Algebra Examples

$\begin{array}{c} x & y \\ 2 & 10 \\ 4 & 5 \\ 5 & 4 \end{array}$

Step 1

Check if the function rule is linear.

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Step 1.1

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

$y = a x + b$

Step 1.2

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

$10 = a (2) + b 5 = a (4) + b 4 = a (5) + b$

Step 1.3

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

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Step 1.3.1

Solve for $b$ in $10 = a (2) + b$ .

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Step 1.3.1.1

Rewrite the equation as $a (2) + b = 10$ .

$a (2) + b = 10$

$5 = a (4) + b$

$4 = a (5) + b$

Step 1.3.1.2

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

$2 a + b = 10$

$5 = a (4) + b$

$4 = a (5) + b$

Step 1.3.1.3

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

$b = 10 - 2 a$

$5 = a (4) + b$

$4 = a (5) + b$

$b = 10 - 2 a$

$5 = a (4) + b$

$4 = a (5) + b$

Step 1.3.2

Replace all occurrences of $b$ with $10 - 2 a$ in each equation.

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Step 1.3.2.1

Replace all occurrences of $b$ in $5 = a (4) + b$ with $10 - 2 a$ .

$5 = a (4) + 10 - 2 a$

$b = 10 - 2 a$

$4 = a (5) + b$

Step 1.3.2.2

Simplify $5 = a (4) + 10 - 2 a$ .

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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.

$5 = a (4) + 10 - 2 a$

$b = 10 - 2 a$

$4 = a (5) + b$

$5 = a (4) + 10 - 2 a$

$b = 10 - 2 a$

$4 = a (5) + b$

Step 1.3.2.2.2

Simplify the right side.

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Step 1.3.2.2.2.1

Simplify $a (4) + 10 - 2 a$ .

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Step 1.3.2.2.2.1.1

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

$5 = 4 a + 10 - 2 a$

$b = 10 - 2 a$

$4 = a (5) + b$

Step 1.3.2.2.2.1.2

Subtract $2 a$ from $4 a$ .

$5 = 2 a + 10$

$b = 10 - 2 a$

$4 = a (5) + b$

$5 = 2 a + 10$

$b = 10 - 2 a$

$4 = a (5) + b$

$5 = 2 a + 10$

$b = 10 - 2 a$

$4 = a (5) + b$

$5 = 2 a + 10$

$b = 10 - 2 a$

$4 = a (5) + b$

Step 1.3.2.3

Replace all occurrences of $b$ in $4 = a (5) + b$ with $10 - 2 a$ .

$4 = a (5) + 10 - 2 a$

$5 = 2 a + 10$

$b = 10 - 2 a$

Step 1.3.2.4

Simplify $4 = a (5) + 10 - 2 a$ .

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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.

$4 = a (5) + 10 - 2 a$

$5 = 2 a + 10$

$b = 10 - 2 a$

$4 = a (5) + 10 - 2 a$

$5 = 2 a + 10$

$b = 10 - 2 a$

Step 1.3.2.4.2

Simplify the right side.

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Step 1.3.2.4.2.1

Simplify $a (5) + 10 - 2 a$ .

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Step 1.3.2.4.2.1.1

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

$4 = 5 a + 10 - 2 a$

$5 = 2 a + 10$

$b = 10 - 2 a$

Step 1.3.2.4.2.1.2

Subtract $2 a$ from $5 a$ .

$4 = 3 a + 10$

$5 = 2 a + 10$

$b = 10 - 2 a$

$4 = 3 a + 10$

$5 = 2 a + 10$

$b = 10 - 2 a$

$4 = 3 a + 10$

$5 = 2 a + 10$

$b = 10 - 2 a$

$4 = 3 a + 10$

$5 = 2 a + 10$

$b = 10 - 2 a$

$4 = 3 a + 10$

$5 = 2 a + 10$

$b = 10 - 2 a$

Step 1.3.3

Solve for $a$ in $4 = 3 a + 10$ .

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Step 1.3.3.1

Rewrite the equation as $3 a + 10 = 4$ .

$3 a + 10 = 4$

$5 = 2 a + 10$

$b = 10 - 2 a$

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 $10$ from both sides of the equation.

$3 a = 4 - 10$

$5 = 2 a + 10$

$b = 10 - 2 a$

Step 1.3.3.2.2

Subtract $10$ from $4$ .

$3 a = - 6$

$5 = 2 a + 10$

$b = 10 - 2 a$

$3 a = - 6$

$5 = 2 a + 10$

$b = 10 - 2 a$

Step 1.3.3.3

Divide each term in $3 a = - 6$ by $3$ and simplify.

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Step 1.3.3.3.1

Divide each term in $3 a = - 6$ by $3$ .

$\frac{3 a}{3} = \frac{- 6}{3}$

$5 = 2 a + 10$

$b = 10 - 2 a$

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.

$\frac{3 a}{3} = \frac{- 6}{3}$

$5 = 2 a + 10$

$b = 10 - 2 a$

Step 1.3.3.3.2.1.2

Divide $a$ by $1$ .

$a = \frac{- 6}{3}$

$5 = 2 a + 10$

$b = 10 - 2 a$

$a = \frac{- 6}{3}$

$5 = 2 a + 10$

$b = 10 - 2 a$

$a = \frac{- 6}{3}$

$5 = 2 a + 10$

$b = 10 - 2 a$

Step 1.3.3.3.3

Simplify the right side.

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Step 1.3.3.3.3.1

Divide $- 6$ by $3$ .

$a = - 2$

$5 = 2 a + 10$

$b = 10 - 2 a$

$a = - 2$

$5 = 2 a + 10$

$b = 10 - 2 a$

$a = - 2$

$5 = 2 a + 10$

$b = 10 - 2 a$

$a = - 2$

$5 = 2 a + 10$

$b = 10 - 2 a$

Step 1.3.4

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

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Step 1.3.4.1

Replace all occurrences of $a$ in $5 = 2 a + 10$ with $- 2$ .

$5 = 2 (- 2) + 10$

$a = - 2$

$b = 10 - 2 a$

Step 1.3.4.2

Simplify the right side.

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Step 1.3.4.2.1

Simplify $2 (- 2) + 10$ .

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Step 1.3.4.2.1.1

Multiply $2$ by $- 2$ .

$5 = - 4 + 10$

$a = - 2$

$b = 10 - 2 a$

Step 1.3.4.2.1.2

Add $- 4$ and $10$ .

$5 = 6$

$a = - 2$

$b = 10 - 2 a$

$5 = 6$

$a = - 2$

$b = 10 - 2 a$

$5 = 6$

$a = - 2$

$b = 10 - 2 a$

Step 1.3.4.3

Replace all occurrences of $a$ in $b = 10 - 2 a$ with $- 2$ .

