over the common denominator.

$c = \frac{4 b - 2 b \cdot 7}{7} + \frac{155}{21}$

$23 = \frac{2 b + 153}{7}$

$a = - \frac{b}{7} + \frac{19}{21}$

$c = \frac{4 b - 2 b \cdot 7}{7} + \frac{155}{21}$

$23 = \frac{2 b + 153}{7}$

$a = - \frac{b}{7} + \frac{19}{21}$

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 $2 b$ out of $4 b - 2 b \cdot 7$ .

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Step 2.3.4.4.1.8.1.1.1

Factor $2 b$ out of $4 b$ .

$c = \frac{2 b (2) - 2 b \cdot 7}{7} + \frac{155}{21}$

$23 = \frac{2 b + 153}{7}$

$a = - \frac{b}{7} + \frac{19}{21}$

Step 2.3.4.4.1.8.1.1.2

Factor $2 b$ out of $- 2 b \cdot 7$ .

$c = \frac{2 b (2) + 2 b (- 1 \cdot 7)}{7} + \frac{155}{21}$

$23 = \frac{2 b + 153}{7}$

$a = - \frac{b}{7} + \frac{19}{21}$

Step 2.3.4.4.1.8.1.1.3

Factor $2 b$ out of $2 b (2) + 2 b (- 1 \cdot 7)$ .

$c = \frac{2 b (2 - 1 \cdot 7)}{7} + \frac{155}{21}$

$23 = \frac{2 b + 153}{7}$

$a = - \frac{b}{7} + \frac{19}{21}$

$c = \frac{2 b (2 - 1 \cdot 7)}{7} + \frac{155}{21}$

$23 = \frac{2 b + 153}{7}$

$a = - \frac{b}{7} + \frac{19}{21}$

Step 2.3.4.4.1.8.1.2

Multiply $- 1$ by $7$ .

$c = \frac{2 b (2 - 7)}{7} + \frac{155}{21}$

$23 = \frac{2 b + 153}{7}$

$a = - \frac{b}{7} + \frac{19}{21}$

Step 2.3.4.4.1.8.1.3

Subtract $7$ from $2$ .

$c = \frac{2 b \cdot - 5}{7} + \frac{155}{21}$

$23 = \frac{2 b + 153}{7}$

$a = - \frac{b}{7} + \frac{19}{21}$

Step 2.3.4.4.1.8.1.4

Multiply $- 5$ by $2$ .

$c = \frac{- 10 b}{7} + \frac{155}{21}$

$23 = \frac{2 b + 153}{7}$

$a = - \frac{b}{7} + \frac{19}{21}$

$c = \frac{- 10 b}{7} + \frac{155}{21}$

$23 = \frac{2 b + 153}{7}$

$a = - \frac{b}{7} + \frac{19}{21}$

Step 2.3.4.4.1.8.2

Move the negative in front of the fraction.

$c = - \frac{10 b}{7} + \frac{155}{21}$

$23 = \frac{2 b + 153}{7}$

$a = - \frac{b}{7} + \frac{19}{21}$

$c = - \frac{10 b}{7} + \frac{155}{21}$

$23 = \frac{2 b + 153}{7}$

$a = - \frac{b}{7} + \frac{19}{21}$

$c = - \frac{10 b}{7} + \frac{155}{21}$

$23 = \frac{2 b + 153}{7}$

$a = - \frac{b}{7} + \frac{19}{21}$

$c = - \frac{10 b}{7} + \frac{155}{21}$

$23 = \frac{2 b + 153}{7}$

$a = - \frac{b}{7} + \frac{19}{21}$

$c = - \frac{10 b}{7} + \frac{155}{21}$

$23 = \frac{2 b + 153}{7}$

$a = - \frac{b}{7} + \frac{19}{21}$

Step 2.3.5

Solve for $b$ in $23 = \frac{2 b + 153}{7}$ .

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Step 2.3.5.1

Rewrite the equation as $\frac{2 b + 153}{7} = 23$ .

$\frac{2 b + 153}{7} = 23$

$c = - \frac{10 b}{7} + \frac{155}{21}$

$a = - \frac{b}{7} + \frac{19}{21}$

Step 2.3.5.2

Multiply both sides by $7$ .

