Algebra Examples

$\begin{array}{c} x & q (x) \\ 1 & - 1 \\ 9 & 159 \\ 2 & 5 \\ 8 & 125 \\ 3 & 15 \end{array}$

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 $q (x) = a x + b$ .

$- 1 = a (1) + b 159 = a (9) + b 5 = a (2) + b 125 = a (8) + b 15 = a (3) + b$

Step 1.3

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

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Step 1.3.1

Solve for $a$ in $- 1 = a + b$ .

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Step 1.3.1.1

Rewrite the equation as $a + b = - 1$ .

$a + b = - 1$

$159 = a (9) + b$

$5 = a (2) + b$

$125 = a (8) + b$

$15 = a (3) + b$

Step 1.3.1.2

Subtract $b$ from both sides of the equation.

$a = - 1 - b$

$159 = a (9) + b$

$5 = a (2) + b$

$125 = a (8) + b$

$15 = a (3) + b$

$a = - 1 - b$

$159 = a (9) + b$

$5 = a (2) + b$

$125 = a (8) + b$

$15 = a (3) + b$

Step 1.3.2

Replace all occurrences of $a$ with $- 1 - b$ in each equation.

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Step 1.3.2.1

Replace all occurrences of $a$ in $159 = a (9) + b$ with $- 1 - b$ .

$159 = (- 1 - b) (9) + b$

$a = - 1 - b$

$5 = a (2) + b$

$125 = a (8) + b$

$15 = a (3) + b$

Step 1.3.2.2

Simplify the right side.

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Step 1.3.2.2.1

Simplify $(- 1 - b) (9) + b$ .

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Step 1.3.2.2.1.1

Simplify each term.

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Step 1.3.2.2.1.1.1

Apply the distributive property.

$159 = - 1 \cdot 9 - b \cdot 9 + b$

$a = - 1 - b$

$5 = a (2) + b$

$125 = a (8) + b$

$15 = a (3) + b$

Step 1.3.2.2.1.1.2

Multiply $- 1$ by $9$ .

$159 = - 9 - b \cdot 9 + b$

$a = - 1 - b$

$5 = a (2) + b$

$125 = a (8) + b$

$15 = a (3) + b$

Step 1.3.2.2.1.1.3

Multiply $9$ by $- 1$ .

$159 = - 9 - 9 b + b$

$a = - 1 - b$

$5 = a (2) + b$

$125 = a (8) + b$

$15 = a (3) + b$

$159 = - 9 - 9 b + b$

$a = - 1 - b$

$5 = a (2) + b$

$125 = a (8) + b$

$15 = a (3) + b$

Step 1.3.2.2.1.2

Add $- 9 b$ and $b$ .

$159 = - 9 - 8 b$

$a = - 1 - b$

$5 = a (2) + b$

$125 = a (8) + b$

$15 = a (3) + b$

$159 = - 9 - 8 b$

$a = - 1 - b$

$5 = a (2) + b$

$125 = a (8) + b$

$15 = a (3) + b$

$159 = - 9 - 8 b$

$a = - 1 - b$

$5 = a (2) + b$

$125 = a (8) + b$

$15 = a (3) + b$

Step 1.3.2.3

Replace all occurrences of $a$ in $5 = a (2) + b$ with $- 1 - b$ .

$5 = (- 1 - b) (2) + b$

$159 = - 9 - 8 b$

$a = - 1 - b$

$125 = a (8) + b$

$15 = a (3) + b$

Step 1.3.2.4

Simplify the right side.

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Step 1.3.2.4.1

Simplify $(- 1 - b) (2) + b$ .

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Step 1.3.2.4.1.1

Simplify each term.

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Step 1.3.2.4.1.1.1

Apply the distributive property.

$5 = - 1 \cdot 2 - b \cdot 2 + b$

$159 = - 9 - 8 b$

$a = - 1 - b$

$125 = a (8) + b$

$15 = a (3) + b$

Step 1.3.2.4.1.1.2

Multiply $- 1$ by $2$ .

$5 = - 2 - b \cdot 2 + b$

$159 = - 9 - 8 b$

$a = - 1 - b$

$125 = a (8) + b$

$15 = a (3) + b$

Step 1.3.2.4.1.1.3

Multiply $2$ by $- 1$ .

$5 = - 2 - 2 b + b$

$159 = - 9 - 8 b$

$a = - 1 - b$

$125 = a (8) + b$

$15 = a (3) + b$

$5 = - 2 - 2 b + b$

$159 = - 9 - 8 b$

$a = - 1 - b$

$125 = a (8) + b$

$15 = a (3) + b$

Step 1.3.2.4.1.2

Add $- 2 b$ and $b$ .

$5 = - 2 - b$

$159 = - 9 - 8 b$

$a = - 1 - b$

$125 = a (8) + b$

$15 = a (3) + b$

$5 = - 2 - b$

$159 = - 9 - 8 b$

$a = - 1 - b$

$125 = a (8) + b$

$15 = a (3) + b$

$5 = - 2 - b$

$159 = - 9 - 8 b$

$a = - 1 - b$

$125 = a (8) + b$

$15 = a (3) + b$

Step 1.3.2.5

Replace all occurrences of $a$ in $125 = a (8) + b$ with $- 1 - b$ .

$125 = (- 1 - b) (8) + b$

$5 = - 2 - b$

$159 = - 9 - 8 b$

$a = - 1 - b$

$15 = a (3) + b$

Step 1.3.2.6

Simplify the right side.

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Step 1.3.2.6.1

Simplify $(- 1 - b) (8) + b$ .

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Step 1.3.2.6.1.1

Simplify each term.

