Pre-Algebra Examples

$\begin{array}{c} x & y \\ - 4 & - 19 \\ 7 & 25 \\ 12 & 33 \end{array}$

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 $y = a x + b$ .

$- 19 = a (- 4) + b 25 = a (7) + b 33 = a (12) + b$

Step 1.3

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

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Step 1.3.1

Solve for $b$ in $- 19 = a (- 4) + b$ .

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Step 1.3.1.1

Rewrite the equation as $a (- 4) + b = - 19$ .

$a (- 4) + b = - 19$

$25 = a (7) + b$

$33 = a (12) + b$

Step 1.3.1.2

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

$- 4 a + b = - 19$

$25 = a (7) + b$

$33 = a (12) + b$

Step 1.3.1.3

Add $4 a$ to both sides of the equation.

$b = - 19 + 4 a$

$25 = a (7) + b$

$33 = a (12) + b$

$b = - 19 + 4 a$

$25 = a (7) + b$

$33 = a (12) + b$

Step 1.3.2

Replace all occurrences of $b$ with $- 19 + 4 a$ in each equation.

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Step 1.3.2.1

Replace all occurrences of $b$ in $25 = a (7) + b$ with $- 19 + 4 a$ .

$25 = a (7) - 19 + 4 a$

$b = - 19 + 4 a$

$33 = a (12) + b$

Step 1.3.2.2

Simplify $25 = a (7) - 19 + 4 a$ .

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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.

$25 = a (7) - 19 + 4 a$

$b = - 19 + 4 a$

$33 = a (12) + b$

$25 = a (7) - 19 + 4 a$

$b = - 19 + 4 a$

$33 = a (12) + b$

Step 1.3.2.2.2

Simplify the right side.

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Step 1.3.2.2.2.1

Simplify $a (7) - 19 + 4 a$ .

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Step 1.3.2.2.2.1.1

Move $7$ to the left of $a$ .

$25 = 7 a - 19 + 4 a$

$b = - 19 + 4 a$

$33 = a (12) + b$

Step 1.3.2.2.2.1.2

Add $7 a$ and $4 a$ .

$25 = 11 a - 19$

$b = - 19 + 4 a$

$33 = a (12) + b$

$25 = 11 a - 19$

$b = - 19 + 4 a$

$33 = a (12) + b$

$25 = 11 a - 19$

$b = - 19 + 4 a$

$33 = a (12) + b$

$25 = 11 a - 19$

$b = - 19 + 4 a$

$33 = a (12) + b$

Step 1.3.2.3

Replace all occurrences of $b$ in $33 = a (12) + b$ with $- 19 + 4 a$ .

$33 = a (12) - 19 + 4 a$

$25 = 11 a - 19$

$b = - 19 + 4 a$

Step 1.3.2.4

Simplify $33 = a (12) - 19 + 4 a$ .

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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.

$33 = a (12) - 19 + 4 a$

$25 = 11 a - 19$

$b = - 19 + 4 a$

$33 = a (12) - 19 + 4 a$

$25 = 11 a - 19$

$b = - 19 + 4 a$

Step 1.3.2.4.2

Simplify the right side.

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Step 1.3.2.4.2.1

Simplify $a (12) - 19 + 4 a$ .

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Step 1.3.2.4.2.1.1

Move $12$ to the left of $a$ .

$33 = 12 a - 19 + 4 a$

$25 = 11 a - 19$

$b = - 19 + 4 a$

Step 1.3.2.4.2.1.2

Add $12 a$ and $4 a$ .

$33 = 16 a - 19$

$25 = 11 a - 19$

$b = - 19 + 4 a$

$33 = 16 a - 19$

$25 = 11 a - 19$

$b = - 19 + 4 a$

$33 = 16 a - 19$

$25 = 11 a - 19$

$b = - 19 + 4 a$

$33 = 16 a - 19$

$25 = 11 a - 19$

$b = - 19 + 4 a$

$33 = 16 a - 19$

$25 = 11 a - 19$

$b = - 19 + 4 a$

Step 1.3.3

Solve for $a$ in $33 = 16 a - 19$ .

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Step 1.3.3.1

Rewrite the equation as $16 a - 19 = 33$ .

$16 a - 19 = 33$

$25 = 11 a - 19$

$b = - 19 + 4 a$

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

Add $19$ to both sides of the equation.

$16 a = 33 + 19$

$25 = 11 a - 19$

$b = - 19 + 4 a$

Step 1.3.3.2.2

Add $33$ and $19$ .

$16 a = 52$

$25 = 11 a - 19$

$b = - 19 + 4 a$

$16 a = 52$

$25 = 11 a - 19$

$b = - 19 + 4 a$

Step 1.3.3.3

Divide each term in $16 a = 52$ by $16$ and simplify.

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Step 1.3.3.3.1

Divide each term in $16 a = 52$ by $16$ .

$\frac{16 a}{16} = \frac{52}{16}$

$25 = 11 a - 19$

$b = - 19 + 4 a$

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 $16$ .

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Step 1.3.3.3.2.1.1

Cancel the common factor.

$\frac{16 a}{16} = \frac{52}{16}$

$25 = 11 a - 19$

$b = - 19 + 4 a$

Step 1.3.3.3.2.1.2

Divide $a$ by $1$ .

$a = \frac{52}{16}$

$25 = 11 a - 19$

$b = - 19 + 4 a$

$a = \frac{52}{16}$

$25 = 11 a - 19$

$b = - 19 + 4 a$

$a = \frac{52}{16}$

$25 = 11 a - 19$

$b = - 19 + 4 a$

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 $52$ and $16$ .

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Step 1.3.3.3.3.1.1

Factor $4$ out of $52$ .

$a = \frac{4 (13)}{16}$

$25 = 11 a - 19$

$b = - 19 + 4 a$

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 $4$ out of $16$ .

$a = \frac{4 \cdot 13}{4 \cdot 4}$

$25 = 11 a - 19$

$b = - 19 + 4 a$

Step 1.3.3.3.3.1.2.2

Cancel the common factor.

$a = \frac{4 \cdot 13}{4 \cdot 4}$

$25 = 11 a - 19$

$b = - 19 + 4 a$

Step 1.3.3.3.3.1.2.3

Rewrite the expression.

$a = \frac{13}{4}$

$25 = 11 a - 19$

$b = - 19 + 4 a$

$a = \frac{13}{4}$

$25 = 11 a - 19$

$b = - 19 + 4 a$

$a = \frac{13}{4}$

$25 = 11 a - 19$

$b = - 19 + 4 a$

$a = \frac{13}{4}$

$25 = 11 a - 19$

$b = - 19 + 4 a$

$a = \frac{13}{4}$

$25 = 11 a - 19$

$b = - 19 + 4 a$

$a = \frac{13}{4}$

$25 = 11 a - 19$

$b = - 19 + 4 a$

Step 1.3.4

Replace all occurrences of $a$ with $\frac{13}{4}$ in each equation.

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Step 1.3.4.1

Replace all occurrences of $a$ in $25 = 11 a - 19$ with $\frac{13}{4}$ .

$25 = 11 (\frac{13}{4}) - 19$

$a = \frac{13}{4}$

$b = - 19 + 4 a$

Step 1.3.4.2

Simplify the right side.

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Step 1.3.4.2.1

Simplify $11 (\frac{13}{4}) - 19$ .

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Step 1.3.4.2.1.1

Multiply $11 (\frac{13}{4})$ .

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Step 1.3.4.2.1.1.1

Combine $11$ and $\frac{13}{4}$ .

$25 = \frac{11 \cdot 13}{4} - 19$

$a = \frac{13}{4}$

$b = - 19 + 4 a$

Step 1.3.4.2.1.1.2

Multiply $11$ by $13$ .

$25 = \frac{143}{4} - 19$

$a = \frac{13}{4}$

$b = - 19 + 4 a$

$25 = \frac{143}{4} - 19$

$a = \frac{13}{4}$

$b = - 19 + 4 a$

Step 1.3.4.2.1.2

To write $- 19$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$25 = \frac{143}{4} - 19 \cdot \frac{4}{4}$

$a = \frac{13}{4}$

$b = - 19 + 4 a$

Step 1.3.4.2.1.3

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

$25 = \frac{143}{4} + \frac{- 19 \cdot 4}{4}$

$a = \frac{13}{4}$

$b = - 19 + 4 a$

Step 1.3.4.2.1.4

Combine the numerators over the common denominator.

$25 = \frac{143 - 19 \cdot 4}{4}$

$a = \frac{13}{4}$

$b = - 19 + 4 a$

Step 1.3.4.2.1.5

Simplify the numerator.

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Step 1.3.4.2.1.5.1

Multiply $- 19$ by $4$ .