$b = 10 - 2 \cdot - 2$

$5 = 6$

$a = - 2$

Step 1.3.4.4

Simplify the right side.

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Step 1.3.4.4.1

Simplify $10 - 2 \cdot - 2$ .

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Step 1.3.4.4.1.1

Multiply $- 2$ by $- 2$ .

$b = 10 + 4$

$5 = 6$

$a = - 2$

Step 1.3.4.4.1.2

Add $10$ and $4$ .

$b = 14$

$5 = 6$

$a = - 2$

$b = 14$

$5 = 6$

$a = - 2$

$b = 14$

$5 = 6$

$a = - 2$

$b = 14$

$5 = 6$

$a = - 2$

Step 1.3.5

Since $5 = 6$ 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 $10 = a \cdot 2^{2} + b (2) + c$ .

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Step 2.3.1.1

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

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

$5 = a \cdot 4^{2} + b (4) + c$

$4 = a \cdot 5^{2} + b (5) + c$

Step 2.3.1.2

Simplify each term.

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Step 2.3.1.2.1

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

$a \cdot 4 + b (2) + c = 10$

$5 = a \cdot 4^{2} + b (4) + c$

$4 = a \cdot 5^{2} + b (5) + c$

Step 2.3.1.2.2

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

$4 \cdot a + b (2) + c = 10$

$5 = a \cdot 4^{2} + b (4) + c$

$4 = a \cdot 5^{2} + b (5) + c$

Step 2.3.1.2.3

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

$4 a + 2 b + c = 10$

$5 = a \cdot 4^{2} + b (4) + c$

$4 = a \cdot 5^{2} + b (5) + c$

$4 a + 2 b + c = 10$

$5 = a \cdot 4^{2} + b (4) + c$

$4 = a \cdot 5^{2} + b (5) + c$

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.

$2 b + c = 10 - 4 a$

$5 = a \cdot 4^{2} + b (4) + c$

$4 = a \cdot 5^{2} + b (5) + c$

Step 2.3.1.3.2

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

$c = 10 - 4 a - 2 b$

$5 = a \cdot 4^{2} + b (4) + c$

$4 = a \cdot 5^{2} + b (5) + c$

$c = 10 - 4 a - 2 b$

$5 = a \cdot 4^{2} + b (4) + c$

$4 = a \cdot 5^{2} + b (5) + c$

$c = 10 - 4 a - 2 b$

$5 = a \cdot 4^{2} + b (4) + c$

$4 = a \cdot 5^{2} + b (5) + c$

Step 2.3.2

Replace all occurrences of $c$ with $10 - 4 a - 2 b$ in each equation.

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Step 2.3.2.1

Replace all occurrences of $c$ in $5 = a \cdot 4^{2} + b (4) + c$ with $10 - 4 a - 2 b$ .

$5 = a \cdot 4^{2} + b (4) + 10 - 4 a - 2 b$

$c = 10 - 4 a - 2 b$

$4 = a \cdot 5^{2} + b (5) + c$

Step 2.3.2.2

Simplify $5 = a \cdot 4^{2} + b (4) + 10 - 4 a - 2 b$ .

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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.

$5 = a \cdot 4^{2} + b (4) + 10 - 4 a - 2 b$

$c = 10 - 4 a - 2 b$

$4 = a \cdot 5^{2} + b (5) + c$

$5 = a \cdot 4^{2} + b (4) + 10 - 4 a - 2 b$

$c = 10 - 4 a - 2 b$

$4 = a \cdot 5^{2} + b (5) + c$

Step 2.3.2.2.2

Simplify the right side.

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Step 2.3.2.2.2.1

Simplify $a \cdot 4^{2} + b (4) + 10 - 4 a - 2 b$ .

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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 $2$ .

$5 = a \cdot 16 + b (4) + 10 - 4 a - 2 b$

$c = 10 - 4 a - 2 b$

$4 = a \cdot 5^{2} + b (5) + c$

Step 2.3.2.2.2.1.1.2

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

$5 = 16 \cdot a + b (4) + 10 - 4 a - 2 b$

$c = 10 - 4 a - 2 b$

$4 = a \cdot 5^{2} + b (5) + c$

Step 2.3.2.2.2.1.1.3

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

$5 = 16 a + 4 b + 10 - 4 a - 2 b$

$c = 10 - 4 a - 2 b$

$4 = a \cdot 5^{2} + b (5) + c$

$5 = 16 a + 4 b + 10 - 4 a - 2 b$

$c = 10 - 4 a - 2 b$

$4 = a \cdot 5^{2} + b (5) + c$

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 $16 a$ .

$5 = 12 a + 4 b + 10 - 2 b$

$c = 10 - 4 a - 2 b$

$4 = a \cdot 5^{2} + b (5) + c$

Step 2.3.2.2.2.1.2.2

Subtract $2 b$ from $4 b$ .

$5 = 12 a + 2 b + 10$

$c = 10 - 4 a - 2 b$

$4 = a \cdot 5^{2} + b (5) + c$

$5 = 12 a + 2 b + 10$

$c = 10 - 4 a - 2 b$

$4 = a \cdot 5^{2} + b (5) + c$

$5 = 12 a + 2 b + 10$

$c = 10 - 4 a - 2 b$

$4 = a \cdot 5^{2} + b (5) + c$

$5 = 12 a + 2 b + 10$

$c = 10 - 4 a - 2 b$

$4 = a \cdot 5^{2} + b (5) + c$

$5 = 12 a + 2 b + 10$

$c = 10 - 4 a - 2 b$

$4 = a \cdot 5^{2} + b (5) + c$

Step 2.3.2.3

Replace all occurrences of $c$ in $4 = a \cdot 5^{2} + b (5) + c$ with $10 - 4 a - 2 b$ .

$4 = a \cdot 5^{2} + b (5) + 10 - 4 a - 2 b$

$5 = 12 a + 2 b + 10$

$c = 10 - 4 a - 2 b$

Step 2.3.2.4

Simplify $4 = a \cdot 5^{2} + b (5) + 10 - 4 a - 2 b$ .

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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.

$4 = a \cdot 5^{2} + b (5) + 10 - 4 a - 2 b$

$5 = 12 a + 2 b + 10$

$c = 10 - 4 a - 2 b$

$4 = a \cdot 5^{2} + b (5) + 10 - 4 a - 2 b$

$5 = 12 a + 2 b + 10$

$c = 10 - 4 a - 2 b$

Step 2.3.2.4.2

Simplify the right side.

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Step 2.3.2.4.2.1

Simplify $a \cdot 5^{2} + b (5) + 10 - 4 a - 2 b$ .

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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 $2$ .