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Step 1.3.2.6.1.1.1

Apply the distributive property.

$125 = - 1 \cdot 8 - b \cdot 8 + b$

$5 = - 2 - b$

$159 = - 9 - 8 b$

$a = - 1 - b$

$15 = a (3) + b$

Step 1.3.2.6.1.1.2

Multiply $- 1$ by $8$ .

$125 = - 8 - b \cdot 8 + b$

$5 = - 2 - b$

$159 = - 9 - 8 b$

$a = - 1 - b$

$15 = a (3) + b$

Step 1.3.2.6.1.1.3

Multiply $8$ by $- 1$ .

$125 = - 8 - 8 b + b$

$5 = - 2 - b$

$159 = - 9 - 8 b$

$a = - 1 - b$

$15 = a (3) + b$

$125 = - 8 - 8 b + b$

$5 = - 2 - b$

$159 = - 9 - 8 b$

$a = - 1 - b$

$15 = a (3) + b$

Step 1.3.2.6.1.2

Add $- 8 b$ and $b$ .

$125 = - 8 - 7 b$

$5 = - 2 - b$

$159 = - 9 - 8 b$

$a = - 1 - b$

$15 = a (3) + b$

$125 = - 8 - 7 b$

$5 = - 2 - b$

$159 = - 9 - 8 b$

$a = - 1 - b$

$15 = a (3) + b$

$125 = - 8 - 7 b$

$5 = - 2 - b$

$159 = - 9 - 8 b$

$a = - 1 - b$

$15 = a (3) + b$

Step 1.3.2.7

Replace all occurrences of $a$ in $15 = a (3) + b$ with $- 1 - b$ .

$15 = (- 1 - b) (3) + b$

$125 = - 8 - 7 b$

$5 = - 2 - b$

$159 = - 9 - 8 b$

$a = - 1 - b$

Step 1.3.2.8

Simplify the right side.

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Step 1.3.2.8.1

Simplify $(- 1 - b) (3) + b$ .

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Step 1.3.2.8.1.1

Simplify each term.

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Step 1.3.2.8.1.1.1

Apply the distributive property.

$15 = - 1 \cdot 3 - b \cdot 3 + b$

$125 = - 8 - 7 b$

$5 = - 2 - b$

$159 = - 9 - 8 b$

$a = - 1 - b$

Step 1.3.2.8.1.1.2

Multiply $- 1$ by $3$ .

$15 = - 3 - b \cdot 3 + b$

$125 = - 8 - 7 b$

$5 = - 2 - b$

$159 = - 9 - 8 b$

$a = - 1 - b$

Step 1.3.2.8.1.1.3

Multiply $3$ by $- 1$ .

$15 = - 3 - 3 b + b$

$125 = - 8 - 7 b$

$5 = - 2 - b$

$159 = - 9 - 8 b$

$a = - 1 - b$

$15 = - 3 - 3 b + b$

$125 = - 8 - 7 b$

$5 = - 2 - b$

$159 = - 9 - 8 b$

$a = - 1 - b$

Step 1.3.2.8.1.2

Add $- 3 b$ and $b$ .

$15 = - 3 - 2 b$

$125 = - 8 - 7 b$

$5 = - 2 - b$

$159 = - 9 - 8 b$

$a = - 1 - b$

$15 = - 3 - 2 b$

$125 = - 8 - 7 b$

$5 = - 2 - b$

$159 = - 9 - 8 b$

$a = - 1 - b$

$15 = - 3 - 2 b$

$125 = - 8 - 7 b$

$5 = - 2 - b$

$159 = - 9 - 8 b$

$a = - 1 - b$

$15 = - 3 - 2 b$

$125 = - 8 - 7 b$

$5 = - 2 - b$

$159 = - 9 - 8 b$

$a = - 1 - b$

Step 1.3.3

Solve for $b$ in $15 = - 3 - 2 b$ .

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Step 1.3.3.1

Rewrite the equation as $- 3 - 2 b = 15$ .

$- 3 - 2 b = 15$

$125 = - 8 - 7 b$

$5 = - 2 - b$

$159 = - 9 - 8 b$

$a = - 1 - b$

Step 1.3.3.2

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

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Step 1.3.3.2.1

Add $3$ to both sides of the equation.

$- 2 b = 15 + 3$

$125 = - 8 - 7 b$

$5 = - 2 - b$

$159 = - 9 - 8 b$

$a = - 1 - b$

Step 1.3.3.2.2

Add $15$ and $3$ .

$- 2 b = 18$

$125 = - 8 - 7 b$

$5 = - 2 - b$

$159 = - 9 - 8 b$

$a = - 1 - b$

$- 2 b = 18$

$125 = - 8 - 7 b$

$5 = - 2 - b$

$159 = - 9 - 8 b$

$a = - 1 - b$

Step 1.3.3.3

Divide each term in $- 2 b = 18$ by $- 2$ and simplify.

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Step 1.3.3.3.1

Divide each term in $- 2 b = 18$ by $- 2$ .

$\frac{- 2 b}{- 2} = \frac{18}{- 2}$

$125 = - 8 - 7 b$

$5 = - 2 - b$

$159 = - 9 - 8 b$

$a = - 1 - b$

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 $- 2$ .

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Step 1.3.3.3.2.1.1

Cancel the common factor.

$\frac{- 2 b}{- 2} = \frac{18}{- 2}$

$125 = - 8 - 7 b$

$5 = - 2 - b$

$159 = - 9 - 8 b$

$a = - 1 - b$

Step 1.3.3.3.2.1.2

Divide $b$ by $1$ .