$25 = \frac{143 - 76}{4}$

$a = \frac{13}{4}$

$b = - 19 + 4 a$

Step 1.3.4.2.1.5.2

Subtract $76$ from $143$ .

$25 = \frac{67}{4}$

$a = \frac{13}{4}$

$b = - 19 + 4 a$

$25 = \frac{67}{4}$

$a = \frac{13}{4}$

$b = - 19 + 4 a$

$25 = \frac{67}{4}$

$a = \frac{13}{4}$

$b = - 19 + 4 a$

$25 = \frac{67}{4}$

$a = \frac{13}{4}$

$b = - 19 + 4 a$

Step 1.3.4.3

Replace all occurrences of $a$ in $b = - 19 + 4 a$ with $\frac{13}{4}$ .

$b = - 19 + 4 (\frac{13}{4})$

$25 = \frac{67}{4}$

$a = \frac{13}{4}$

Step 1.3.4.4

Simplify the right side.

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Step 1.3.4.4.1

Simplify $- 19 + 4 (\frac{13}{4})$ .

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Step 1.3.4.4.1.1

Cancel the common factor of $4$ .

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Step 1.3.4.4.1.1.1

Cancel the common factor.

$b = - 19 + 4 (\frac{13}{4})$

$25 = \frac{67}{4}$

$a = \frac{13}{4}$

Step 1.3.4.4.1.1.2

Rewrite the expression.

$b = - 19 + 13$

$25 = \frac{67}{4}$

$a = \frac{13}{4}$

$b = - 19 + 13$

$25 = \frac{67}{4}$

$a = \frac{13}{4}$

Step 1.3.4.4.1.2

Add $- 19$ and $13$ .

$b = - 6$

$25 = \frac{67}{4}$

$a = \frac{13}{4}$

$b = - 6$

$25 = \frac{67}{4}$

$a = \frac{13}{4}$

$b = - 6$

$25 = \frac{67}{4}$

$a = \frac{13}{4}$

$b = - 6$

$25 = \frac{67}{4}$

$a = \frac{13}{4}$

Step 1.3.5

Since $25 = \frac{67}{4}$ 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 $- 19 = a {(- 4)}^{2} + b (- 4) + c$ .

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Step 2.3.1.1

Rewrite the equation as $a {(- 4)}^{2} + b (- 4) + c = - 19$ .

$a {(- 4)}^{2} + b (- 4) + c = - 19$

$25 = a \cdot 7^{2} + b (7) + c$

$33 = a \cdot 12^{2} + b (12) + c$

Step 2.3.1.2

Simplify each term.

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Step 2.3.1.2.1

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

$a \cdot 16 + b (- 4) + c = - 19$

$25 = a \cdot 7^{2} + b (7) + c$

$33 = a \cdot 12^{2} + b (12) + c$

Step 2.3.1.2.2

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

$16 \cdot a + b (- 4) + c = - 19$

$25 = a \cdot 7^{2} + b (7) + c$

$33 = a \cdot 12^{2} + b (12) + c$

Step 2.3.1.2.3

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

$16 a - 4 b + c = - 19$

$25 = a \cdot 7^{2} + b (7) + c$

$33 = a \cdot 12^{2} + b (12) + c$

$16 a - 4 b + c = - 19$

$25 = a \cdot 7^{2} + b (7) + c$

$33 = a \cdot 12^{2} + b (12) + c$

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 $16 a$ from both sides of the equation.

$- 4 b + c = - 19 - 16 a$

$25 = a \cdot 7^{2} + b (7) + c$

$33 = a \cdot 12^{2} + b (12) + c$

Step 2.3.1.3.2

Add $4 b$ to both sides of the equation.

$c = - 19 - 16 a + 4 b$

$25 = a \cdot 7^{2} + b (7) + c$

$33 = a \cdot 12^{2} + b (12) + c$

$c = - 19 - 16 a + 4 b$

$25 = a \cdot 7^{2} + b (7) + c$

$33 = a \cdot 12^{2} + b (12) + c$

$c = - 19 - 16 a + 4 b$

$25 = a \cdot 7^{2} + b (7) + c$

$33 = a \cdot 12^{2} + b (12) + c$

Step 2.3.2

Replace all occurrences of $c$ with $- 19 - 16 a + 4 b$ in each equation.

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Step 2.3.2.1

Replace all occurrences of $c$ in $25 = a \cdot 7^{2} + b (7) + c$ with $- 19 - 16 a + 4 b$ .

$25 = a \cdot 7^{2} + b (7) - 19 - 16 a + 4 b$

$c = - 19 - 16 a + 4 b$

$33 = a \cdot 12^{2} + b (12) + c$

Step 2.3.2.2

Simplify $25 = a \cdot 7^{2} + b (7) - 19 - 16 a + 4 b$ .

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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.

$25 = a \cdot 7^{2} + b (7) - 19 - 16 a + 4 b$

$c = - 19 - 16 a + 4 b$

$33 = a \cdot 12^{2} + b (12) + c$

$25 = a \cdot 7^{2} + b (7) - 19 - 16 a + 4 b$

$c = - 19 - 16 a + 4 b$

$33 = a \cdot 12^{2} + b (12) + c$

Step 2.3.2.2.2

Simplify the right side.

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Step 2.3.2.2.2.1

Simplify $a \cdot 7^{2} + b (7) - 19 - 16 a + 4 b$ .

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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 $7$ to the power of $2$ .

$25 = a \cdot 49 + b (7) - 19 - 16 a + 4 b$

$c = - 19 - 16 a + 4 b$

$33 = a \cdot 12^{2} + b (12) + c$

Step 2.3.2.2.2.1.1.2

Move $49$ to the left of $a$ .

$25 = 49 \cdot a + b (7) - 19 - 16 a + 4 b$

$c = - 19 - 16 a + 4 b$

$33 = a \cdot 12^{2} + b (12) + c$

Step 2.3.2.2.2.1.1.3

Move $7$ to the left of $b$ .

$25 = 49 a + 7 b - 19 - 16 a + 4 b$

$c = - 19 - 16 a + 4 b$

$33 = a \cdot 12^{2} + b (12) + c$

$25 = 49 a + 7 b - 19 - 16 a + 4 b$

$c = - 19 - 16 a + 4 b$

$33 = a \cdot 12^{2} + b (12) + c$

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 $16 a$ from $49 a$ .

$25 = 33 a + 7 b - 19 + 4 b$

$c = - 19 - 16 a + 4 b$

$33 = a \cdot 12^{2} + b (12) + c$

Step 2.3.2.2.2.1.2.2

Add $7 b$ and $4 b$ .

$25 = 33 a + 11 b - 19$

$c = - 19 - 16 a + 4 b$

$33 = a \cdot 12^{2} + b (12) + c$

$25 = 33 a + 11 b - 19$

$c = - 19 - 16 a + 4 b$

$33 = a \cdot 12^{2} + b (12) + c$

$25 = 33 a + 11 b - 19$

$c = - 19 - 16 a + 4 b$

$33 = a \cdot 12^{2} + b (12) + c$

$25 = 33 a + 11 b - 19$

$c = - 19 - 16 a + 4 b$

$33 = a \cdot 12^{2} + b (12) + c$

$25 = 33 a + 11 b - 19$

$c = - 19 - 16 a + 4 b$

$33 = a \cdot 12^{2} + b (12) + c$

Step 2.3.2.3

Replace all occurrences of $c$ in $33 = a \cdot 12^{2} + b (12) + c$ with $- 19 - 16 a + 4 b$ .

$33 = a \cdot 12^{2} + b (12) - 19 - 16 a + 4 b$

$25 = 33 a + 11 b - 19$

$c = - 19 - 16 a + 4 b$

Step 2.3.2.4

Simplify $33 = a \cdot 12^{2} + b (12) - 19 - 16 a + 4 b$ .

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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.

$33 = a \cdot 12^{2} + b (12) - 19 - 16 a + 4 b$

$25 = 33 a + 11 b - 19$

$c = - 19 - 16 a + 4 b$

$33 = a \cdot 12^{2} + b (12) - 19 - 16 a + 4 b$

$25 = 33 a + 11 b - 19$

$c = - 19 - 16 a + 4 b$

Step 2.3.2.4.2

Simplify the right side.

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Step 2.3.2.4.2.1

Simplify $a \cdot 12^{2} + b (12) - 19 - 16 a + 4 b$ .

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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 $12$ to the power of $2$ .

$33 = a \cdot 144 + b (12) - 19 - 16 a + 4 b$

$25 = 33 a + 11 b - 19$

$c = - 19 - 16 a + 4 b$

Step 2.3.2.4.2.1.1.2

Move $144$ to the left of $a$ .