$\frac{2 b + 153}{7} \cdot 7 = 23 \cdot 7$

$c = - \frac{10 b}{7} + \frac{155}{21}$

$a = - \frac{b}{7} + \frac{19}{21}$

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 $7$ .

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Step 2.3.5.3.1.1.1

Cancel the common factor.

$\frac{2 b + 153}{7} \cdot 7 = 23 \cdot 7$

$c = - \frac{10 b}{7} + \frac{155}{21}$

$a = - \frac{b}{7} + \frac{19}{21}$

Step 2.3.5.3.1.1.2

Rewrite the expression.

$2 b + 153 = 23 \cdot 7$

$c = - \frac{10 b}{7} + \frac{155}{21}$

$a = - \frac{b}{7} + \frac{19}{21}$

$2 b + 153 = 23 \cdot 7$

$c = - \frac{10 b}{7} + \frac{155}{21}$

$a = - \frac{b}{7} + \frac{19}{21}$

$2 b + 153 = 23 \cdot 7$

$c = - \frac{10 b}{7} + \frac{155}{21}$

$a = - \frac{b}{7} + \frac{19}{21}$

Step 2.3.5.3.2

Simplify the right side.

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Step 2.3.5.3.2.1

Multiply $23$ by $7$ .

$2 b + 153 = 161$

$c = - \frac{10 b}{7} + \frac{155}{21}$

$a = - \frac{b}{7} + \frac{19}{21}$

$2 b + 153 = 161$

$c = - \frac{10 b}{7} + \frac{155}{21}$

$a = - \frac{b}{7} + \frac{19}{21}$

$2 b + 153 = 161$

$c = - \frac{10 b}{7} + \frac{155}{21}$

$a = - \frac{b}{7} + \frac{19}{21}$

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 $153$ from both sides of the equation.

$2 b = 161 - 153$

$c = - \frac{10 b}{7} + \frac{155}{21}$

$a = - \frac{b}{7} + \frac{19}{21}$

Step 2.3.5.4.1.2

Subtract $153$ from $161$ .

$2 b = 8$

$c = - \frac{10 b}{7} + \frac{155}{21}$

$a = - \frac{b}{7} + \frac{19}{21}$

$2 b = 8$

$c = - \frac{10 b}{7} + \frac{155}{21}$

$a = - \frac{b}{7} + \frac{19}{21}$

Step 2.3.5.4.2

Divide each term in $2 b = 8$ by $2$ and simplify.

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Step 2.3.5.4.2.1

Divide each term in $2 b = 8$ by $2$ .

$\frac{2 b}{2} = \frac{8}{2}$

$c = - \frac{10 b}{7} + \frac{155}{21}$

$a = - \frac{b}{7} + \frac{19}{21}$

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 $2$ .

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Step 2.3.5.4.2.2.1.1

Cancel the common factor.

$\frac{2 b}{2} = \frac{8}{2}$

$c = - \frac{10 b}{7} + \frac{155}{21}$

$a = - \frac{b}{7} + \frac{19}{21}$

Step 2.3.5.4.2.2.1.2

Divide $b$ by $1$ .

$b = \frac{8}{2}$

$c = - \frac{10 b}{7} + \frac{155}{21}$

$a = - \frac{b}{7} + \frac{19}{21}$

$b = \frac{8}{2}$

$c = - \frac{10 b}{7} + \frac{155}{21}$

$a = - \frac{b}{7} + \frac{19}{21}$

$b = \frac{8}{2}$

$c = - \frac{10 b}{7} + \frac{155}{21}$

$a = - \frac{b}{7} + \frac{19}{21}$

Step 2.3.5.4.2.3

Simplify the right side.

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Step 2.3.5.4.2.3.1

Divide $8$ by $2$ .