$4 = a \cdot 25 + b (5) + 10 - 4 a - 2 b$

$5 = 12 a + 2 b + 10$

$c = 10 - 4 a - 2 b$

Step 2.3.2.4.2.1.1.2

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

$4 = 25 \cdot a + b (5) + 10 - 4 a - 2 b$

$5 = 12 a + 2 b + 10$

$c = 10 - 4 a - 2 b$

Step 2.3.2.4.2.1.1.3

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

$4 = 25 a + 5 b + 10 - 4 a - 2 b$

$5 = 12 a + 2 b + 10$

$c = 10 - 4 a - 2 b$

$4 = 25 a + 5 b + 10 - 4 a - 2 b$

$5 = 12 a + 2 b + 10$

$c = 10 - 4 a - 2 b$

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 $25 a$ .

$4 = 21 a + 5 b + 10 - 2 b$

$5 = 12 a + 2 b + 10$

$c = 10 - 4 a - 2 b$

Step 2.3.2.4.2.1.2.2

Subtract $2 b$ from $5 b$ .

$4 = 21 a + 3 b + 10$

$5 = 12 a + 2 b + 10$

$c = 10 - 4 a - 2 b$

$4 = 21 a + 3 b + 10$

$5 = 12 a + 2 b + 10$

$c = 10 - 4 a - 2 b$

$4 = 21 a + 3 b + 10$

$5 = 12 a + 2 b + 10$

$c = 10 - 4 a - 2 b$

$4 = 21 a + 3 b + 10$

$5 = 12 a + 2 b + 10$

$c = 10 - 4 a - 2 b$

$4 = 21 a + 3 b + 10$

$5 = 12 a + 2 b + 10$

$c = 10 - 4 a - 2 b$

$4 = 21 a + 3 b + 10$

$5 = 12 a + 2 b + 10$

$c = 10 - 4 a - 2 b$

Step 2.3.3

Solve for $a$ in $4 = 21 a + 3 b + 10$ .

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Step 2.3.3.1

Rewrite the equation as $21 a + 3 b + 10 = 4$ .

$21 a + 3 b + 10 = 4$

$5 = 12 a + 2 b + 10$

$c = 10 - 4 a - 2 b$

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.

$21 a + 10 = 4 - 3 b$

$5 = 12 a + 2 b + 10$

$c = 10 - 4 a - 2 b$

Step 2.3.3.2.2

Subtract $10$ from both sides of the equation.

$21 a = 4 - 3 b - 10$

$5 = 12 a + 2 b + 10$

$c = 10 - 4 a - 2 b$

Step 2.3.3.2.3

Subtract $10$ from $4$ .

$21 a = - 3 b - 6$

$5 = 12 a + 2 b + 10$

$c = 10 - 4 a - 2 b$

$21 a = - 3 b - 6$

$5 = 12 a + 2 b + 10$

$c = 10 - 4 a - 2 b$

Step 2.3.3.3

Divide each term in $21 a = - 3 b - 6$ by $21$ and simplify.

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Step 2.3.3.3.1

Divide each term in $21 a = - 3 b - 6$ by $21$ .

$\frac{21 a}{21} = \frac{- 3 b}{21} + \frac{- 6}{21}$

$5 = 12 a + 2 b + 10$

$c = 10 - 4 a - 2 b$

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.

$\frac{21 a}{21} = \frac{- 3 b}{21} + \frac{- 6}{21}$

$5 = 12 a + 2 b + 10$

$c = 10 - 4 a - 2 b$

Step 2.3.3.3.2.1.2

Divide $a$ by $1$ .

$a = \frac{- 3 b}{21} + \frac{- 6}{21}$

$5 = 12 a + 2 b + 10$

$c = 10 - 4 a - 2 b$

$a = \frac{- 3 b}{21} + \frac{- 6}{21}$

$5 = 12 a + 2 b + 10$

$c = 10 - 4 a - 2 b$

$a = \frac{- 3 b}{21} + \frac{- 6}{21}$

$5 = 12 a + 2 b + 10$

$c = 10 - 4 a - 2 b$

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 $- 3 b$ .

$a = \frac{3 (- b)}{21} + \frac{- 6}{21}$

$5 = 12 a + 2 b + 10$

$c = 10 - 4 a - 2 b$

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 $21$ .

$a = \frac{3 (- b)}{3 (7)} + \frac{- 6}{21}$

$5 = 12 a + 2 b + 10$

$c = 10 - 4 a - 2 b$

Step 2.3.3.3.3.1.1.2.2

Cancel the common factor.

$a = \frac{3 (- b)}{3 \cdot 7} + \frac{- 6}{21}$

$5 = 12 a + 2 b + 10$

$c = 10 - 4 a - 2 b$

Step 2.3.3.3.3.1.1.2.3

Rewrite the expression.

$a = \frac{- b}{7} + \frac{- 6}{21}$

$5 = 12 a + 2 b + 10$

$c = 10 - 4 a - 2 b$

$a = \frac{- b}{7} + \frac{- 6}{21}$

$5 = 12 a + 2 b + 10$

$c = 10 - 4 a - 2 b$

$a = \frac{- b}{7} + \frac{- 6}{21}$

$5 = 12 a + 2 b + 10$

$c = 10 - 4 a - 2 b$

Step 2.3.3.3.3.1.2

Move the negative in front of the fraction.

$a = - \frac{b}{7} + \frac{- 6}{21}$

$5 = 12 a + 2 b + 10$

$c = 10 - 4 a - 2 b$

Step 2.3.3.3.3.1.3

Cancel the common factor of $- 6$ and $21$ .

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Step 2.3.3.3.3.1.3.1

Factor $3$ out of $- 6$ .

$a = - \frac{b}{7} + \frac{3 (- 2)}{21}$

$5 = 12 a + 2 b + 10$

$c = 10 - 4 a - 2 b$

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 $3$ out of $21$ .

$a = - \frac{b}{7} + \frac{3 \cdot - 2}{3 \cdot 7}$

$5 = 12 a + 2 b + 10$

$c = 10 - 4 a - 2 b$

Step 2.3.3.3.3.1.3.2.2

Cancel the common factor.

$a = - \frac{b}{7} + \frac{3 \cdot - 2}{3 \cdot 7}$

$5 = 12 a + 2 b + 10$

$c = 10 - 4 a - 2 b$

Step 2.3.3.3.3.1.3.2.3

Rewrite the expression.

$a = - \frac{b}{7} + \frac{- 2}{7}$

$5 = 12 a + 2 b + 10$

$c = 10 - 4 a - 2 b$

$a = - \frac{b}{7} + \frac{- 2}{7}$

$5 = 12 a + 2 b + 10$

$c = 10 - 4 a - 2 b$

$a = - \frac{b}{7} + \frac{- 2}{7}$

$5 = 12 a + 2 b + 10$

$c = 10 - 4 a - 2 b$

Step 2.3.3.3.3.1.4

Move the negative in front of the fraction.