$33 = 144 \cdot a + b (12) - 19 - 16 a + 4 b$

$25 = 33 a + 11 b - 19$

$c = - 19 - 16 a + 4 b$

Step 2.3.2.4.2.1.1.3

Move $12$ to the left of $b$ .

$33 = 144 a + 12 b - 19 - 16 a + 4 b$

$25 = 33 a + 11 b - 19$

$c = - 19 - 16 a + 4 b$

$33 = 144 a + 12 b - 19 - 16 a + 4 b$

$25 = 33 a + 11 b - 19$

$c = - 19 - 16 a + 4 b$

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 $16 a$ from $144 a$ .

$33 = 128 a + 12 b - 19 + 4 b$

$25 = 33 a + 11 b - 19$

$c = - 19 - 16 a + 4 b$

Step 2.3.2.4.2.1.2.2

Add $12 b$ and $4 b$ .

$33 = 128 a + 16 b - 19$

$25 = 33 a + 11 b - 19$

$c = - 19 - 16 a + 4 b$

$33 = 128 a + 16 b - 19$

$25 = 33 a + 11 b - 19$

$c = - 19 - 16 a + 4 b$

$33 = 128 a + 16 b - 19$

$25 = 33 a + 11 b - 19$

$c = - 19 - 16 a + 4 b$

$33 = 128 a + 16 b - 19$

$25 = 33 a + 11 b - 19$

$c = - 19 - 16 a + 4 b$

$33 = 128 a + 16 b - 19$

$25 = 33 a + 11 b - 19$

$c = - 19 - 16 a + 4 b$

$33 = 128 a + 16 b - 19$

$25 = 33 a + 11 b - 19$

$c = - 19 - 16 a + 4 b$

Step 2.3.3

Solve for $a$ in $33 = 128 a + 16 b - 19$ .

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Step 2.3.3.1

Rewrite the equation as $128 a + 16 b - 19 = 33$ .

$128 a + 16 b - 19 = 33$

$25 = 33 a + 11 b - 19$

$c = - 19 - 16 a + 4 b$

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 $16 b$ from both sides of the equation.

$128 a - 19 = 33 - 16 b$

$25 = 33 a + 11 b - 19$

$c = - 19 - 16 a + 4 b$

Step 2.3.3.2.2

Add $19$ to both sides of the equation.

$128 a = 33 - 16 b + 19$

$25 = 33 a + 11 b - 19$

$c = - 19 - 16 a + 4 b$

Step 2.3.3.2.3

Add $33$ and $19$ .

$128 a = - 16 b + 52$

$25 = 33 a + 11 b - 19$

$c = - 19 - 16 a + 4 b$

$128 a = - 16 b + 52$

$25 = 33 a + 11 b - 19$

$c = - 19 - 16 a + 4 b$

Step 2.3.3.3

Divide each term in $128 a = - 16 b + 52$ by $128$ and simplify.

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Step 2.3.3.3.1

Divide each term in $128 a = - 16 b + 52$ by $128$ .

$\frac{128 a}{128} = \frac{- 16 b}{128} + \frac{52}{128}$

$25 = 33 a + 11 b - 19$

$c = - 19 - 16 a + 4 b$

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 $128$ .

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Step 2.3.3.3.2.1.1

Cancel the common factor.

$\frac{128 a}{128} = \frac{- 16 b}{128} + \frac{52}{128}$

$25 = 33 a + 11 b - 19$

$c = - 19 - 16 a + 4 b$

Step 2.3.3.3.2.1.2

Divide $a$ by $1$ .

$a = \frac{- 16 b}{128} + \frac{52}{128}$

$25 = 33 a + 11 b - 19$

$c = - 19 - 16 a + 4 b$

$a = \frac{- 16 b}{128} + \frac{52}{128}$

$25 = 33 a + 11 b - 19$

$c = - 19 - 16 a + 4 b$

$a = \frac{- 16 b}{128} + \frac{52}{128}$

$25 = 33 a + 11 b - 19$

$c = - 19 - 16 a + 4 b$

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 $- 16$ and $128$ .

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Step 2.3.3.3.3.1.1.1

Factor $16$ out of $- 16 b$ .

$a = \frac{16 (- b)}{128} + \frac{52}{128}$

$25 = 33 a + 11 b - 19$

$c = - 19 - 16 a + 4 b$

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 $16$ out of $128$ .

$a = \frac{16 (- b)}{16 (8)} + \frac{52}{128}$

$25 = 33 a + 11 b - 19$

$c = - 19 - 16 a + 4 b$

Step 2.3.3.3.3.1.1.2.2

Cancel the common factor.

$a = \frac{16 (- b)}{16 \cdot 8} + \frac{52}{128}$

$25 = 33 a + 11 b - 19$

$c = - 19 - 16 a + 4 b$

Step 2.3.3.3.3.1.1.2.3

Rewrite the expression.

$a = \frac{- b}{8} + \frac{52}{128}$

$25 = 33 a + 11 b - 19$

$c = - 19 - 16 a + 4 b$

$a = \frac{- b}{8} + \frac{52}{128}$

$25 = 33 a + 11 b - 19$

$c = - 19 - 16 a + 4 b$

$a = \frac{- b}{8} + \frac{52}{128}$

$25 = 33 a + 11 b - 19$

$c = - 19 - 16 a + 4 b$

Step 2.3.3.3.3.1.2

Move the negative in front of the fraction.

$a = - \frac{b}{8} + \frac{52}{128}$

$25 = 33 a + 11 b - 19$

$c = - 19 - 16 a + 4 b$

Step 2.3.3.3.3.1.3

Cancel the common factor of $52$ and $128$ .

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Step 2.3.3.3.3.1.3.1

Factor $4$ out of $52$ .

$a = - \frac{b}{8} + \frac{4 (13)}{128}$

$25 = 33 a + 11 b - 19$

$c = - 19 - 16 a + 4 b$

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 $4$ out of $128$ .

$a = - \frac{b}{8} + \frac{4 \cdot 13}{4 \cdot 32}$

$25 = 33 a + 11 b - 19$

$c = - 19 - 16 a + 4 b$

Step 2.3.3.3.3.1.3.2.2

Cancel the common factor.

$a = - \frac{b}{8} + \frac{4 \cdot 13}{4 \cdot 32}$

$25 = 33 a + 11 b - 19$

$c = - 19 - 16 a + 4 b$

Step 2.3.3.3.3.1.3.2.3

Rewrite the expression.

$a = - \frac{b}{8} + \frac{13}{32}$

$25 = 33 a + 11 b - 19$

$c = - 19 - 16 a + 4 b$

$a = - \frac{b}{8} + \frac{13}{32}$

$25 = 33 a + 11 b - 19$

$c = - 19 - 16 a + 4 b$

$a = - \frac{b}{8} + \frac{13}{32}$

$25 = 33 a + 11 b - 19$

$c = - 19 - 16 a + 4 b$

$a = - \frac{b}{8} + \frac{13}{32}$

$25 = 33 a + 11 b - 19$

$c = - 19 - 16 a + 4 b$

$a = - \frac{b}{8} + \frac{13}{32}$

$25 = 33 a + 11 b - 19$

$c = - 19 - 16 a + 4 b$

$a = - \frac{b}{8} + \frac{13}{32}$

$25 = 33 a + 11 b - 19$

$c = - 19 - 16 a + 4 b$

$a = - \frac{b}{8} + \frac{13}{32}$

$25 = 33 a + 11 b - 19$

$c = - 19 - 16 a + 4 b$

Step 2.3.4

Replace all occurrences of $a$ with $- \frac{b}{8} + \frac{13}{32}$ in each equation.

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Step 2.3.4.1

Replace all occurrences of $a$ in $25 = 33 a + 11 b - 19$ with $- \frac{b}{8} + \frac{13}{32}$ .

$25 = 33 (- \frac{b}{8} + \frac{13}{32}) + 11 b - 19$

$a = - \frac{b}{8} + \frac{13}{32}$

$c = - 19 - 16 a + 4 b$

Step 2.3.4.2

Simplify the right side.

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Step 2.3.4.2.1

Simplify $33 (- \frac{b}{8} + \frac{13}{32}) + 11 b - 19$ .

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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.

$25 = 33 (- \frac{b}{8}) + 33 (\frac{13}{32}) + 11 b - 19$

$a = - \frac{b}{8} + \frac{13}{32}$

$c = - 19 - 16 a + 4 b$

Step 2.3.4.2.1.1.2

Multiply $33 (- \frac{b}{8})$ .

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Step 2.3.4.2.1.1.2.1

Multiply $- 1$ by $33$ .

$25 = - 33 \frac{b}{8} + 33 (\frac{13}{32}) + 11 b - 19$

$a = - \frac{b}{8} + \frac{13}{32}$

$c = - 19 - 16 a + 4 b$

Step 2.3.4.2.1.1.2.2

Combine $- 33$ and $\frac{b}{8}$ .