$b = 4$

$c = - \frac{10 b}{7} + \frac{155}{21}$

$a = - \frac{b}{7} + \frac{19}{21}$

$b = 4$

$c = - \frac{10 b}{7} + \frac{155}{21}$

$a = - \frac{b}{7} + \frac{19}{21}$

$b = 4$

$c = - \frac{10 b}{7} + \frac{155}{21}$

$a = - \frac{b}{7} + \frac{19}{21}$

$b = 4$

$c = - \frac{10 b}{7} + \frac{155}{21}$

$a = - \frac{b}{7} + \frac{19}{21}$

$b = 4$

$c = - \frac{10 b}{7} + \frac{155}{21}$

$a = - \frac{b}{7} + \frac{19}{21}$

Step 2.3.6

Replace all occurrences of $b$ with $4$ in each equation.

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Step 2.3.6.1

Replace all occurrences of $b$ in $c = - \frac{10 b}{7} + \frac{155}{21}$ with $4$ .

$c = - \frac{10 (4)}{7} + \frac{155}{21}$

$b = 4$

$a = - \frac{b}{7} + \frac{19}{21}$

Step 2.3.6.2

Simplify the right side.

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Step 2.3.6.2.1

Simplify $- \frac{10 (4)}{7} + \frac{155}{21}$ .

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Step 2.3.6.2.1.1

Multiply $10$ by $4$ .

$c = - \frac{40}{7} + \frac{155}{21}$

$b = 4$

$a = - \frac{b}{7} + \frac{19}{21}$

Step 2.3.6.2.1.2

To write $- \frac{40}{7}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$c = - \frac{40}{7} \cdot \frac{3}{3} + \frac{155}{21}$

$b = 4$

$a = - \frac{b}{7} + \frac{19}{21}$

Step 2.3.6.2.1.3

Write each expression with a common denominator of $21$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.6.2.1.3.1

Multiply $\frac{40}{7}$ by $\frac{3}{3}$ .

$c = - \frac{40 \cdot 3}{7 \cdot 3} + \frac{155}{21}$

$b = 4$

$a = - \frac{b}{7} + \frac{19}{21}$

Step 2.3.6.2.1.3.2

Multiply $7$ by $3$ .

$c = - \frac{40 \cdot 3}{21} + \frac{155}{21}$

$b = 4$

$a = - \frac{b}{7} + \frac{19}{21}$

$c = - \frac{40 \cdot 3}{21} + \frac{155}{21}$

$b = 4$

$a = - \frac{b}{7} + \frac{19}{21}$

Step 2.3.6.2.1.4

Combine the numerators over the common denominator.

$c = \frac{- 40 \cdot 3 + 155}{21}$

$b = 4$

$a = - \frac{b}{7} + \frac{19}{21}$

Step 2.3.6.2.1.5

Simplify the numerator.

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Step 2.3.6.2.1.5.1

Multiply $- 40$ by $3$ .

$c = \frac{- 120 + 155}{21}$

$b = 4$

$a = - \frac{b}{7} + \frac{19}{21}$

Step 2.3.6.2.1.5.2

Add $- 120$ and $155$ .

$c = \frac{35}{21}$

$b = 4$

$a = - \frac{b}{7} + \frac{19}{21}$

$c = \frac{35}{21}$

$b = 4$

$a = - \frac{b}{7} + \frac{19}{21}$

Step 2.3.6.2.1.6

Cancel the common factor of $35$ and $21$ .

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Step 2.3.6.2.1.6.1

Factor $7$ out of $35$ .

$c = \frac{7 (5)}{21}$

$b = 4$

$a = - \frac{b}{7} + \frac{19}{21}$

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 $7$ out of $21$ .

$c = \frac{7 \cdot 5}{7 \cdot 3}$

$b = 4$

$a = - \frac{b}{7} + \frac{19}{21}$

Step 2.3.6.2.1.6.2.2

Cancel the common factor.

$c = \frac{7 \cdot 5}{7 \cdot 3}$

$b = 4$

$a = - \frac{b}{7} + \frac{19}{21}$

Step 2.3.6.2.1.6.2.3

Rewrite the expression.