$a = - \frac{b}{7} - \frac{2}{7}$

$5 = 12 a + 2 b + 10$

$c = 10 - 4 a - 2 b$

$a = - \frac{b}{7} - \frac{2}{7}$

$5 = 12 a + 2 b + 10$

$c = 10 - 4 a - 2 b$

$a = - \frac{b}{7} - \frac{2}{7}$

$5 = 12 a + 2 b + 10$

$c = 10 - 4 a - 2 b$

$a = - \frac{b}{7} - \frac{2}{7}$

$5 = 12 a + 2 b + 10$

$c = 10 - 4 a - 2 b$

$a = - \frac{b}{7} - \frac{2}{7}$

$5 = 12 a + 2 b + 10$

$c = 10 - 4 a - 2 b$

Step 2.3.4

Replace all occurrences of $a$ with $- \frac{b}{7} - \frac{2}{7}$ in each equation.

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Step 2.3.4.1

Replace all occurrences of $a$ in $5 = 12 a + 2 b + 10$ with $- \frac{b}{7} - \frac{2}{7}$ .

$5 = 12 (- \frac{b}{7} - \frac{2}{7}) + 2 b + 10$

$a = - \frac{b}{7} - \frac{2}{7}$

$c = 10 - 4 a - 2 b$

Step 2.3.4.2

Simplify the right side.

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Step 2.3.4.2.1

Simplify $12 (- \frac{b}{7} - \frac{2}{7}) + 2 b + 10$ .

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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.

$5 = 12 (- \frac{b}{7}) + 12 (- \frac{2}{7}) + 2 b + 10$

$a = - \frac{b}{7} - \frac{2}{7}$

$c = 10 - 4 a - 2 b$

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 $12$ .

$5 = - 12 \frac{b}{7} + 12 (- \frac{2}{7}) + 2 b + 10$

$a = - \frac{b}{7} - \frac{2}{7}$

$c = 10 - 4 a - 2 b$

Step 2.3.4.2.1.1.2.2

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

$5 = \frac{- 12 b}{7} + 12 (- \frac{2}{7}) + 2 b + 10$

$a = - \frac{b}{7} - \frac{2}{7}$

$c = 10 - 4 a - 2 b$

$5 = \frac{- 12 b}{7} + 12 (- \frac{2}{7}) + 2 b + 10$

$a = - \frac{b}{7} - \frac{2}{7}$

$c = 10 - 4 a - 2 b$

Step 2.3.4.2.1.1.3

Multiply $12 (- \frac{2}{7})$ .

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Step 2.3.4.2.1.1.3.1

Multiply $- 1$ by $12$ .

$5 = \frac{- 12 b}{7} - 12 (\frac{2}{7}) + 2 b + 10$

$a = - \frac{b}{7} - \frac{2}{7}$

$c = 10 - 4 a - 2 b$

Step 2.3.4.2.1.1.3.2

Combine $- 12$ and $\frac{2}{7}$ .

$5 = \frac{- 12 b}{7} + \frac{- 12 \cdot 2}{7} + 2 b + 10$

$a = - \frac{b}{7} - \frac{2}{7}$

$c = 10 - 4 a - 2 b$

Step 2.3.4.2.1.1.3.3

Multiply $- 12$ by $2$ .

$5 = \frac{- 12 b}{7} + \frac{- 24}{7} + 2 b + 10$

$a = - \frac{b}{7} - \frac{2}{7}$

$c = 10 - 4 a - 2 b$

$5 = \frac{- 12 b}{7} + \frac{- 24}{7} + 2 b + 10$

$a = - \frac{b}{7} - \frac{2}{7}$

$c = 10 - 4 a - 2 b$

Step 2.3.4.2.1.1.4

Simplify each term.

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Step 2.3.4.2.1.1.4.1

Move the negative in front of the fraction.

$5 = - \frac{(12) b}{7} + \frac{- 24}{7} + 2 b + 10$

$a = - \frac{b}{7} - \frac{2}{7}$

$c = 10 - 4 a - 2 b$

Step 2.3.4.2.1.1.4.2

Move the negative in front of the fraction.

$5 = - \frac{12 b}{7} - \frac{24}{7} + 2 b + 10$

$a = - \frac{b}{7} - \frac{2}{7}$

$c = 10 - 4 a - 2 b$

$5 = - \frac{12 b}{7} - \frac{24}{7} + 2 b + 10$

$a = - \frac{b}{7} - \frac{2}{7}$

$c = 10 - 4 a - 2 b$

$5 = - \frac{12 b}{7} - \frac{24}{7} + 2 b + 10$

$a = - \frac{b}{7} - \frac{2}{7}$

$c = 10 - 4 a - 2 b$

Step 2.3.4.2.1.2

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

$5 = - \frac{12 b}{7} + 2 b \cdot \frac{7}{7} - \frac{24}{7} + 10$

$a = - \frac{b}{7} - \frac{2}{7}$

$c = 10 - 4 a - 2 b$

Step 2.3.4.2.1.3

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

$5 = - \frac{12 b}{7} + \frac{2 b \cdot 7}{7} - \frac{24}{7} + 10$

$a = - \frac{b}{7} - \frac{2}{7}$

$c = 10 - 4 a - 2 b$

Step 2.3.4.2.1.4

Combine the numerators over the common denominator.

$5 = \frac{- 12 b + 2 b \cdot 7}{7} - \frac{24}{7} + 10$

$a = - \frac{b}{7} - \frac{2}{7}$

$c = 10 - 4 a - 2 b$

Step 2.3.4.2.1.5

Combine the numerators over the common denominator.

$5 = 10 + \frac{- 12 b + 2 b \cdot 7 - 24}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

$c = 10 - 4 a - 2 b$

Step 2.3.4.2.1.6

Multiply $7$ by $2$ .

$5 = 10 + \frac{- 12 b + 14 b - 24}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

$c = 10 - 4 a - 2 b$

Step 2.3.4.2.1.7

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

$5 = 10 + \frac{2 b - 24}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

$c = 10 - 4 a - 2 b$

Step 2.3.4.2.1.8

Factor $2$ out of $2 b - 24$ .

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Step 2.3.4.2.1.8.1

Factor $2$ out of $2 b$ .

$5 = 10 + \frac{2 (b) - 24}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

$c = 10 - 4 a - 2 b$

Step 2.3.4.2.1.8.2

Factor $2$ out of $- 24$ .