$b = \frac{18}{- 2}$

$125 = - 8 - 7 b$

$5 = - 2 - b$

$159 = - 9 - 8 b$

$a = - 1 - b$

$b = \frac{18}{- 2}$

$125 = - 8 - 7 b$

$5 = - 2 - b$

$159 = - 9 - 8 b$

$a = - 1 - b$

$b = \frac{18}{- 2}$

$125 = - 8 - 7 b$

$5 = - 2 - b$

$159 = - 9 - 8 b$

$a = - 1 - b$

Step 1.3.3.3.3

Simplify the right side.

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Step 1.3.3.3.3.1

Divide $18$ by $- 2$ .

$b = - 9$

$125 = - 8 - 7 b$

$5 = - 2 - b$

$159 = - 9 - 8 b$

$a = - 1 - b$

$b = - 9$

$125 = - 8 - 7 b$

$5 = - 2 - b$

$159 = - 9 - 8 b$

$a = - 1 - b$

$b = - 9$

$125 = - 8 - 7 b$

$5 = - 2 - b$

$159 = - 9 - 8 b$

$a = - 1 - b$

$b = - 9$

$125 = - 8 - 7 b$

$5 = - 2 - b$

$159 = - 9 - 8 b$

$a = - 1 - b$

Step 1.3.4

Replace all occurrences of $b$ with $- 9$ in each equation.

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Step 1.3.4.1

Replace all occurrences of $b$ in $125 = - 8 - 7 b$ with $- 9$ .

$125 = - 8 - 7 \cdot - 9$

$b = - 9$

$5 = - 2 - b$

$159 = - 9 - 8 b$

$a = - 1 - b$

Step 1.3.4.2

Simplify the right side.

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Step 1.3.4.2.1

Simplify $- 8 - 7 \cdot - 9$ .

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Step 1.3.4.2.1.1

Multiply $- 7$ by $- 9$ .

$125 = - 8 + 63$

$b = - 9$

$5 = - 2 - b$

$159 = - 9 - 8 b$

$a = - 1 - b$

Step 1.3.4.2.1.2

Add $- 8$ and $63$ .

$125 = 55$

$b = - 9$

$5 = - 2 - b$

$159 = - 9 - 8 b$

$a = - 1 - b$

$125 = 55$

$b = - 9$

$5 = - 2 - b$

$159 = - 9 - 8 b$

$a = - 1 - b$

$125 = 55$

$b = - 9$

$5 = - 2 - b$

$159 = - 9 - 8 b$

$a = - 1 - b$

Step 1.3.4.3

Replace all occurrences of $b$ in $5 = - 2 - b$ with $- 9$ .

$5 = - 2 - (- 9)$

$125 = 55$

$b = - 9$

$159 = - 9 - 8 b$

$a = - 1 - b$

Step 1.3.4.4

Simplify the right side.

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Step 1.3.4.4.1

Simplify $- 2 - (- 9)$ .

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Step 1.3.4.4.1.1

Multiply $- 1$ by $- 9$ .

$5 = - 2 + 9$

$125 = 55$

$b = - 9$

$159 = - 9 - 8 b$

$a = - 1 - b$

Step 1.3.4.4.1.2

Add $- 2$ and $9$ .

$5 = 7$

$125 = 55$

$b = - 9$

$159 = - 9 - 8 b$

$a = - 1 - b$

$5 = 7$

$125 = 55$

$b = - 9$

$159 = - 9 - 8 b$

$a = - 1 - b$

$5 = 7$

$125 = 55$

$b = - 9$

$159 = - 9 - 8 b$

$a = - 1 - b$

Step 1.3.4.5

Replace all occurrences of $b$ in $159 = - 9 - 8 b$ with $- 9$ .

$159 = - 9 - 8 \cdot - 9$

$5 = 7$

$125 = 55$

$b = - 9$

$a = - 1 - b$

Step 1.3.4.6

Simplify the right side.

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Step 1.3.4.6.1

Simplify $- 9 - 8 \cdot - 9$ .

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Step 1.3.4.6.1.1

Multiply $- 8$ by $- 9$ .

$159 = - 9 + 72$

$5 = 7$

$125 = 55$

$b = - 9$

$a = - 1 - b$

Step 1.3.4.6.1.2

Add $- 9$ and $72$ .

$159 = 63$

$5 = 7$

$125 = 55$

$b = - 9$

$a = - 1 - b$

$159 = 63$

$5 = 7$

$125 = 55$

$b = - 9$

$a = - 1 - b$

$159 = 63$

$5 = 7$

$125 = 55$

$b = - 9$

$a = - 1 - b$

Step 1.3.4.7

Replace all occurrences of $b$ in $a = - 1 - b$ with $- 9$ .

$a = - 1 - (- 9)$

$159 = 63$

$5 = 7$

$125 = 55$

$b = - 9$

Step 1.3.4.8

Simplify the right side.

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Step 1.3.4.8.1

Simplify $- 1 - (- 9)$ .

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Step 1.3.4.8.1.1

Multiply $- 1$ by $- 9$ .

$a = - 1 + 9$

$159 = 63$

$5 = 7$

$125 = 55$

$b = - 9$

Step 1.3.4.8.1.2

Add $- 1$ and $9$ .

$a = 8$

$159 = 63$

$5 = 7$

$125 = 55$

$b = - 9$

$a = 8$

$159 = 63$

$5 = 7$

$125 = 55$

$b = - 9$

$a = 8$

$159 = 63$

$5 = 7$

$125 = 55$

$b = - 9$

$a = 8$

$159 = 63$

$5 = 7$

$125 = 55$

$b = - 9$

Step 1.3.5

Since $159 = 63$ is not true, there is no solution.