$25 = \frac{- 33 b}{8} + 33 (\frac{13}{32}) + 11 b - 19$

$a = - \frac{b}{8} + \frac{13}{32}$

$c = - 19 - 16 a + 4 b$

$25 = \frac{- 33 b}{8} + 33 (\frac{13}{32}) + 11 b - 19$

$a = - \frac{b}{8} + \frac{13}{32}$

$c = - 19 - 16 a + 4 b$

Step 2.3.4.2.1.1.3

Multiply $33 (\frac{13}{32})$ .

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Step 2.3.4.2.1.1.3.1

Combine $33$ and $\frac{13}{32}$ .

$25 = \frac{- 33 b}{8} + \frac{33 \cdot 13}{32} + 11 b - 19$

$a = - \frac{b}{8} + \frac{13}{32}$

$c = - 19 - 16 a + 4 b$

Step 2.3.4.2.1.1.3.2

Multiply $33$ by $13$ .

$25 = \frac{- 33 b}{8} + \frac{429}{32} + 11 b - 19$

$a = - \frac{b}{8} + \frac{13}{32}$

$c = - 19 - 16 a + 4 b$

$25 = \frac{- 33 b}{8} + \frac{429}{32} + 11 b - 19$

$a = - \frac{b}{8} + \frac{13}{32}$

$c = - 19 - 16 a + 4 b$

Step 2.3.4.2.1.1.4

Move the negative in front of the fraction.

$25 = - \frac{33 b}{8} + \frac{429}{32} + 11 b - 19$

$a = - \frac{b}{8} + \frac{13}{32}$

$c = - 19 - 16 a + 4 b$

$25 = - \frac{33 b}{8} + \frac{429}{32} + 11 b - 19$

$a = - \frac{b}{8} + \frac{13}{32}$

$c = - 19 - 16 a + 4 b$

Step 2.3.4.2.1.2

To write $11 b$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$25 = - \frac{33 b}{8} + 11 b \cdot \frac{8}{8} + \frac{429}{32} - 19$

$a = - \frac{b}{8} + \frac{13}{32}$

$c = - 19 - 16 a + 4 b$

Step 2.3.4.2.1.3

Combine $11 b$ and $\frac{8}{8}$ .

$25 = - \frac{33 b}{8} + \frac{11 b \cdot 8}{8} + \frac{429}{32} - 19$

$a = - \frac{b}{8} + \frac{13}{32}$

$c = - 19 - 16 a + 4 b$

Step 2.3.4.2.1.4

Combine the numerators over the common denominator.

$25 = \frac{- 33 b + 11 b \cdot 8}{8} + \frac{429}{32} - 19$

$a = - \frac{b}{8} + \frac{13}{32}$

$c = - 19 - 16 a + 4 b$

Step 2.3.4.2.1.5

Find the common denominator.

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Step 2.3.4.2.1.5.1

Multiply $\frac{- 33 b + 11 b \cdot 8}{8}$ by $\frac{4}{4}$ .

$25 = \frac{- 33 b + 11 b \cdot 8}{8} \cdot \frac{4}{4} + \frac{429}{32} - 19$

$a = - \frac{b}{8} + \frac{13}{32}$

$c = - 19 - 16 a + 4 b$

Step 2.3.4.2.1.5.2

Multiply $\frac{- 33 b + 11 b \cdot 8}{8}$ by $\frac{4}{4}$ .

$25 = \frac{(- 33 b + 11 b \cdot 8) \cdot 4}{8 \cdot 4} + \frac{429}{32} - 19$

$a = - \frac{b}{8} + \frac{13}{32}$

$c = - 19 - 16 a + 4 b$

Step 2.3.4.2.1.5.3

Write $- 19$ as a fraction with denominator $1$ .

$25 = \frac{(- 33 b + 11 b \cdot 8) \cdot 4}{8 \cdot 4} + \frac{429}{32} + \frac{- 19}{1}$

$a = - \frac{b}{8} + \frac{13}{32}$

$c = - 19 - 16 a + 4 b$

Step 2.3.4.2.1.5.4

Multiply $\frac{- 19}{1}$ by $\frac{32}{32}$ .

$25 = \frac{(- 33 b + 11 b \cdot 8) \cdot 4}{8 \cdot 4} + \frac{429}{32} + \frac{- 19}{1} \cdot \frac{32}{32}$

$a = - \frac{b}{8} + \frac{13}{32}$

$c = - 19 - 16 a + 4 b$

Step 2.3.4.2.1.5.5

Multiply $\frac{- 19}{1}$ by $\frac{32}{32}$ .

$25 = \frac{(- 33 b + 11 b \cdot 8) \cdot 4}{8 \cdot 4} + \frac{429}{32} + \frac{- 19 \cdot 32}{32}$

$a = - \frac{b}{8} + \frac{13}{32}$

$c = - 19 - 16 a + 4 b$

Step 2.3.4.2.1.5.6

Reorder the factors of $8 \cdot 4$ .

$25 = \frac{(- 33 b + 11 b \cdot 8) \cdot 4}{4 \cdot 8} + \frac{429}{32} + \frac{- 19 \cdot 32}{32}$

$a = - \frac{b}{8} + \frac{13}{32}$

$c = - 19 - 16 a + 4 b$

Step 2.3.4.2.1.5.7

Multiply $4$ by $8$ .

$25 = \frac{(- 33 b + 11 b \cdot 8) \cdot 4}{32} + \frac{429}{32} + \frac{- 19 \cdot 32}{32}$

$a = - \frac{b}{8} + \frac{13}{32}$

$c = - 19 - 16 a + 4 b$

$25 = \frac{(- 33 b + 11 b \cdot 8) \cdot 4}{32} + \frac{429}{32} + \frac{- 19 \cdot 32}{32}$

$a = - \frac{b}{8} + \frac{13}{32}$

$c = - 19 - 16 a + 4 b$

Step 2.3.4.2.1.6

Combine the numerators over the common denominator.

$25 = \frac{(- 33 b + 11 b \cdot 8) \cdot 4 + 429 - 19 \cdot 32}{32}$

$a = - \frac{b}{8} + \frac{13}{32}$

$c = - 19 - 16 a + 4 b$

Step 2.3.4.2.1.7

Simplify each term.

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Step 2.3.4.2.1.7.1

Multiply $8$ by $11$ .

$25 = \frac{(- 33 b + 88 b) \cdot 4 + 429 - 19 \cdot 32}{32}$

$a = - \frac{b}{8} + \frac{13}{32}$

$c = - 19 - 16 a + 4 b$

Step 2.3.4.2.1.7.2

Add $- 33 b$ and $88 b$ .

$25 = \frac{55 b \cdot 4 + 429 - 19 \cdot 32}{32}$

$a = - \frac{b}{8} + \frac{13}{32}$

$c = - 19 - 16 a + 4 b$

Step 2.3.4.2.1.7.3

Multiply $4$ by $55$ .

$25 = \frac{220 b + 429 - 19 \cdot 32}{32}$

$a = - \frac{b}{8} + \frac{13}{32}$

$c = - 19 - 16 a + 4 b$

Step 2.3.4.2.1.7.4

Multiply $- 19$ by $32$ .

$25 = \frac{220 b + 429 - 608}{32}$

$a = - \frac{b}{8} + \frac{13}{32}$

$c = - 19 - 16 a + 4 b$

$25 = \frac{220 b + 429 - 608}{32}$

$a = - \frac{b}{8} + \frac{13}{32}$

$c = - 19 - 16 a + 4 b$

Step 2.3.4.2.1.8

Subtract $608$ from $429$ .

$25 = \frac{220 b - 179}{32}$

$a = - \frac{b}{8} + \frac{13}{32}$

$c = - 19 - 16 a + 4 b$

$25 = \frac{220 b - 179}{32}$

$a = - \frac{b}{8} + \frac{13}{32}$

$c = - 19 - 16 a + 4 b$

$25 = \frac{220 b - 179}{32}$

$a = - \frac{b}{8} + \frac{13}{32}$

$c = - 19 - 16 a + 4 b$

Step 2.3.4.3

Replace all occurrences of $a$ in $c = - 19 - 16 a + 4 b$ with $- \frac{b}{8} + \frac{13}{32}$ .

$c = - 19 - 16 (- \frac{b}{8} + \frac{13}{32}) + 4 b$

$25 = \frac{220 b - 179}{32}$

$a = - \frac{b}{8} + \frac{13}{32}$

Step 2.3.4.4

Simplify the right side.

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Step 2.3.4.4.1

Simplify $- 19 - 16 (- \frac{b}{8} + \frac{13}{32}) + 4 b$ .