$5 = 10 + \frac{2 b + 2 \cdot - 12}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

$c = 10 - 4 a - 2 b$

Step 2.3.4.2.1.8.3

Factor $2$ out of $2 b + 2 \cdot - 12$ .

$5 = 10 + \frac{2 (b - 12)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

$c = 10 - 4 a - 2 b$

$5 = 10 + \frac{2 (b - 12)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

$c = 10 - 4 a - 2 b$

Step 2.3.4.2.1.9

To write $10$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$5 = 10 \cdot \frac{7}{7} + \frac{2 (b - 12)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

$c = 10 - 4 a - 2 b$

Step 2.3.4.2.1.10

Simplify terms.

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Step 2.3.4.2.1.10.1

Combine $10$ and $\frac{7}{7}$ .

$5 = \frac{10 \cdot 7}{7} + \frac{2 (b - 12)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

$c = 10 - 4 a - 2 b$

Step 2.3.4.2.1.10.2

Combine the numerators over the common denominator.

$5 = \frac{10 \cdot 7 + 2 (b - 12)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

$c = 10 - 4 a - 2 b$

$5 = \frac{10 \cdot 7 + 2 (b - 12)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

$c = 10 - 4 a - 2 b$

Step 2.3.4.2.1.11

Simplify the numerator.

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Step 2.3.4.2.1.11.1

Factor $2$ out of $10 \cdot 7 + 2 (b - 12)$ .

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Step 2.3.4.2.1.11.1.1

Factor $2$ out of $10 \cdot 7$ .

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

$a = - \frac{b}{7} - \frac{2}{7}$

$c = 10 - 4 a - 2 b$

Step 2.3.4.2.1.11.1.2

Factor $2$ out of $2 (5 \cdot 7) + 2 (b - 12)$ .

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

$a = - \frac{b}{7} - \frac{2}{7}$

$c = 10 - 4 a - 2 b$

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

$a = - \frac{b}{7} - \frac{2}{7}$

$c = 10 - 4 a - 2 b$

Step 2.3.4.2.1.11.2

Multiply $5$ by $7$ .

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

$a = - \frac{b}{7} - \frac{2}{7}$

$c = 10 - 4 a - 2 b$

Step 2.3.4.2.1.11.3

Subtract $12$ from $35$ .

$5 = \frac{2 (b + 23)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

$c = 10 - 4 a - 2 b$

$5 = \frac{2 (b + 23)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

$c = 10 - 4 a - 2 b$

$5 = \frac{2 (b + 23)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

$c = 10 - 4 a - 2 b$

$5 = \frac{2 (b + 23)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

$c = 10 - 4 a - 2 b$

Step 2.3.4.3

Replace all occurrences of $a$ in $c = 10 - 4 a - 2 b$ with $- \frac{b}{7} - \frac{2}{7}$ .

$c = 10 - 4 (- \frac{b}{7} - \frac{2}{7}) - 2 b$

$5 = \frac{2 (b + 23)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.4.4

Simplify the right side.

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Step 2.3.4.4.1

Simplify $10 - 4 (- \frac{b}{7} - \frac{2}{7}) - 2 b$ .

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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.

$c = 10 - 4 (- \frac{b}{7}) - 4 (- \frac{2}{7}) - 2 b$

$5 = \frac{2 (b + 23)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

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 $- 4$ .

$c = 10 + 4 (\frac{b}{7}) - 4 (- \frac{2}{7}) - 2 b$

$5 = \frac{2 (b + 23)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.4.4.1.1.2.2

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

$c = 10 + \frac{4 b}{7} - 4 (- \frac{2}{7}) - 2 b$

$5 = \frac{2 (b + 23)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

$c = 10 + \frac{4 b}{7} - 4 (- \frac{2}{7}) - 2 b$

$5 = \frac{2 (b + 23)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.4.4.1.1.3

Multiply $- 4 (- \frac{2}{7})$ .

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Step 2.3.4.4.1.1.3.1

Multiply $- 1$ by $- 4$ .

$c = 10 + \frac{4 b}{7} + 4 (\frac{2}{7}) - 2 b$

$5 = \frac{2 (b + 23)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.4.4.1.1.3.2

Combine $4$ and $\frac{2}{7}$ .

$c = 10 + \frac{4 b}{7} + \frac{4 \cdot 2}{7} - 2 b$

$5 = \frac{2 (b + 23)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.4.4.1.1.3.3

Multiply $4$ by $2$ .

$c = 10 + \frac{4 b}{7} + \frac{8}{7} - 2 b$

$5 = \frac{2 (b + 23)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

$c = 10 + \frac{4 b}{7} + \frac{8}{7} - 2 b$

$5 = \frac{2 (b + 23)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

$c = 10 + \frac{4 b}{7} + \frac{8}{7} - 2 b$

$5 = \frac{2 (b + 23)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.4.4.1.2

To write $10$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$c = \frac{4 b}{7} + 10 \cdot \frac{7}{7} + \frac{8}{7} - 2 b$

$5 = \frac{2 (b + 23)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.4.4.1.3

Combine $10$ and $\frac{7}{7}$ .

$c = \frac{4 b}{7} + \frac{10 \cdot 7}{7} + \frac{8}{7} - 2 b$

$5 = \frac{2 (b + 23)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.4.4.1.4

Combine the numerators over the common denominator.

$c = \frac{4 b}{7} + \frac{10 \cdot 7 + 8}{7} - 2 b$

$5 = \frac{2 (b + 23)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.4.4.1.5

Simplify the numerator.

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Step 2.3.4.4.1.5.1

Multiply $10$ by $7$ .

$c = \frac{4 b}{7} + \frac{70 + 8}{7} - 2 b$

$5 = \frac{2 (b + 23)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.4.4.1.5.2

Add $70$ and $8$ .

$c = \frac{4 b}{7} + \frac{78}{7} - 2 b$

$5 = \frac{2 (b + 23)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

$c = \frac{4 b}{7} + \frac{78}{7} - 2 b$

$5 = \frac{2 (b + 23)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.4.4.1.6

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

$c = \frac{4 b}{7} - 2 b \cdot \frac{7}{7} + \frac{78}{7}$

$5 = \frac{2 (b + 23)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.4.4.1.7

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

$c = \frac{4 b}{7} + \frac{- 2 b \cdot 7}{7} + \frac{78}{7}$

$5 = \frac{2 (b + 23)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.4.4.1.8

Combine the numerators over the common denominator.

$c = \frac{4 b - 2 b \cdot 7}{7} + \frac{78}{7}$

$5 = \frac{2 (b + 23)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.4.4.1.9

Combine the numerators over the common denominator.

$c = \frac{4 b - 2 b \cdot 7 + 78}{7}$

$5 = \frac{2 (b + 23)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.4.4.1.10

Multiply $7$ by $- 2$ .