No solution

Step 1.4

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

The function is not linear

Step 2

Check if the function rule is quadratic.

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Step 2.1

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

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

Step 2.2

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

Step 2.3

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

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Step 2.3.1

Solve for $a$ in $- 1 = a + b + c$ .

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Step 2.3.1.1

Rewrite the equation as $a + b + c = - 1$ .

$a + b + c = - 1$

$159 = a \cdot 9^{2} + b (9) + c$

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

$125 = a \cdot 8^{2} + b (8) + c$

$15 = a \cdot 3^{2} + b (3) + c$

Step 2.3.1.2

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

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Step 2.3.1.2.1

Subtract $b$ from both sides of the equation.

$a + c = - 1 - b$

$159 = a \cdot 9^{2} + b (9) + c$

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

$125 = a \cdot 8^{2} + b (8) + c$

$15 = a \cdot 3^{2} + b (3) + c$

Step 2.3.1.2.2

Subtract $c$ from both sides of the equation.

$a = - 1 - b - c$

$159 = a \cdot 9^{2} + b (9) + c$

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

$125 = a \cdot 8^{2} + b (8) + c$

$15 = a \cdot 3^{2} + b (3) + c$

$a = - 1 - b - c$

$159 = a \cdot 9^{2} + b (9) + c$

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

$125 = a \cdot 8^{2} + b (8) + c$

$15 = a \cdot 3^{2} + b (3) + c$

$a = - 1 - b - c$

$159 = a \cdot 9^{2} + b (9) + c$

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

$125 = a \cdot 8^{2} + b (8) + c$

$15 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2

Replace all occurrences of $a$ with $- 1 - b - c$ in each equation.

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Step 2.3.2.1

Replace all occurrences of $a$ in $159 = a \cdot 9^{2} + b (9) + c$ with $- 1 - b - c$ .

$159 = (- 1 - b - c) \cdot 9^{2} + b (9) + c$

$a = - 1 - b - c$

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

$125 = a \cdot 8^{2} + b (8) + c$

$15 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.2

Simplify the right side.

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Step 2.3.2.2.1

Simplify $(- 1 - b - c) \cdot 9^{2} + b (9) + c$ .

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Step 2.3.2.2.1.1

Simplify each term.

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Step 2.3.2.2.1.1.1

Raise $9$ to the power of $2$ .

$159 = (- 1 - b - c) \cdot 81 + b (9) + c$

$a = - 1 - b - c$

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

$125 = a \cdot 8^{2} + b (8) + c$

$15 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.2.1.1.2

Apply the distributive property.

$159 = - 1 \cdot 81 - b \cdot 81 - c \cdot 81 + b (9) + c$

$a = - 1 - b - c$

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

$125 = a \cdot 8^{2} + b (8) + c$

$15 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.2.1.1.3

Simplify.

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Step 2.3.2.2.1.1.3.1

Multiply $- 1$ by $81$ .

$159 = - 81 - b \cdot 81 - c \cdot 81 + b (9) + c$

$a = - 1 - b - c$

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

$125 = a \cdot 8^{2} + b (8) + c$

$15 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.2.1.1.3.2

Multiply $81$ by $- 1$ .

$159 = - 81 - 81 b - c \cdot 81 + b (9) + c$

$a = - 1 - b - c$

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

$125 = a \cdot 8^{2} + b (8) + c$

$15 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.2.1.1.3.3

Multiply $81$ by $- 1$ .

$159 = - 81 - 81 b - 81 c + b (9) + c$

$a = - 1 - b - c$

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

$125 = a \cdot 8^{2} + b (8) + c$

$15 = a \cdot 3^{2} + b (3) + c$

$159 = - 81 - 81 b - 81 c + b (9) + c$

$a = - 1 - b - c$

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

$125 = a \cdot 8^{2} + b (8) + c$

$15 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.2.1.1.4

Move $9$ to the left of $b$ .

$159 = - 81 - 81 b - 81 c + 9 b + c$

$a = - 1 - b - c$

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

$125 = a \cdot 8^{2} + b (8) + c$

$15 = a \cdot 3^{2} + b (3) + c$

$159 = - 81 - 81 b - 81 c + 9 b + c$

$a = - 1 - b - c$

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

$125 = a \cdot 8^{2} + b (8) + c$

$15 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.2.1.2

Simplify by adding terms.

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Step 2.3.2.2.1.2.1

Add $- 81 b$ and $9 b$ .

$159 = - 81 - 72 b - 81 c + c$

$a = - 1 - b - c$

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

$125 = a \cdot 8^{2} + b (8) + c$

$15 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.2.1.2.2

Add $- 81 c$ and $c$ .

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

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

$125 = a \cdot 8^{2} + b (8) + c$

$15 = a \cdot 3^{2} + b (3) + c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

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

$125 = a \cdot 8^{2} + b (8) + c$

$15 = a \cdot 3^{2} + b (3) + c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

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

$125 = a \cdot 8^{2} + b (8) + c$

$15 = a \cdot 3^{2} + b (3) + c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

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

$125 = a \cdot 8^{2} + b (8) + c$

$15 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.3

Replace all occurrences of $a$ in $5 = a \cdot 2^{2} + b (2) + c$ with $- 1 - b - c$ .

$5 = (- 1 - b - c) \cdot 2^{2} + b (2) + c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$125 = a \cdot 8^{2} + b (8) + c$

$15 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.4

Simplify the right side.

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Step 2.3.2.4.1

Simplify $(- 1 - b - c) \cdot 2^{2} + b (2) + c$ .

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Step 2.3.2.4.1.1

Simplify each term.