$c = \frac{5}{3}$

$b = 4$

$a = - \frac{b}{7} + \frac{19}{21}$

$c = \frac{5}{3}$

$b = 4$

$a = - \frac{b}{7} + \frac{19}{21}$

$c = \frac{5}{3}$

$b = 4$

$a = - \frac{b}{7} + \frac{19}{21}$

$c = \frac{5}{3}$

$b = 4$

$a = - \frac{b}{7} + \frac{19}{21}$

$c = \frac{5}{3}$

$b = 4$

$a = - \frac{b}{7} + \frac{19}{21}$

Step 2.3.6.3

Replace all occurrences of $b$ in $a = - \frac{b}{7} + \frac{19}{21}$ with $4$ .

$a = - \frac{4}{7} + \frac{19}{21}$

$c = \frac{5}{3}$

$b = 4$

Step 2.3.6.4

Simplify the right side.

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Step 2.3.6.4.1

Simplify $- \frac{4}{7} + \frac{19}{21}$ .

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Step 2.3.6.4.1.1

To write $- \frac{4}{7}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$a = - \frac{4}{7} \cdot \frac{3}{3} + \frac{19}{21}$

$c = \frac{5}{3}$

$b = 4$

Step 2.3.6.4.1.2

Write each expression with a common denominator of $21$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.6.4.1.2.1

Multiply $\frac{4}{7}$ by $\frac{3}{3}$ .

$a = - \frac{4 \cdot 3}{7 \cdot 3} + \frac{19}{21}$

$c = \frac{5}{3}$

$b = 4$

Step 2.3.6.4.1.2.2

Multiply $7$ by $3$ .

$a = - \frac{4 \cdot 3}{21} + \frac{19}{21}$

$c = \frac{5}{3}$

$b = 4$

$a = - \frac{4 \cdot 3}{21} + \frac{19}{21}$

$c = \frac{5}{3}$

$b = 4$

Step 2.3.6.4.1.3

Combine the numerators over the common denominator.

$a = \frac{- 4 \cdot 3 + 19}{21}$

$c = \frac{5}{3}$

$b = 4$

Step 2.3.6.4.1.4

Simplify the numerator.

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Step 2.3.6.4.1.4.1

Multiply $- 4$ by $3$ .

$a = \frac{- 12 + 19}{21}$

$c = \frac{5}{3}$

$b = 4$

Step 2.3.6.4.1.4.2

Add $- 12$ and $19$ .

$a = \frac{7}{21}$

$c = \frac{5}{3}$

$b = 4$

$a = \frac{7}{21}$

$c = \frac{5}{3}$

$b = 4$

Step 2.3.6.4.1.5

Cancel the common factor of $7$ and $21$ .

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Step 2.3.6.4.1.5.1

Factor $7$ out of $7$ .

$a = \frac{7 (1)}{21}$

$c = \frac{5}{3}$

$b = 4$

Step 2.3.6.4.1.5.2

Cancel the common factors.

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Step 2.3.6.4.1.5.2.1

Factor $7$ out of $21$ .

$a = \frac{7 \cdot 1}{7 \cdot 3}$

$c = \frac{5}{3}$

$b = 4$

Step 2.3.6.4.1.5.2.2

Cancel the common factor.

$a = \frac{7 \cdot 1}{7 \cdot 3}$

$c = \frac{5}{3}$

$b = 4$

Step 2.3.6.4.1.5.2.3

Rewrite the expression.

$a = \frac{1}{3}$

$c = \frac{5}{3}$

$b = 4$

$a = \frac{1}{3}$

$c = \frac{5}{3}$

$b = 4$

$a = \frac{1}{3}$

$c = \frac{5}{3}$

$b = 4$

$a = \frac{1}{3}$

$c = \frac{5}{3}$

$b = 4$

$a = \frac{1}{3}$

$c = \frac{5}{3}$

$b = 4$

$a = \frac{1}{3}$

$c = \frac{5}{3}$

$b = 4$

Step 2.3.7

List all of the solutions.

$a = \frac{1}{3}, c = \frac{5}{3}, b = 4$

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 $a = \frac{1}{3}$ , $b = 4$ , $c = \frac{5}{3}$ , and $x = 2$ .

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Step 2.4.1.1

Simplify each term.