$c = \frac{4 b - 14 b + 78}{7}$

$5 = \frac{2 (b + 23)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.4.4.1.11

Subtract $14 b$ from $4 b$ .

$c = \frac{- 10 b + 78}{7}$

$5 = \frac{2 (b + 23)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.4.4.1.12

Factor $2$ out of $- 10 b + 78$ .

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Step 2.3.4.4.1.12.1

Factor $2$ out of $- 10 b$ .

$c = \frac{2 (- 5 b) + 78}{7}$

$5 = \frac{2 (b + 23)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.4.4.1.12.2

Factor $2$ out of $78$ .

$c = \frac{2 (- 5 b) + 2 (39)}{7}$

$5 = \frac{2 (b + 23)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.4.4.1.12.3

Factor $2$ out of $2 (- 5 b) + 2 (39)$ .

$c = \frac{2 (- 5 b + 39)}{7}$

$5 = \frac{2 (b + 23)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

$c = \frac{2 (- 5 b + 39)}{7}$

$5 = \frac{2 (b + 23)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.4.4.1.13

Factor $- 1$ out of $- 5 b$ .

$c = \frac{2 (- (5 b) + 39)}{7}$

$5 = \frac{2 (b + 23)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.4.4.1.14

Rewrite $39$ as $- 1 (- 39)$ .

$c = \frac{2 (- (5 b) - 1 \cdot - 39)}{7}$

$5 = \frac{2 (b + 23)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.4.4.1.15

Factor $- 1$ out of $- (5 b) - 1 (- 39)$ .

$c = \frac{2 (- (5 b - 39))}{7}$

$5 = \frac{2 (b + 23)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.4.4.1.16

Simplify the expression.

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Step 2.3.4.4.1.16.1

Rewrite $- (5 b - 39)$ as $- 1 (5 b - 39)$ .

$c = \frac{2 (- 1 (5 b - 39))}{7}$

$5 = \frac{2 (b + 23)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.4.4.1.16.2

Move the negative in front of the fraction.

$c = - \frac{2 (5 b - 39)}{7}$

$5 = \frac{2 (b + 23)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

$c = - \frac{2 (5 b - 39)}{7}$

$5 = \frac{2 (b + 23)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

$c = - \frac{2 (5 b - 39)}{7}$

$5 = \frac{2 (b + 23)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

$c = - \frac{2 (5 b - 39)}{7}$

$5 = \frac{2 (b + 23)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

$c = - \frac{2 (5 b - 39)}{7}$

$5 = \frac{2 (b + 23)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.5

Solve for $b$ in $5 = \frac{2 (b + 23)}{7}$ .

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Step 2.3.5.1

Rewrite the equation as $\frac{2 (b + 23)}{7} = 5$ .

$\frac{2 (b + 23)}{7} = 5$

$c = - \frac{2 (5 b - 39)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.5.2

Multiply both sides of the equation by $\frac{7}{2}$ .

$\frac{7}{2} \cdot \frac{2 (b + 23)}{7} = \frac{7}{2} \cdot 5$

$c = - \frac{2 (5 b - 39)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.5.3

Simplify both sides of the equation.

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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

Simplify $\frac{7}{2} \cdot \frac{2 (b + 23)}{7}$ .

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Step 2.3.5.3.1.1.1

Cancel the common factor of $7$ .

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Step 2.3.5.3.1.1.1.1

Cancel the common factor.

$\frac{7}{2} \cdot \frac{2 (b + 23)}{7} = \frac{7}{2} \cdot 5$

$c = - \frac{2 (5 b - 39)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.5.3.1.1.1.2

Rewrite the expression.

$\frac{1}{2} \cdot (2 (b + 23)) = \frac{7}{2} \cdot 5$

$c = - \frac{2 (5 b - 39)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

$\frac{1}{2} \cdot (2 (b + 23)) = \frac{7}{2} \cdot 5$

$c = - \frac{2 (5 b - 39)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.5.3.1.1.2

Cancel the common factor of $2$ .

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Step 2.3.5.3.1.1.2.1

Cancel the common factor.

$\frac{1}{2} \cdot (2 (b + 23)) = \frac{7}{2} \cdot 5$

$c = - \frac{2 (5 b - 39)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.5.3.1.1.2.2

Rewrite the expression.

$b + 23 = \frac{7}{2} \cdot 5$

$c = - \frac{2 (5 b - 39)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

$b + 23 = \frac{7}{2} \cdot 5$

$c = - \frac{2 (5 b - 39)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

$b + 23 = \frac{7}{2} \cdot 5$

$c = - \frac{2 (5 b - 39)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

$b + 23 = \frac{7}{2} \cdot 5$

$c = - \frac{2 (5 b - 39)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.5.3.2

Simplify the right side.

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Step 2.3.5.3.2.1

Multiply $\frac{7}{2} \cdot 5$ .

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Step 2.3.5.3.2.1.1

Combine $\frac{7}{2}$ and $5$ .

$b + 23 = \frac{7 \cdot 5}{2}$

$c = - \frac{2 (5 b - 39)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.5.3.2.1.2

Multiply $7$ by $5$ .

$b + 23 = \frac{35}{2}$

$c = - \frac{2 (5 b - 39)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

$b + 23 = \frac{35}{2}$

$c = - \frac{2 (5 b - 39)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

$b + 23 = \frac{35}{2}$

$c = - \frac{2 (5 b - 39)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

$b + 23 = \frac{35}{2}$

$c = - \frac{2 (5 b - 39)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.5.4

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

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Step 2.3.5.4.1

Subtract $23$ from both sides of the equation.

$b = \frac{35}{2} - 23$

$c = - \frac{2 (5 b - 39)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.5.4.2

To write $- 23$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$b = \frac{35}{2} - 23 \cdot \frac{2}{2}$

$c = - \frac{2 (5 b - 39)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.5.4.3

Combine $- 23$ and $\frac{2}{2}$ .

$b = \frac{35}{2} + \frac{- 23 \cdot 2}{2}$

$c = - \frac{2 (5 b - 39)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.5.4.4

Combine the numerators over the common denominator.

$b = \frac{35 - 23 \cdot 2}{2}$

$c = - \frac{2 (5 b - 39)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.5.4.5

Simplify the numerator.

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Step 2.3.5.4.5.1

Multiply $- 23$ by $2$ .

$b = \frac{35 - 46}{2}$

$c = - \frac{2 (5 b - 39)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.5.4.5.2

Subtract $46$ from $35$ .

$b = \frac{- 11}{2}$

$c = - \frac{2 (5 b - 39)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

$b = \frac{- 11}{2}$

$c = - \frac{2 (5 b - 39)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.5.4.6

Move the negative in front of the fraction.