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Step 2.3.2.4.1.1.1

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

$5 = (- 1 - b - c) \cdot 4 + b (2) + c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$125 = a \cdot 8^{2} + b (8) + c$

$15 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.4.1.1.2

Apply the distributive property.

$5 = - 1 \cdot 4 - b \cdot 4 - c \cdot 4 + b (2) + c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$125 = a \cdot 8^{2} + b (8) + c$

$15 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.4.1.1.3

Simplify.

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Step 2.3.2.4.1.1.3.1

Multiply $- 1$ by $4$ .

$5 = - 4 - b \cdot 4 - c \cdot 4 + b (2) + c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$125 = a \cdot 8^{2} + b (8) + c$

$15 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.4.1.1.3.2

Multiply $4$ by $- 1$ .

$5 = - 4 - 4 b - c \cdot 4 + b (2) + c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$125 = a \cdot 8^{2} + b (8) + c$

$15 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.4.1.1.3.3

Multiply $4$ by $- 1$ .

$5 = - 4 - 4 b - 4 c + b (2) + c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$125 = a \cdot 8^{2} + b (8) + c$

$15 = a \cdot 3^{2} + b (3) + c$

$5 = - 4 - 4 b - 4 c + b (2) + c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$125 = a \cdot 8^{2} + b (8) + c$

$15 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.4.1.1.4

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

$5 = - 4 - 4 b - 4 c + 2 b + c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$125 = a \cdot 8^{2} + b (8) + c$

$15 = a \cdot 3^{2} + b (3) + c$

$5 = - 4 - 4 b - 4 c + 2 b + c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$125 = a \cdot 8^{2} + b (8) + c$

$15 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.4.1.2

Simplify by adding terms.

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Step 2.3.2.4.1.2.1

Add $- 4 b$ and $2 b$ .

$5 = - 4 - 2 b - 4 c + c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$125 = a \cdot 8^{2} + b (8) + c$

$15 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.4.1.2.2

Add $- 4 c$ and $c$ .

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$125 = a \cdot 8^{2} + b (8) + c$

$15 = a \cdot 3^{2} + b (3) + c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$125 = a \cdot 8^{2} + b (8) + c$

$15 = a \cdot 3^{2} + b (3) + c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$125 = a \cdot 8^{2} + b (8) + c$

$15 = a \cdot 3^{2} + b (3) + c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$125 = a \cdot 8^{2} + b (8) + c$

$15 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.5

Replace all occurrences of $a$ in $125 = a \cdot 8^{2} + b (8) + c$ with $- 1 - b - c$ .

$125 = (- 1 - b - c) \cdot 8^{2} + b (8) + c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$15 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.6

Simplify the right side.

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Step 2.3.2.6.1

Simplify $(- 1 - b - c) \cdot 8^{2} + b (8) + c$ .

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Step 2.3.2.6.1.1

Simplify each term.

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Step 2.3.2.6.1.1.1

Raise $8$ to the power of $2$ .

$125 = (- 1 - b - c) \cdot 64 + b (8) + c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$15 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.6.1.1.2

Apply the distributive property.

$125 = - 1 \cdot 64 - b \cdot 64 - c \cdot 64 + b (8) + c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$15 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.6.1.1.3

Simplify.

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Step 2.3.2.6.1.1.3.1

Multiply $- 1$ by $64$ .

$125 = - 64 - b \cdot 64 - c \cdot 64 + b (8) + c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$15 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.6.1.1.3.2

Multiply $64$ by $- 1$ .

$125 = - 64 - 64 b - c \cdot 64 + b (8) + c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$15 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.6.1.1.3.3

Multiply $64$ by $- 1$ .

$125 = - 64 - 64 b - 64 c + b (8) + c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$15 = a \cdot 3^{2} + b (3) + c$

$125 = - 64 - 64 b - 64 c + b (8) + c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$15 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.6.1.1.4

Move $8$ to the left of $b$ .

$125 = - 64 - 64 b - 64 c + 8 b + c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$15 = a \cdot 3^{2} + b (3) + c$

$125 = - 64 - 64 b - 64 c + 8 b + c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$15 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.6.1.2

Simplify by adding terms.

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Step 2.3.2.6.1.2.1

Add $- 64 b$ and $8 b$ .

$125 = - 64 - 56 b - 64 c + c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$15 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.6.1.2.2

Add $- 64 c$ and $c$ .

$125 = - 64 - 56 b - 63 c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$15 = a \cdot 3^{2} + b (3) + c$

$125 = - 64 - 56 b - 63 c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$15 = a \cdot 3^{2} + b (3) + c$

$125 = - 64 - 56 b - 63 c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$15 = a \cdot 3^{2} + b (3) + c$

$125 = - 64 - 56 b - 63 c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$15 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.7

Replace all occurrences of $a$ in $15 = a \cdot 3^{2} + b (3) + c$ with $- 1 - b - c$ .

$15 = (- 1 - b - c) \cdot 3^{2} + b (3) + c$

$125 = - 64 - 56 b - 63 c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.2.8

Simplify the right side.

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Step 2.3.2.8.1

Simplify $(- 1 - b - c) \cdot 3^{2} + b (3) + c$ .

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Step 2.3.2.8.1.1

Simplify each term.

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Step 2.3.2.8.1.1.1

Raise $3$ to the power of $2$ .

$15 = (- 1 - b - c) \cdot 9 + b (3) + c$

$125 = - 64 - 56 b - 63 c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.2.8.1.1.2

Apply the distributive property.

$15 = - 1 \cdot 9 - b \cdot 9 - c \cdot 9 + b (3) + c$

$125 = - 64 - 56 b - 63 c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.2.8.1.1.3

Simplify.