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Step 2.4.1.1.1

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

$y = \frac{1}{3} \cdot 4 + (4) \cdot (2) + \frac{5}{3}$

Step 2.4.1.1.2

Combine $\frac{1}{3}$ and $4$ .

$y = \frac{4}{3} + (4) \cdot (2) + \frac{5}{3}$

Step 2.4.1.1.3

Multiply $4$ by $2$ .

$y = \frac{4}{3} + 8 + \frac{5}{3}$

Step 2.4.1.2

Combine fractions.

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Step 2.4.1.2.1

Combine the numerators over the common denominator.

$y = 8 + \frac{4 + 5}{3}$

Step 2.4.1.2.2

Simplify the expression.

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Step 2.4.1.2.2.1

Add $4$ and $5$ .

$y = 8 + \frac{9}{3}$

Step 2.4.1.2.2.2

Divide $9$ by $3$ .

$y = 8 + 3$

Step 2.4.1.2.2.3

Add $8$ and $3$ .

$y = 11$

Step 2.4.2

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = 2$ . This check passes since $y = 11$ and $y = 11$ .

$11 = 11$

Step 2.4.3

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = \frac{1}{3}$ , $b = 4$ , $c = \frac{5}{3}$ , and $x = 4$ .

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Step 2.4.3.1

Simplify each term.

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Step 2.4.3.1.1

Raise $4$ to the power of $2$ .

$y = \frac{1}{3} \cdot 16 + (4) \cdot (4) + \frac{5}{3}$

Step 2.4.3.1.2

Combine $\frac{1}{3}$ and $16$ .

$y = \frac{16}{3} + (4) \cdot (4) + \frac{5}{3}$

Step 2.4.3.1.3

Multiply $4$ by $4$ .

$y = \frac{16}{3} + 16 + \frac{5}{3}$

Step 2.4.3.2

Combine fractions.

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Step 2.4.3.2.1

Combine the numerators over the common denominator.

$y = 16 + \frac{16 + 5}{3}$

Step 2.4.3.2.2

Simplify the expression.

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Step 2.4.3.2.2.1

Add $16$ and $5$ .

$y = 16 + \frac{21}{3}$

Step 2.4.3.2.2.2

Divide $21$ by $3$ .

$y = 16 + 7$

Step 2.4.3.2.2.3

Add $16$ and $7$ .

$y = 23$

Step 2.4.4

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = 4$ . This check passes since $y = 23$ and $y = 23$ .

$23 = 23$

Step 2.4.5

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = \frac{1}{3}$ , $b = 4$ , $c = \frac{5}{3}$ , and $x = 5$ .

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Step 2.4.5.1

Simplify each term.

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Step 2.4.5.1.1

Raise $5$ to the power of $2$ .

$y = \frac{1}{3} \cdot 25 + (4) \cdot (5) + \frac{5}{3}$

Step 2.4.5.1.2

Combine $\frac{1}{3}$ and $25$ .

$y = \frac{25}{3} + (4) \cdot (5) + \frac{5}{3}$

Step 2.4.5.1.3

Multiply $4$ by $5$ .

$y = \frac{25}{3} + 20 + \frac{5}{3}$

Step 2.4.5.2

Combine fractions.

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Step 2.4.5.2.1

Combine the numerators over the common denominator.

$y = 20 + \frac{25 + 5}{3}$

Step 2.4.5.2.2

Simplify the expression.

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Step 2.4.5.2.2.1

Add $25$ and $5$ .

$y = 20 + \frac{30}{3}$

Step 2.4.5.2.2.2

Divide $30$ by $3$ .

$y = 20 + 10$

Step 2.4.5.2.2.3

Add $20$ and $10$ .

$y = 30$

Step 2.4.6

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = 5$ . This check passes since $y = 30$ and $y = 30$ .

$30 = 30$

Step 2.4.7

Since $y = y$ for the corresponding $x$ values, the function is quadratic.

The function is quadratic

Step 3

Since all $y = y$ , the function is quadratic and follows the form $y = \frac{x^{2}}{3} + 4 x + \frac{5}{3}$ .

$y = \frac{x^{2}}{3} + 4 x + \frac{5}{3}$