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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.

$c = - 19 - 16 (- \frac{b}{8}) - 16 (\frac{13}{32}) + 4 b$

$25 = \frac{220 b - 179}{32}$

$a = - \frac{b}{8} + \frac{13}{32}$

Step 2.3.4.4.1.1.2

Cancel the common factor of $8$ .

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Step 2.3.4.4.1.1.2.1

Move the leading negative in $- \frac{b}{8}$ into the numerator.

$c = - 19 - 16 \frac{- b}{8} - 16 (\frac{13}{32}) + 4 b$

$25 = \frac{220 b - 179}{32}$

$a = - \frac{b}{8} + \frac{13}{32}$

Step 2.3.4.4.1.1.2.2

Factor $8$ out of $- 16$ .

$c = - 19 + 8 (- 2) (\frac{- b}{8}) - 16 (\frac{13}{32}) + 4 b$

$25 = \frac{220 b - 179}{32}$

$a = - \frac{b}{8} + \frac{13}{32}$

Step 2.3.4.4.1.1.2.3

Cancel the common factor.

$c = - 19 + 8 \cdot (- 2 \frac{- b}{8}) - 16 (\frac{13}{32}) + 4 b$

$25 = \frac{220 b - 179}{32}$

$a = - \frac{b}{8} + \frac{13}{32}$

Step 2.3.4.4.1.1.2.4

Rewrite the expression.

$c = - 19 - 2 (- b) - 16 (\frac{13}{32}) + 4 b$

$25 = \frac{220 b - 179}{32}$

$a = - \frac{b}{8} + \frac{13}{32}$

$c = - 19 - 2 (- b) - 16 (\frac{13}{32}) + 4 b$

$25 = \frac{220 b - 179}{32}$

$a = - \frac{b}{8} + \frac{13}{32}$

Step 2.3.4.4.1.1.3

Multiply $- 1$ by $- 2$ .

$c = - 19 + 2 b - 16 (\frac{13}{32}) + 4 b$

$25 = \frac{220 b - 179}{32}$

$a = - \frac{b}{8} + \frac{13}{32}$

Step 2.3.4.4.1.1.4

Cancel the common factor of $16$ .

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Step 2.3.4.4.1.1.4.1

Factor $16$ out of $- 16$ .

$c = - 19 + 2 b + 16 (- 1) (\frac{13}{32}) + 4 b$

$25 = \frac{220 b - 179}{32}$

$a = - \frac{b}{8} + \frac{13}{32}$

Step 2.3.4.4.1.1.4.2

Factor $16$ out of $32$ .

$c = - 19 + 2 b + 16 \cdot (- 1 \frac{13}{16 \cdot 2}) + 4 b$

$25 = \frac{220 b - 179}{32}$

$a = - \frac{b}{8} + \frac{13}{32}$

Step 2.3.4.4.1.1.4.3

Cancel the common factor.

$c = - 19 + 2 b + 16 \cdot (- 1 \frac{13}{16 \cdot 2}) + 4 b$

$25 = \frac{220 b - 179}{32}$

$a = - \frac{b}{8} + \frac{13}{32}$

Step 2.3.4.4.1.1.4.4

Rewrite the expression.

$c = - 19 + 2 b - 1 (\frac{13}{2}) + 4 b$

$25 = \frac{220 b - 179}{32}$

$a = - \frac{b}{8} + \frac{13}{32}$

$c = - 19 + 2 b - 1 (\frac{13}{2}) + 4 b$

$25 = \frac{220 b - 179}{32}$

$a = - \frac{b}{8} + \frac{13}{32}$

Step 2.3.4.4.1.1.5

Rewrite $- 1 (\frac{13}{2})$ as $- (\frac{13}{2})$ .

$c = - 19 + 2 b - \frac{13}{2} + 4 b$

$25 = \frac{220 b - 179}{32}$

$a = - \frac{b}{8} + \frac{13}{32}$

$c = - 19 + 2 b - \frac{13}{2} + 4 b$

$25 = \frac{220 b - 179}{32}$

$a = - \frac{b}{8} + \frac{13}{32}$

Step 2.3.4.4.1.2

To write $- 19$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$c = 2 b - 19 \cdot \frac{2}{2} - \frac{13}{2} + 4 b$

$25 = \frac{220 b - 179}{32}$

$a = - \frac{b}{8} + \frac{13}{32}$

Step 2.3.4.4.1.3

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

$c = 2 b + \frac{- 19 \cdot 2}{2} - \frac{13}{2} + 4 b$

$25 = \frac{220 b - 179}{32}$

$a = - \frac{b}{8} + \frac{13}{32}$

Step 2.3.4.4.1.4

Combine the numerators over the common denominator.

$c = 2 b + \frac{- 19 \cdot 2 - 13}{2} + 4 b$

$25 = \frac{220 b - 179}{32}$

$a = - \frac{b}{8} + \frac{13}{32}$

Step 2.3.4.4.1.5

Simplify the numerator.

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Step 2.3.4.4.1.5.1

Multiply $- 19$ by $2$ .

$c = 2 b + \frac{- 38 - 13}{2} + 4 b$

$25 = \frac{220 b - 179}{32}$

$a = - \frac{b}{8} + \frac{13}{32}$

Step 2.3.4.4.1.5.2

Subtract $13$ from $- 38$ .

$c = 2 b + \frac{- 51}{2} + 4 b$

$25 = \frac{220 b - 179}{32}$

$a = - \frac{b}{8} + \frac{13}{32}$

$c = 2 b + \frac{- 51}{2} + 4 b$

$25 = \frac{220 b - 179}{32}$

$a = - \frac{b}{8} + \frac{13}{32}$

Step 2.3.4.4.1.6

Move the negative in front of the fraction.

$c = 2 b - \frac{51}{2} + 4 b$

$25 = \frac{220 b - 179}{32}$

$a = - \frac{b}{8} + \frac{13}{32}$

Step 2.3.4.4.1.7

Add $2 b$ and $4 b$ .

$c = 6 b - \frac{51}{2}$

$25 = \frac{220 b - 179}{32}$

$a = - \frac{b}{8} + \frac{13}{32}$

$c = 6 b - \frac{51}{2}$

$25 = \frac{220 b - 179}{32}$

$a = - \frac{b}{8} + \frac{13}{32}$

$c = 6 b - \frac{51}{2}$

$25 = \frac{220 b - 179}{32}$

$a = - \frac{b}{8} + \frac{13}{32}$

$c = 6 b - \frac{51}{2}$

$25 = \frac{220 b - 179}{32}$

$a = - \frac{b}{8} + \frac{13}{32}$

Step 2.3.5

Solve for $b$ in $25 = \frac{220 b - 179}{32}$ .

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Step 2.3.5.1

Rewrite the equation as $\frac{220 b - 179}{32} = 25$ .

$\frac{220 b - 179}{32} = 25$

$c = 6 b - \frac{51}{2}$

$a = - \frac{b}{8} + \frac{13}{32}$

Step 2.3.5.2

Multiply both sides by $32$ .

$\frac{220 b - 179}{32} \cdot 32 = 25 \cdot 32$

$c = 6 b - \frac{51}{2}$

$a = - \frac{b}{8} + \frac{13}{32}$

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 $32$ .

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Step 2.3.5.3.1.1.1

Cancel the common factor.

$\frac{220 b - 179}{32} \cdot 32 = 25 \cdot 32$

$c = 6 b - \frac{51}{2}$

$a = - \frac{b}{8} + \frac{13}{32}$

Step 2.3.5.3.1.1.2

Rewrite the expression.

$220 b - 179 = 25 \cdot 32$

$c = 6 b - \frac{51}{2}$

$a = - \frac{b}{8} + \frac{13}{32}$

$220 b - 179 = 25 \cdot 32$

$c = 6 b - \frac{51}{2}$

$a = - \frac{b}{8} + \frac{13}{32}$

$220 b - 179 = 25 \cdot 32$

$c = 6 b - \frac{51}{2}$

$a = - \frac{b}{8} + \frac{13}{32}$

Step 2.3.5.3.2

Simplify the right side.

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Step 2.3.5.3.2.1

Multiply $25$ by $32$ .

$220 b - 179 = 800$

$c = 6 b - \frac{51}{2}$

$a = - \frac{b}{8} + \frac{13}{32}$

$220 b - 179 = 800$

$c = 6 b - \frac{51}{2}$

$a = - \frac{b}{8} + \frac{13}{32}$

$220 b - 179 = 800$

$c = 6 b - \frac{51}{2}$

$a = - \frac{b}{8} + \frac{13}{32}$

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

Add $179$ to both sides of the equation.