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Step 2.3.2.8.1.1.3.1

Multiply $- 1$ by $9$ .

$15 = - 9 - b \cdot 9 - c \cdot 9 + b (3) + c$

$125 = - 64 - 56 b - 63 c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.2.8.1.1.3.2

Multiply $9$ by $- 1$ .

$15 = - 9 - 9 b - c \cdot 9 + b (3) + c$

$125 = - 64 - 56 b - 63 c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.2.8.1.1.3.3

Multiply $9$ by $- 1$ .

$15 = - 9 - 9 b - 9 c + b (3) + c$

$125 = - 64 - 56 b - 63 c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$15 = - 9 - 9 b - 9 c + b (3) + c$

$125 = - 64 - 56 b - 63 c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.2.8.1.1.4

Move $3$ to the left of $b$ .

$15 = - 9 - 9 b - 9 c + 3 b + c$

$125 = - 64 - 56 b - 63 c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$15 = - 9 - 9 b - 9 c + 3 b + c$

$125 = - 64 - 56 b - 63 c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.2.8.1.2

Simplify by adding terms.

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Step 2.3.2.8.1.2.1

Add $- 9 b$ and $3 b$ .

$15 = - 9 - 6 b - 9 c + c$

$125 = - 64 - 56 b - 63 c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.2.8.1.2.2

Add $- 9 c$ and $c$ .

$15 = - 9 - 6 b - 8 c$

$125 = - 64 - 56 b - 63 c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$15 = - 9 - 6 b - 8 c$

$125 = - 64 - 56 b - 63 c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$15 = - 9 - 6 b - 8 c$

$125 = - 64 - 56 b - 63 c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$15 = - 9 - 6 b - 8 c$

$125 = - 64 - 56 b - 63 c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$15 = - 9 - 6 b - 8 c$

$125 = - 64 - 56 b - 63 c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.3

Solve for $b$ in $15 = - 9 - 6 b - 8 c$ .

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Step 2.3.3.1

Rewrite the equation as $- 9 - 6 b - 8 c = 15$ .

$- 9 - 6 b - 8 c = 15$

$125 = - 64 - 56 b - 63 c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.3.2

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

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Step 2.3.3.2.1

Add $9$ to both sides of the equation.

$- 6 b - 8 c = 15 + 9$

$125 = - 64 - 56 b - 63 c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.3.2.2

Add $8 c$ to both sides of the equation.

$- 6 b = 15 + 9 + 8 c$

$125 = - 64 - 56 b - 63 c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.3.2.3

Add $15$ and $9$ .

$- 6 b = 24 + 8 c$

$125 = - 64 - 56 b - 63 c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$- 6 b = 24 + 8 c$

$125 = - 64 - 56 b - 63 c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.3.3

Divide each term in $- 6 b = 24 + 8 c$ by $- 6$ and simplify.

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Step 2.3.3.3.1

Divide each term in $- 6 b = 24 + 8 c$ by $- 6$ .

$\frac{- 6 b}{- 6} = \frac{24}{- 6} + \frac{8 c}{- 6}$

$125 = - 64 - 56 b - 63 c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

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 $- 6$ .

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Step 2.3.3.3.2.1.1

Cancel the common factor.

$\frac{- 6 b}{- 6} = \frac{24}{- 6} + \frac{8 c}{- 6}$

$125 = - 64 - 56 b - 63 c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.3.3.2.1.2

Divide $b$ by $1$ .

$b = \frac{24}{- 6} + \frac{8 c}{- 6}$

$125 = - 64 - 56 b - 63 c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$b = \frac{24}{- 6} + \frac{8 c}{- 6}$

$125 = - 64 - 56 b - 63 c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$b = \frac{24}{- 6} + \frac{8 c}{- 6}$

$125 = - 64 - 56 b - 63 c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

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

Divide $24$ by $- 6$ .

$b = - 4 + \frac{8 c}{- 6}$

$125 = - 64 - 56 b - 63 c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.3.3.3.1.2

Cancel the common factor of $8$ and $- 6$ .

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Step 2.3.3.3.3.1.2.1

Factor $2$ out of $8 c$ .

$b = - 4 + \frac{2 (4 c)}{- 6}$

$125 = - 64 - 56 b - 63 c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.3.3.3.1.2.2

Cancel the common factors.

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Step 2.3.3.3.3.1.2.2.1

Factor $2$ out of $- 6$ .

$b = - 4 + \frac{2 (4 c)}{2 (- 3)}$

$125 = - 64 - 56 b - 63 c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.3.3.3.1.2.2.2

Cancel the common factor.

$b = - 4 + \frac{2 (4 c)}{2 \cdot - 3}$

$125 = - 64 - 56 b - 63 c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.3.3.3.1.2.2.3

Rewrite the expression.

$b = - 4 + \frac{4 c}{- 3}$

$125 = - 64 - 56 b - 63 c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$b = - 4 + \frac{4 c}{- 3}$

$125 = - 64 - 56 b - 63 c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$b = - 4 + \frac{4 c}{- 3}$

$125 = - 64 - 56 b - 63 c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.3.3.3.1.3

Move the negative in front of the fraction.

$b = - 4 - \frac{4 c}{3}$

$125 = - 64 - 56 b - 63 c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$b = - 4 - \frac{4 c}{3}$

$125 = - 64 - 56 b - 63 c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$b = - 4 - \frac{4 c}{3}$

$125 = - 64 - 56 b - 63 c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$b = - 4 - \frac{4 c}{3}$

$125 = - 64 - 56 b - 63 c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$b = - 4 - \frac{4 c}{3}$

$125 = - 64 - 56 b - 63 c$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.4

Replace all occurrences of $b$ with $- 4 - \frac{4 c}{3}$ in each equation.