$b = - \frac{11}{2}$

$c = - \frac{2 (5 b - 39)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

$b = - \frac{11}{2}$

$c = - \frac{2 (5 b - 39)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

$b = - \frac{11}{2}$

$c = - \frac{2 (5 b - 39)}{7}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.6

Replace all occurrences of $b$ with $- \frac{11}{2}$ in each equation.

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Step 2.3.6.1

Replace all occurrences of $b$ in $c = - \frac{2 (5 b - 39)}{7}$ with $- \frac{11}{2}$ .

$c = - \frac{2 (5 (- \frac{11}{2}) - 39)}{7}$

$b = - \frac{11}{2}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.6.2

Simplify the right side.

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Step 2.3.6.2.1

Simplify $- \frac{2 (5 (- \frac{11}{2}) - 39)}{7}$ .

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Step 2.3.6.2.1.1

Simplify the numerator.

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Step 2.3.6.2.1.1.1

Multiply $5 (- \frac{11}{2})$ .

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Step 2.3.6.2.1.1.1.1

Multiply $- 1$ by $5$ .

$c = - \frac{2 (- 5 (\frac{11}{2}) - 39)}{7}$

$b = - \frac{11}{2}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.6.2.1.1.1.2

Combine $- 5$ and $\frac{11}{2}$ .

$c = - \frac{2 (\frac{- 5 \cdot 11}{2} - 39)}{7}$

$b = - \frac{11}{2}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.6.2.1.1.1.3

Multiply $- 5$ by $11$ .

$c = - \frac{2 (\frac{- 55}{2} - 39)}{7}$

$b = - \frac{11}{2}$

$a = - \frac{b}{7} - \frac{2}{7}$

$c = - \frac{2 (\frac{- 55}{2} - 39)}{7}$

$b = - \frac{11}{2}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.6.2.1.1.2

Move the negative in front of the fraction.

$c = - \frac{2 (- \frac{55}{2} - 39)}{7}$

$b = - \frac{11}{2}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.6.2.1.1.3

To write $- 39$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$c = - \frac{2 (- \frac{55}{2} - 39 \cdot \frac{2}{2})}{7}$

$b = - \frac{11}{2}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.6.2.1.1.4

Combine $- 39$ and $\frac{2}{2}$ .

$c = - \frac{2 (- \frac{55}{2} + \frac{- 39 \cdot 2}{2})}{7}$

$b = - \frac{11}{2}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.6.2.1.1.5

Combine the numerators over the common denominator.

$c = - \frac{2 (\frac{- 55 - 39 \cdot 2}{2})}{7}$

$b = - \frac{11}{2}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.6.2.1.1.6

Simplify the numerator.

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Step 2.3.6.2.1.1.6.1

Multiply $- 39$ by $2$ .

$c = - \frac{2 (\frac{- 55 - 78}{2})}{7}$

$b = - \frac{11}{2}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.6.2.1.1.6.2

Subtract $78$ from $- 55$ .

$c = - \frac{2 (\frac{- 133}{2})}{7}$

$b = - \frac{11}{2}$

$a = - \frac{b}{7} - \frac{2}{7}$

$c = - \frac{2 (\frac{- 133}{2})}{7}$

$b = - \frac{11}{2}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.6.2.1.1.7

Move the negative in front of the fraction.

$c = - \frac{2 \cdot (- 1 (\frac{133}{2}))}{7}$

$b = - \frac{11}{2}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.6.2.1.1.8

Combine exponents.

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Step 2.3.6.2.1.1.8.1

Factor out negative.

$c = - \frac{- (2 (\frac{133}{2}))}{7}$

$b = - \frac{11}{2}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.6.2.1.1.8.2

Combine $2$ and $\frac{133}{2}$ .

$c = - \frac{- \frac{2 \cdot 133}{2}}{7}$

$b = - \frac{11}{2}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.6.2.1.1.8.3

Multiply $2$ by $133$ .

$c = - \frac{- \frac{266}{2}}{7}$

$b = - \frac{11}{2}$

$a = - \frac{b}{7} - \frac{2}{7}$

$c = - \frac{- \frac{266}{2}}{7}$

$b = - \frac{11}{2}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.6.2.1.1.9

Divide $266$ by $2$ .

$c = - \frac{- 1 \cdot 133}{7}$

$b = - \frac{11}{2}$

$a = - \frac{b}{7} - \frac{2}{7}$

$c = - \frac{- 1 \cdot 133}{7}$

$b = - \frac{11}{2}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.6.2.1.2

Simplify the expression.

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Step 2.3.6.2.1.2.1

Multiply $- 1$ by $133$ .

$c = - \frac{- 133}{7}$

$b = - \frac{11}{2}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.6.2.1.2.2

Divide $- 133$ by $7$ .

$c = 19$

$b = - \frac{11}{2}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.6.2.1.2.3

Multiply $- 1$ by $- 19$ .

$c = 19$

$b = - \frac{11}{2}$

$a = - \frac{b}{7} - \frac{2}{7}$

$c = 19$

$b = - \frac{11}{2}$

$a = - \frac{b}{7} - \frac{2}{7}$

$c = 19$

$b = - \frac{11}{2}$

$a = - \frac{b}{7} - \frac{2}{7}$

$c = 19$

$b = - \frac{11}{2}$

$a = - \frac{b}{7} - \frac{2}{7}$

Step 2.3.6.3

Replace all occurrences of $b$ in $a = - \frac{b}{7} - \frac{2}{7}$ with $- \frac{11}{2}$ .

$a = - \frac{- \frac{11}{2}}{7} - \frac{2}{7}$

$c = 19$

$b = - \frac{11}{2}$

Step 2.3.6.4

Simplify the right side.

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Step 2.3.6.4.1

Simplify $- \frac{- \frac{11}{2}}{7} - \frac{2}{7}$ .

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Step 2.3.6.4.1.1

Combine the numerators over the common denominator.

$a = \frac{\frac{11}{2} - 2}{7}$

$c = 19$

$b = - \frac{11}{2}$

Step 2.3.6.4.1.2

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

$a = \frac{\frac{11}{2} - 2 \cdot \frac{2}{2}}{7}$

$c = 19$

$b = - \frac{11}{2}$

Step 2.3.6.4.1.3

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

$a = \frac{\frac{11}{2} + \frac{- 2 \cdot 2}{2}}{7}$

$c = 19$

$b = - \frac{11}{2}$

Step 2.3.6.4.1.4

Combine the numerators over the common denominator.

$a = \frac{\frac{11 - 2 \cdot 2}{2}}{7}$

$c = 19$

$b = - \frac{11}{2}$

Step 2.3.6.4.1.5

Simplify the numerator.

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Step 2.3.6.4.1.5.1

Multiply $- 2$ by $2$ .