$220 b = 800 + 179$

$c = 6 b - \frac{51}{2}$

$a = - \frac{b}{8} + \frac{13}{32}$

Step 2.3.5.4.1.2

Add $800$ and $179$ .

$220 b = 979$

$c = 6 b - \frac{51}{2}$

$a = - \frac{b}{8} + \frac{13}{32}$

$220 b = 979$

$c = 6 b - \frac{51}{2}$

$a = - \frac{b}{8} + \frac{13}{32}$

Step 2.3.5.4.2

Divide each term in $220 b = 979$ by $220$ and simplify.

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Step 2.3.5.4.2.1

Divide each term in $220 b = 979$ by $220$ .

$\frac{220 b}{220} = \frac{979}{220}$

$c = 6 b - \frac{51}{2}$

$a = - \frac{b}{8} + \frac{13}{32}$

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 $220$ .

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Step 2.3.5.4.2.2.1.1

Cancel the common factor.

$\frac{220 b}{220} = \frac{979}{220}$

$c = 6 b - \frac{51}{2}$

$a = - \frac{b}{8} + \frac{13}{32}$

Step 2.3.5.4.2.2.1.2

Divide $b$ by $1$ .

$b = \frac{979}{220}$

$c = 6 b - \frac{51}{2}$

$a = - \frac{b}{8} + \frac{13}{32}$

$b = \frac{979}{220}$

$c = 6 b - \frac{51}{2}$

$a = - \frac{b}{8} + \frac{13}{32}$

$b = \frac{979}{220}$

$c = 6 b - \frac{51}{2}$

$a = - \frac{b}{8} + \frac{13}{32}$

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 $979$ and $220$ .

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Step 2.3.5.4.2.3.1.1

Factor $11$ out of $979$ .

$b = \frac{11 (89)}{220}$

$c = 6 b - \frac{51}{2}$

$a = - \frac{b}{8} + \frac{13}{32}$

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 $11$ out of $220$ .

$b = \frac{11 \cdot 89}{11 \cdot 20}$

$c = 6 b - \frac{51}{2}$

$a = - \frac{b}{8} + \frac{13}{32}$

Step 2.3.5.4.2.3.1.2.2

Cancel the common factor.

$b = \frac{11 \cdot 89}{11 \cdot 20}$

$c = 6 b - \frac{51}{2}$

$a = - \frac{b}{8} + \frac{13}{32}$

Step 2.3.5.4.2.3.1.2.3

Rewrite the expression.

$b = \frac{89}{20}$

$c = 6 b - \frac{51}{2}$

$a = - \frac{b}{8} + \frac{13}{32}$

$b = \frac{89}{20}$

$c = 6 b - \frac{51}{2}$

$a = - \frac{b}{8} + \frac{13}{32}$

$b = \frac{89}{20}$

$c = 6 b - \frac{51}{2}$

$a = - \frac{b}{8} + \frac{13}{32}$

$b = \frac{89}{20}$

$c = 6 b - \frac{51}{2}$

$a = - \frac{b}{8} + \frac{13}{32}$

$b = \frac{89}{20}$

$c = 6 b - \frac{51}{2}$

$a = - \frac{b}{8} + \frac{13}{32}$

$b = \frac{89}{20}$

$c = 6 b - \frac{51}{2}$

$a = - \frac{b}{8} + \frac{13}{32}$

$b = \frac{89}{20}$

$c = 6 b - \frac{51}{2}$

$a = - \frac{b}{8} + \frac{13}{32}$

Step 2.3.6

Replace all occurrences of $b$ with $\frac{89}{20}$ in each equation.

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Step 2.3.6.1

Replace all occurrences of $b$ in $c = 6 b - \frac{51}{2}$ with $\frac{89}{20}$ .

$c = 6 (\frac{89}{20}) - \frac{51}{2}$

$b = \frac{89}{20}$

$a = - \frac{b}{8} + \frac{13}{32}$

Step 2.3.6.2

Simplify the right side.

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Step 2.3.6.2.1

Simplify $6 (\frac{89}{20}) - \frac{51}{2}$ .

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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

Cancel the common factor of $2$ .

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Step 2.3.6.2.1.1.1.1

Factor $2$ out of $6$ .

$c = 2 (3) (\frac{89}{20}) - \frac{51}{2}$

$b = \frac{89}{20}$

$a = - \frac{b}{8} + \frac{13}{32}$

Step 2.3.6.2.1.1.1.2

Factor $2$ out of $20$ .

$c = 2 \cdot (3 (\frac{89}{2 \cdot 10})) - \frac{51}{2}$

$b = \frac{89}{20}$

$a = - \frac{b}{8} + \frac{13}{32}$

Step 2.3.6.2.1.1.1.3

Cancel the common factor.

$c = 2 \cdot (3 (\frac{89}{2 \cdot 10})) - \frac{51}{2}$

$b = \frac{89}{20}$

$a = - \frac{b}{8} + \frac{13}{32}$

Step 2.3.6.2.1.1.1.4

Rewrite the expression.

$c = 3 (\frac{89}{10}) - \frac{51}{2}$

$b = \frac{89}{20}$

$a = - \frac{b}{8} + \frac{13}{32}$

$c = 3 (\frac{89}{10}) - \frac{51}{2}$

$b = \frac{89}{20}$

$a = - \frac{b}{8} + \frac{13}{32}$

Step 2.3.6.2.1.1.2

Combine $3$ and $\frac{89}{10}$ .

$c = \frac{3 \cdot 89}{10} - \frac{51}{2}$

$b = \frac{89}{20}$

$a = - \frac{b}{8} + \frac{13}{32}$

Step 2.3.6.2.1.1.3

Multiply $3$ by $89$ .

$c = \frac{267}{10} - \frac{51}{2}$

$b = \frac{89}{20}$

$a = - \frac{b}{8} + \frac{13}{32}$

$c = \frac{267}{10} - \frac{51}{2}$

$b = \frac{89}{20}$

$a = - \frac{b}{8} + \frac{13}{32}$

Step 2.3.6.2.1.2

To write $- \frac{51}{2}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$c = \frac{267}{10} - \frac{51}{2} \cdot \frac{5}{5}$

$b = \frac{89}{20}$

$a = - \frac{b}{8} + \frac{13}{32}$

Step 2.3.6.2.1.3

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.6.2.1.3.1

Multiply $\frac{51}{2}$ by $\frac{5}{5}$ .

$c = \frac{267}{10} - \frac{51 \cdot 5}{2 \cdot 5}$

$b = \frac{89}{20}$

$a = - \frac{b}{8} + \frac{13}{32}$

Step 2.3.6.2.1.3.2

Multiply $2$ by $5$ .

$c = \frac{267}{10} - \frac{51 \cdot 5}{10}$

$b = \frac{89}{20}$

$a = - \frac{b}{8} + \frac{13}{32}$

$c = \frac{267}{10} - \frac{51 \cdot 5}{10}$

$b = \frac{89}{20}$

$a = - \frac{b}{8} + \frac{13}{32}$

Step 2.3.6.2.1.4

Combine the numerators over the common denominator.

$c = \frac{267 - 51 \cdot 5}{10}$

$b = \frac{89}{20}$

$a = - \frac{b}{8} + \frac{13}{32}$

Step 2.3.6.2.1.5

Simplify the numerator.

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Step 2.3.6.2.1.5.1

Multiply $- 51$ by $5$ .

$c = \frac{267 - 255}{10}$

$b = \frac{89}{20}$

$a = - \frac{b}{8} + \frac{13}{32}$

Step 2.3.6.2.1.5.2

Subtract $255$ from $267$ .

$c = \frac{12}{10}$

$b = \frac{89}{20}$

$a = - \frac{b}{8} + \frac{13}{32}$

$c = \frac{12}{10}$

$b = \frac{89}{20}$

$a = - \frac{b}{8} + \frac{13}{32}$

Step 2.3.6.2.1.6

Cancel the common factor of $12$ and $10$ .

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Step 2.3.6.2.1.6.1

Factor $2$ out of $12$ .

$c = \frac{2 (6)}{10}$

$b = \frac{89}{20}$

$a = - \frac{b}{8} + \frac{13}{32}$

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 $2$ out of $10$ .

$c = \frac{2 \cdot 6}{2 \cdot 5}$

$b = \frac{89}{20}$

$a = - \frac{b}{8} + \frac{13}{32}$

Step 2.3.6.2.1.6.2.2

Cancel the common factor.

$c = \frac{2 \cdot 6}{2 \cdot 5}$

$b = \frac{89}{20}$

$a = - \frac{b}{8} + \frac{13}{32}$

Step 2.3.6.2.1.6.2.3

Rewrite the expression.