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Step 2.3.4.1

Replace all occurrences of $b$ in $125 = - 64 - 56 b - 63 c$ with $- 4 - \frac{4 c}{3}$ .

$125 = - 64 - 56 (- 4 - \frac{4 c}{3}) - 63 c$

$b = - 4 - \frac{4 c}{3}$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.4.2

Simplify the right side.

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Step 2.3.4.2.1

Simplify $- 64 - 56 (- 4 - \frac{4 c}{3}) - 63 c$ .

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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.

$125 = - 64 - 56 \cdot - 4 - 56 (- \frac{4 c}{3}) - 63 c$

$b = - 4 - \frac{4 c}{3}$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.4.2.1.1.2

Multiply $- 56$ by $- 4$ .

$125 = - 64 + 224 - 56 (- \frac{4 c}{3}) - 63 c$

$b = - 4 - \frac{4 c}{3}$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.4.2.1.1.3

Multiply $- 56 (- \frac{4 c}{3})$ .

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Step 2.3.4.2.1.1.3.1

Multiply $- 1$ by $- 56$ .

$125 = - 64 + 224 + 56 (\frac{4 c}{3}) - 63 c$

$b = - 4 - \frac{4 c}{3}$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.4.2.1.1.3.2

Combine $56$ and $\frac{4 c}{3}$ .

$125 = - 64 + 224 + \frac{56 (4 c)}{3} - 63 c$

$b = - 4 - \frac{4 c}{3}$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.4.2.1.1.3.3

Multiply $4$ by $56$ .

$125 = - 64 + 224 + \frac{224 c}{3} - 63 c$

$b = - 4 - \frac{4 c}{3}$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$125 = - 64 + 224 + \frac{224 c}{3} - 63 c$

$b = - 4 - \frac{4 c}{3}$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$125 = - 64 + 224 + \frac{224 c}{3} - 63 c$

$b = - 4 - \frac{4 c}{3}$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.4.2.1.2

Add $- 64$ and $224$ .

$125 = 160 + \frac{224 c}{3} - 63 c$

$b = - 4 - \frac{4 c}{3}$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.4.2.1.3

To write $- 63 c$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$125 = 160 + \frac{224 c}{3} - 63 c \cdot \frac{3}{3}$

$b = - 4 - \frac{4 c}{3}$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.4.2.1.4

Simplify terms.

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Step 2.3.4.2.1.4.1

Combine $- 63 c$ and $\frac{3}{3}$ .

$125 = 160 + \frac{224 c}{3} + \frac{- 63 c \cdot 3}{3}$

$b = - 4 - \frac{4 c}{3}$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.4.2.1.4.2

Combine the numerators over the common denominator.

$125 = 160 + \frac{224 c - 63 c \cdot 3}{3}$

$b = - 4 - \frac{4 c}{3}$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$125 = 160 + \frac{224 c - 63 c \cdot 3}{3}$

$b = - 4 - \frac{4 c}{3}$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.4.2.1.5

Simplify the numerator.

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Step 2.3.4.2.1.5.1

Factor $7 c$ out of $224 c - 63 c \cdot 3$ .

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Step 2.3.4.2.1.5.1.1

Factor $7 c$ out of $224 c$ .

$125 = 160 + \frac{7 c (32) - 63 c \cdot 3}{3}$

$b = - 4 - \frac{4 c}{3}$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.4.2.1.5.1.2

Factor $7 c$ out of $- 63 c \cdot 3$ .

$125 = 160 + \frac{7 c (32) + 7 c (- 9 \cdot 3)}{3}$

$b = - 4 - \frac{4 c}{3}$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.4.2.1.5.1.3

Factor $7 c$ out of $7 c (32) + 7 c (- 9 \cdot 3)$ .

$125 = 160 + \frac{7 c (32 - 9 \cdot 3)}{3}$

$b = - 4 - \frac{4 c}{3}$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$125 = 160 + \frac{7 c (32 - 9 \cdot 3)}{3}$

$b = - 4 - \frac{4 c}{3}$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.4.2.1.5.2

Multiply $- 9$ by $3$ .

$125 = 160 + \frac{7 c (32 - 27)}{3}$

$b = - 4 - \frac{4 c}{3}$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.4.2.1.5.3

Subtract $27$ from $32$ .

$125 = 160 + \frac{7 c \cdot 5}{3}$

$b = - 4 - \frac{4 c}{3}$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.4.2.1.5.4

Multiply $5$ by $7$ .

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$5 = - 4 - 2 b - 3 c$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.4.3

Replace all occurrences of $b$ in $5 = - 4 - 2 b - 3 c$ with $- 4 - \frac{4 c}{3}$ .

$5 = - 4 - 2 (- 4 - \frac{4 c}{3}) - 3 c$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.4.4

Simplify the right side.

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Step 2.3.4.4.1

Simplify $- 4 - 2 (- 4 - \frac{4 c}{3}) - 3 c$ .

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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.

$5 = - 4 - 2 \cdot - 4 - 2 (- \frac{4 c}{3}) - 3 c$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.4.4.1.1.2

Multiply $- 2$ by $- 4$ .

$5 = - 4 + 8 - 2 (- \frac{4 c}{3}) - 3 c$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.4.4.1.1.3

Multiply $- 2 (- \frac{4 c}{3})$ .

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Step 2.3.4.4.1.1.3.1

Multiply $- 1$ by $- 2$ .

$5 = - 4 + 8 + 2 (\frac{4 c}{3}) - 3 c$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.4.4.1.1.3.2

Combine $2$ and $\frac{4 c}{3}$ .