$a = \frac{\frac{11 - 4}{2}}{7}$

$c = 19$

$b = - \frac{11}{2}$

Step 2.3.6.4.1.5.2

Subtract $4$ from $11$ .

$a = \frac{\frac{7}{2}}{7}$

$c = 19$

$b = - \frac{11}{2}$

$a = \frac{\frac{7}{2}}{7}$

$c = 19$

$b = - \frac{11}{2}$

Step 2.3.6.4.1.6

Multiply the numerator by the reciprocal of the denominator.

$a = \frac{7}{2} \cdot \frac{1}{7}$

$c = 19$

$b = - \frac{11}{2}$

Step 2.3.6.4.1.7

Cancel the common factor of $7$ .

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Step 2.3.6.4.1.7.1

Cancel the common factor.

$a = \frac{7}{2} \cdot \frac{1}{7}$

$c = 19$

$b = - \frac{11}{2}$

Step 2.3.6.4.1.7.2

Rewrite the expression.

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

$c = 19$

$b = - \frac{11}{2}$

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

$c = 19$

$b = - \frac{11}{2}$

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

$c = 19$

$b = - \frac{11}{2}$

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

$c = 19$

$b = - \frac{11}{2}$

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

$c = 19$

$b = - \frac{11}{2}$

Step 2.3.7

List all of the solutions.

$a = \frac{1}{2}, c = 19, b = - \frac{11}{2}$

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 $a = \frac{1}{2}$ , $b = - \frac{11}{2}$ , $c = 19$ , 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

Cancel the common factor of $2$ .

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Step 2.4.1.1.1.1

Factor $2$ out of ${(2)}^{2}$ .

$y = \frac{1}{2} \cdot (2 \cdot 2) + (- \frac{11}{2}) \cdot (2) + 19$

Step 2.4.1.1.1.2

Cancel the common factor.

$y = \frac{1}{2} \cdot (2 \cdot 2) + (- \frac{11}{2}) \cdot (2) + 19$

Step 2.4.1.1.1.3

Rewrite the expression.

$y = 2 + (- \frac{11}{2}) \cdot (2) + 19$

Step 2.4.1.1.2

Cancel the common factor of $2$ .

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Step 2.4.1.1.2.1

Move the leading negative in $- \frac{11}{2}$ into the numerator.

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

Step 2.4.1.1.2.2

Cancel the common factor.

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

Step 2.4.1.1.2.3

Rewrite the expression.

$y = 2 - 11 + 19$

Step 2.4.1.2

Simplify by adding and subtracting.

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Step 2.4.1.2.1

Subtract $11$ from $2$ .

$y = - 9 + 19$

Step 2.4.1.2.2

Add $- 9$ and $19$ .

$y = 10$

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 $y = 10$ and $y = 10$ .

$10 = 10$

Step 2.4.3

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = \frac{1}{2}$ , $b = - \frac{11}{2}$ , $c = 19$ , 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 $2$ .

$y = \frac{1}{2} \cdot 16 + (- \frac{11}{2}) \cdot (4) + 19$

Step 2.4.3.1.2

Cancel the common factor of $2$ .

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Step 2.4.3.1.2.1

Factor $2$ out of $16$ .

$y = \frac{1}{2} \cdot (2 (8)) + (- \frac{11}{2}) \cdot (4) + 19$

Step 2.4.3.1.2.2

Cancel the common factor.

$y = \frac{1}{2} \cdot (2 \cdot 8) + (- \frac{11}{2}) \cdot (4) + 19$

Step 2.4.3.1.2.3

Rewrite the expression.

$y = 8 + (- \frac{11}{2}) \cdot (4) + 19$

Step 2.4.3.1.3

Cancel the common factor of $2$ .

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Step 2.4.3.1.3.1

Move the leading negative in $- \frac{11}{2}$ into the numerator.

$y = 8 + \frac{- 11}{2} \cdot 4 + 19$

Step 2.4.3.1.3.2

Factor $2$ out of $4$ .

$y = 8 + \frac{- 11}{2} \cdot (2 (2)) + 19$

Step 2.4.3.1.3.3

Cancel the common factor.

$y = 8 + \frac{- 11}{2} \cdot (2 \cdot 2) + 19$

Step 2.4.3.1.3.4

Rewrite the expression.

$y = 8 - 11 \cdot 2 + 19$

Step 2.4.3.1.4

Multiply $- 11$ by $2$ .

$y = 8 - 22 + 19$

Step 2.4.3.2

Simplify by adding and subtracting.

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Step 2.4.3.2.1

Subtract $22$ from $8$ .

$y = - 14 + 19$

Step 2.4.3.2.2

Add $- 14$ and $19$ .

$y = 5$

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 $y = 5$ and $y = 5$ .

$5 = 5$

Step 2.4.5

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = \frac{1}{2}$ , $b = - \frac{11}{2}$ , $c = 19$ , 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 $2$ .

$y = \frac{1}{2} \cdot 25 + (- \frac{11}{2}) \cdot (5) + 19$

Step 2.4.5.1.2

Combine $\frac{1}{2}$ and $25$ .

$y = \frac{25}{2} + (- \frac{11}{2}) \cdot (5) + 19$

Step 2.4.5.1.3

Multiply $(- \frac{11}{2}) (5)$ .

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Step 2.4.5.1.3.1

Multiply $5$ by $- 1$ .

$y = \frac{25}{2} - 5 (\frac{11}{2}) + 19$

Step 2.4.5.1.3.2

Combine $- 5$ and $\frac{11}{2}$ .

$y = \frac{25}{2} + \frac{- 5 \cdot 11}{2} + 19$

Step 2.4.5.1.3.3

Multiply $- 5$ by $11$ .

$y = \frac{25}{2} + \frac{- 55}{2} + 19$

Step 2.4.5.1.4

Move the negative in front of the fraction.

$y = \frac{25}{2} - \frac{55}{2} + 19$

Step 2.4.5.2

Combine fractions.

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Step 2.4.5.2.1

Combine the numerators over the common denominator.

$y = 19 + \frac{25 - 55}{2}$

Step 2.4.5.2.2

Simplify the expression.

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Step 2.4.5.2.2.1

Subtract $55$ from $25$ .

$y = 19 + \frac{- 30}{2}$

Step 2.4.5.2.2.2

Divide $- 30$ by $2$ .

$y = 19 - 15$

Step 2.4.5.2.2.3

Subtract $15$ from $19$ .

$y = 4$

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 $y = 4$ and $y = 4$ .

$4 = 4$

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 $y = \frac{x^{2}}{2} - \frac{11 x}{2} + 19$ .

$y = \frac{x^{2}}{2} - \frac{11 x}{2} + 19$