$c = \frac{6}{5}$

$b = \frac{89}{20}$

$a = - \frac{b}{8} + \frac{13}{32}$

$c = \frac{6}{5}$

$b = \frac{89}{20}$

$a = - \frac{b}{8} + \frac{13}{32}$

$c = \frac{6}{5}$

$b = \frac{89}{20}$

$a = - \frac{b}{8} + \frac{13}{32}$

$c = \frac{6}{5}$

$b = \frac{89}{20}$

$a = - \frac{b}{8} + \frac{13}{32}$

$c = \frac{6}{5}$

$b = \frac{89}{20}$

$a = - \frac{b}{8} + \frac{13}{32}$

Step 2.3.6.3

Replace all occurrences of $b$ in $a = - \frac{b}{8} + \frac{13}{32}$ with $\frac{89}{20}$ .

$a = - \frac{\frac{89}{20}}{8} + \frac{13}{32}$

$c = \frac{6}{5}$

$b = \frac{89}{20}$

Step 2.3.6.4

Simplify the right side.

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Step 2.3.6.4.1

Simplify $- \frac{\frac{89}{20}}{8} + \frac{13}{32}$ .

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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.

$a = - (\frac{89}{20} \cdot \frac{1}{8}) + \frac{13}{32}$

$c = \frac{6}{5}$

$b = \frac{89}{20}$

Step 2.3.6.4.1.1.2

Multiply $\frac{89}{20} \cdot \frac{1}{8}$ .

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Step 2.3.6.4.1.1.2.1

Multiply $\frac{89}{20}$ by $\frac{1}{8}$ .

$a = - \frac{89}{20 \cdot 8} + \frac{13}{32}$

$c = \frac{6}{5}$

$b = \frac{89}{20}$

Step 2.3.6.4.1.1.2.2

Multiply $20$ by $8$ .

$a = - \frac{89}{160} + \frac{13}{32}$

$c = \frac{6}{5}$

$b = \frac{89}{20}$

$a = - \frac{89}{160} + \frac{13}{32}$

$c = \frac{6}{5}$

$b = \frac{89}{20}$

$a = - \frac{89}{160} + \frac{13}{32}$

$c = \frac{6}{5}$

$b = \frac{89}{20}$

Step 2.3.6.4.1.2

To write $\frac{13}{32}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$a = - \frac{89}{160} + \frac{13}{32} \cdot \frac{5}{5}$

$c = \frac{6}{5}$

$b = \frac{89}{20}$

Step 2.3.6.4.1.3

Write each expression with a common denominator of $160$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.6.4.1.3.1

Multiply $\frac{13}{32}$ by $\frac{5}{5}$ .

$a = - \frac{89}{160} + \frac{13 \cdot 5}{32 \cdot 5}$

$c = \frac{6}{5}$

$b = \frac{89}{20}$

Step 2.3.6.4.1.3.2

Multiply $32$ by $5$ .

$a = - \frac{89}{160} + \frac{13 \cdot 5}{160}$

$c = \frac{6}{5}$

$b = \frac{89}{20}$

$a = - \frac{89}{160} + \frac{13 \cdot 5}{160}$

$c = \frac{6}{5}$

$b = \frac{89}{20}$

Step 2.3.6.4.1.4

Combine the numerators over the common denominator.

$a = \frac{- 89 + 13 \cdot 5}{160}$

$c = \frac{6}{5}$

$b = \frac{89}{20}$

Step 2.3.6.4.1.5

Simplify the numerator.

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Step 2.3.6.4.1.5.1

Multiply $13$ by $5$ .

$a = \frac{- 89 + 65}{160}$

$c = \frac{6}{5}$

$b = \frac{89}{20}$

Step 2.3.6.4.1.5.2

Add $- 89$ and $65$ .

$a = \frac{- 24}{160}$

$c = \frac{6}{5}$

$b = \frac{89}{20}$

$a = \frac{- 24}{160}$

$c = \frac{6}{5}$

$b = \frac{89}{20}$

Step 2.3.6.4.1.6

Cancel the common factor of $- 24$ and $160$ .

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Step 2.3.6.4.1.6.1

Factor $8$ out of $- 24$ .

$a = \frac{8 (- 3)}{160}$

$c = \frac{6}{5}$

$b = \frac{89}{20}$

Step 2.3.6.4.1.6.2

Cancel the common factors.

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Step 2.3.6.4.1.6.2.1

Factor $8$ out of $160$ .

$a = \frac{8 \cdot - 3}{8 \cdot 20}$

$c = \frac{6}{5}$

$b = \frac{89}{20}$

Step 2.3.6.4.1.6.2.2

Cancel the common factor.

$a = \frac{8 \cdot - 3}{8 \cdot 20}$

$c = \frac{6}{5}$

$b = \frac{89}{20}$

Step 2.3.6.4.1.6.2.3

Rewrite the expression.

$a = \frac{- 3}{20}$

$c = \frac{6}{5}$

$b = \frac{89}{20}$

$a = \frac{- 3}{20}$

$c = \frac{6}{5}$

$b = \frac{89}{20}$

$a = \frac{- 3}{20}$

$c = \frac{6}{5}$

$b = \frac{89}{20}$

Step 2.3.6.4.1.7

Move the negative in front of the fraction.

$a = - \frac{3}{20}$

$c = \frac{6}{5}$

$b = \frac{89}{20}$

$a = - \frac{3}{20}$

$c = \frac{6}{5}$

$b = \frac{89}{20}$

$a = - \frac{3}{20}$

$c = \frac{6}{5}$

$b = \frac{89}{20}$

$a = - \frac{3}{20}$

$c = \frac{6}{5}$

$b = \frac{89}{20}$

Step 2.3.7

List all of the solutions.

$a = - \frac{3}{20}, c = \frac{6}{5}, b = \frac{89}{20}$

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 $a = - \frac{3}{20}$ , $b = \frac{89}{20}$ , $c = \frac{6}{5}$ , and $x = - 4$ .

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Step 2.4.1.1

Simplify each term.

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Step 2.4.1.1.1

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

$y = - \frac{3}{20} \cdot 16 + (\frac{89}{20}) \cdot (- 4) + \frac{6}{5}$

Step 2.4.1.1.2

Cancel the common factor of $4$ .

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Step 2.4.1.1.2.1

Move the leading negative in $- \frac{3}{20}$ into the numerator.

$y = \frac{- 3}{20} \cdot 16 + (\frac{89}{20}) \cdot (- 4) + \frac{6}{5}$

Step 2.4.1.1.2.2

Factor $4$ out of $20$ .

$y = \frac{- 3}{4 (5)} \cdot 16 + (\frac{89}{20}) \cdot (- 4) + \frac{6}{5}$

Step 2.4.1.1.2.3

Factor $4$ out of $16$ .

$y = \frac{- 3}{4 \cdot 5} \cdot (4 \cdot 4) + (\frac{89}{20}) \cdot (- 4) + \frac{6}{5}$

Step 2.4.1.1.2.4

Cancel the common factor.

$y = \frac{- 3}{4 \cdot 5} \cdot (4 \cdot 4) + (\frac{89}{20}) \cdot (- 4) + \frac{6}{5}$

Step 2.4.1.1.2.5

Rewrite the expression.

$y = \frac{- 3}{5} \cdot 4 + (\frac{89}{20}) \cdot (- 4) + \frac{6}{5}$

Step 2.4.1.1.3

Combine $\frac{- 3}{5}$ and $4$ .

$y = \frac{- 3 \cdot 4}{5} + (\frac{89}{20}) \cdot (- 4) + \frac{6}{5}$

Step 2.4.1.1.4

Multiply $- 3$ by $4$ .

$y = \frac{- 12}{5} + (\frac{89}{20}) \cdot (- 4) + \frac{6}{5}$

Step 2.4.1.1.5

Move the negative in front of the fraction.

$y = - \frac{12}{5} + (\frac{89}{20}) \cdot (- 4) + \frac{6}{5}$

Step 2.4.1.1.6

Cancel the common factor of $4$ .

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Step 2.4.1.1.6.1

Factor $4$ out of $20$ .

$y = - \frac{12}{5} + \frac{89}{4 (5)} \cdot - 4 + \frac{6}{5}$

Step 2.4.1.1.6.2

Factor $4$ out of $- 4$ .

$y = - \frac{12}{5} + \frac{89}{4 \cdot 5} \cdot (4 \cdot - 1) + \frac{6}{5}$

Step 2.4.1.1.6.3

Cancel the common factor.

$y = - \frac{12}{5} + \frac{89}{4 \cdot 5} \cdot (4 \cdot - 1) + \frac{6}{5}$

Step 2.4.1.1.6.4

Rewrite the expression.

$y = - \frac{12}{5} + \frac{89}{5} \cdot - 1 + \frac{6}{5}$

Step 2.4.1.1.7

Combine $\frac{89}{5}$ and $- 1$ .

$y = - \frac{12}{5} + \frac{89 \cdot - 1}{5} + \frac{6}{5}$

Step 2.4.1.1.8

Multiply $89$ by $- 1$ .

$y = - \frac{12}{5} + \frac{- 89}{5} + \frac{6}{5}$

Step 2.4.1.1.9

Move the negative in front of the fraction.

$y = - \frac{12}{5} - \frac{89}{5} + \frac{6}{5}$

Step 2.4.1.2

Combine fractions.

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Step 2.4.1.2.1

Combine the numerators over the common denominator.

$y = \frac{- 12 - 89 + 6}{5}$

Step 2.4.1.2.2

Simplify the expression.

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Step 2.4.1.2.2.1

Subtract $89$ from $- 12$ .

$y = \frac{- 101 + 6}{5}$

Step 2.4.1.2.2.2

Add $- 101$ and $6$ .

$y = \frac{- 95}{5}$

Step 2.4.1.2.2.3

Divide $- 95$ by $5$ .

$y = - 19$

Step 2.4.2

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = - 4$ . This check passes since $y = - 19$ and $y = - 19$ .

$- 19 = - 19$

Step 2.4.3

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = - \frac{3}{20}$ , $b = \frac{89}{20}$ , $c = \frac{6}{5}$ , and $x = 7$ .

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Step 2.4.3.1

Simplify each term.

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Step 2.4.3.1.1

Raise $7$ to the power of $2$ .

$y = - \frac{3}{20} \cdot 49 + (\frac{89}{20}) \cdot (7) + \frac{6}{5}$

Step 2.4.3.1.2

Multiply $- \frac{3}{20} \cdot 49$ .

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Step 2.4.3.1.2.1

Multiply $49$ by $- 1$ .

$y = - 49 (\frac{3}{20}) + (\frac{89}{20}) \cdot (7) + \frac{6}{5}$

Step 2.4.3.1.2.2

Combine $- 49$ and $\frac{3}{20}$ .

$y = \frac{- 49 \cdot 3}{20} + (\frac{89}{20}) \cdot (7) + \frac{6}{5}$

Step 2.4.3.1.2.3

Multiply $- 49$ by $3$ .

$y = \frac{- 147}{20} + (\frac{89}{20}) \cdot (7) + \frac{6}{5}$

Step 2.4.3.1.3

Move the negative in front of the fraction.

$y = - \frac{147}{20} + (\frac{89}{20}) \cdot (7) + \frac{6}{5}$

Step 2.4.3.1.4

Multiply $(\frac{89}{20}) (7)$ .

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Step 2.4.3.1.4.1

Combine $\frac{89}{20}$ and $7$ .

$y = - \frac{147}{20} + \frac{89 \cdot 7}{20} + \frac{6}{5}$

Step 2.4.3.1.4.2

Multiply $89$ by $7$ .

$y = - \frac{147}{20} + \frac{623}{20} + \frac{6}{5}$

Step 2.4.3.2

Simplify terms.

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Step 2.4.3.2.1

Combine the numerators over the common denominator.

$y = \frac{- 147 + 623}{20} + \frac{6}{5}$

Step 2.4.3.2.2

Add $- 147$ and $623$ .

$y = \frac{476}{20} + \frac{6}{5}$

Step 2.4.3.2.3

Cancel the common factor of $476$ and $20$ .

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Step 2.4.3.2.3.1

Factor $4$ out of $476$ .

$y = \frac{4 (119)}{20} + \frac{6}{5}$

Step 2.4.3.2.3.2

Cancel the common factors.

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Step 2.4.3.2.3.2.1

Factor $4$ out of $20$ .

$y = \frac{4 \cdot 119}{4 \cdot 5} + \frac{6}{5}$

Step 2.4.3.2.3.2.2

Cancel the common factor.

$y = \frac{4 \cdot 119}{4 \cdot 5} + \frac{6}{5}$

Step 2.4.3.2.3.2.3

Rewrite the expression.

$y = \frac{119}{5} + \frac{6}{5}$

Step 2.4.3.2.4

Combine the numerators over the common denominator.

$y = \frac{119 + 6}{5}$

Step 2.4.3.2.5

Simplify the expression.

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Step 2.4.3.2.5.1

Add $119$ and $6$ .

$y = \frac{125}{5}$

Step 2.4.3.2.5.2

Divide $125$ by $5$ .

$y = 25$

Step 2.4.4

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = 7$ . This check passes since $y = 25$ and $y = 25$ .

$25 = 25$

Step 2.4.5

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = - \frac{3}{20}$ , $b = \frac{89}{20}$ , $c = \frac{6}{5}$ , and $x = 12$ .

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Step 2.4.5.1

Simplify each term.

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Step 2.4.5.1.1

Raise $12$ to the power of $2$ .

$y = - \frac{3}{20} \cdot 144 + (\frac{89}{20}) \cdot (12) + \frac{6}{5}$

Step 2.4.5.1.2

Cancel the common factor of $4$ .

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Step 2.4.5.1.2.1

Move the leading negative in $- \frac{3}{20}$ into the numerator.

$y = \frac{- 3}{20} \cdot 144 + (\frac{89}{20}) \cdot (12) + \frac{6}{5}$

Step 2.4.5.1.2.2

Factor $4$ out of $20$ .

$y = \frac{- 3}{4 (5)} \cdot 144 + (\frac{89}{20}) \cdot (12) + \frac{6}{5}$

Step 2.4.5.1.2.3

Factor $4$ out of $144$ .

$y = \frac{- 3}{4 \cdot 5} \cdot (4 \cdot 36) + (\frac{89}{20}) \cdot (12) + \frac{6}{5}$

Step 2.4.5.1.2.4

Cancel the common factor.

$y = \frac{- 3}{4 \cdot 5} \cdot (4 \cdot 36) + (\frac{89}{20}) \cdot (12) + \frac{6}{5}$

Step 2.4.5.1.2.5

Rewrite the expression.

$y = \frac{- 3}{5} \cdot 36 + (\frac{89}{20}) \cdot (12) + \frac{6}{5}$

Step 2.4.5.1.3

Combine $\frac{- 3}{5}$ and $36$ .

$y = \frac{- 3 \cdot 36}{5} + (\frac{89}{20}) \cdot (12) + \frac{6}{5}$

Step 2.4.5.1.4

Multiply $- 3$ by $36$ .

$y = \frac{- 108}{5} + (\frac{89}{20}) \cdot (12) + \frac{6}{5}$

Step 2.4.5.1.5

Move the negative in front of the fraction.

$y = - \frac{108}{5} + (\frac{89}{20}) \cdot (12) + \frac{6}{5}$

Step 2.4.5.1.6

Cancel the common factor of $4$ .

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Step 2.4.5.1.6.1

Factor $4$ out of $20$ .

$y = - \frac{108}{5} + \frac{89}{4 (5)} \cdot 12 + \frac{6}{5}$

Step 2.4.5.1.6.2

Factor $4$ out of $12$ .

$y = - \frac{108}{5} + \frac{89}{4 \cdot 5} \cdot (4 \cdot 3) + \frac{6}{5}$

Step 2.4.5.1.6.3

Cancel the common factor.

$y = - \frac{108}{5} + \frac{89}{4 \cdot 5} \cdot (4 \cdot 3) + \frac{6}{5}$

Step 2.4.5.1.6.4

Rewrite the expression.

$y = - \frac{108}{5} + \frac{89}{5} \cdot 3 + \frac{6}{5}$

Step 2.4.5.1.7

Combine $\frac{89}{5}$ and $3$ .

$y = - \frac{108}{5} + \frac{89 \cdot 3}{5} + \frac{6}{5}$

Step 2.4.5.1.8

Multiply $89$ by $3$ .

$y = - \frac{108}{5} + \frac{267}{5} + \frac{6}{5}$

Step 2.4.5.2

Combine fractions.

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Step 2.4.5.2.1

Combine the numerators over the common denominator.

$y = \frac{- 108 + 267 + 6}{5}$

Step 2.4.5.2.2

Simplify the expression.

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Step 2.4.5.2.2.1

Add $- 108$ and $267$ .

$y = \frac{159 + 6}{5}$

Step 2.4.5.2.2.2

Add $159$ and $6$ .

$y = \frac{165}{5}$

Step 2.4.5.2.2.3

Divide $165$ by $5$ .

$y = 33$

Step 2.4.6

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = 12$ . This check passes since $y = 33$ and $y = 33$ .

$33 = 33$

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 $y = - \frac{3 x^{2}}{20} + \frac{89 x}{20} + \frac{6}{5}$ .

$y = - \frac{3 x^{2}}{20} + \frac{89 x}{20} + \frac{6}{5}$