$5 = - 4 + 8 + \frac{2 (4 c)}{3} - 3 c$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.4.4.1.1.3.3

Multiply $4$ by $2$ .

$5 = - 4 + 8 + \frac{8 c}{3} - 3 c$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$5 = - 4 + 8 + \frac{8 c}{3} - 3 c$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$5 = - 4 + 8 + \frac{8 c}{3} - 3 c$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.4.4.1.2

Add $- 4$ and $8$ .

$5 = 4 + \frac{8 c}{3} - 3 c$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.4.4.1.3

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

$5 = 4 + \frac{8 c}{3} - 3 c \cdot \frac{3}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.4.4.1.4

Simplify terms.

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Step 2.3.4.4.1.4.1

Combine $- 3 c$ and $\frac{3}{3}$ .

$5 = 4 + \frac{8 c}{3} + \frac{- 3 c \cdot 3}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.4.4.1.4.2

Combine the numerators over the common denominator.

$5 = 4 + \frac{8 c - 3 c \cdot 3}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$5 = 4 + \frac{8 c - 3 c \cdot 3}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.4.4.1.5

Simplify each term.

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Step 2.3.4.4.1.5.1

Simplify the numerator.

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Step 2.3.4.4.1.5.1.1

Factor $c$ out of $8 c - 3 c \cdot 3$ .

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Step 2.3.4.4.1.5.1.1.1

Factor $c$ out of $8 c$ .

$5 = 4 + \frac{c \cdot 8 - 3 c \cdot 3}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.4.4.1.5.1.1.2

Factor $c$ out of $- 3 c \cdot 3$ .

$5 = 4 + \frac{c \cdot 8 + c (- 3 \cdot 3)}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.4.4.1.5.1.1.3

Factor $c$ out of $c \cdot 8 + c (- 3 \cdot 3)$ .

$5 = 4 + \frac{c (8 - 3 \cdot 3)}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$5 = 4 + \frac{c (8 - 3 \cdot 3)}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.4.4.1.5.1.2

Multiply $- 3$ by $3$ .

$5 = 4 + \frac{c (8 - 9)}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.4.4.1.5.1.3

Subtract $9$ from $8$ .

$5 = 4 + \frac{c \cdot - 1}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$5 = 4 + \frac{c \cdot - 1}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.4.4.1.5.2

Move $- 1$ to the left of $c$ .

$5 = 4 + \frac{- 1 \cdot c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.4.4.1.5.3

Move the negative in front of the fraction.

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$159 = - 81 - 72 b - 80 c$

$a = - 1 - b - c$

Step 2.3.4.5

Replace all occurrences of $b$ in $159 = - 81 - 72 b - 80 c$ with $- 4 - \frac{4 c}{3}$ .

$159 = - 81 - 72 (- 4 - \frac{4 c}{3}) - 80 c$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$a = - 1 - b - c$

Step 2.3.4.6

Simplify the right side.

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Step 2.3.4.6.1

Simplify $- 81 - 72 (- 4 - \frac{4 c}{3}) - 80 c$ .

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Step 2.3.4.6.1.1

Simplify each term.

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Step 2.3.4.6.1.1.1

Apply the distributive property.

$159 = - 81 - 72 \cdot - 4 - 72 (- \frac{4 c}{3}) - 80 c$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$a = - 1 - b - c$

Step 2.3.4.6.1.1.2

Multiply $- 72$ by $- 4$ .

$159 = - 81 + 288 - 72 (- \frac{4 c}{3}) - 80 c$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$a = - 1 - b - c$

Step 2.3.4.6.1.1.3

Cancel the common factor of $3$ .

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Step 2.3.4.6.1.1.3.1

Move the leading negative in $- \frac{4 c}{3}$ into the numerator.

$159 = - 81 + 288 - 72 \frac{- 4 c}{3} - 80 c$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$a = - 1 - b - c$

Step 2.3.4.6.1.1.3.2

Factor $3$ out of $- 72$ .

$159 = - 81 + 288 + 3 (- 24) (\frac{- 4 c}{3}) - 80 c$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$a = - 1 - b - c$

Step 2.3.4.6.1.1.3.3

Cancel the common factor.

$159 = - 81 + 288 + 3 \cdot (- 24 \frac{- 4 c}{3}) - 80 c$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$a = - 1 - b - c$

Step 2.3.4.6.1.1.3.4

Rewrite the expression.

$159 = - 81 + 288 - 24 (- 4 c) - 80 c$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$a = - 1 - b - c$

$159 = - 81 + 288 - 24 (- 4 c) - 80 c$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$a = - 1 - b - c$

Step 2.3.4.6.1.1.4

Multiply $- 4$ by $- 24$ .

$159 = - 81 + 288 + 96 c - 80 c$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$a = - 1 - b - c$

$159 = - 81 + 288 + 96 c - 80 c$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$a = - 1 - b - c$

Step 2.3.4.6.1.2

Simplify by adding terms.

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Step 2.3.4.6.1.2.1

Add $- 81$ and $288$ .

$159 = 207 + 96 c - 80 c$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$a = - 1 - b - c$

Step 2.3.4.6.1.2.2

Subtract $80 c$ from $96 c$ .

$159 = 207 + 16 c$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$a = - 1 - b - c$

$159 = 207 + 16 c$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$a = - 1 - b - c$

$159 = 207 + 16 c$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$a = - 1 - b - c$

$159 = 207 + 16 c$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$a = - 1 - b - c$

Step 2.3.4.7

Replace all occurrences of $b$ in $a = - 1 - b - c$ with $- 4 - \frac{4 c}{3}$ .

$a = - 1 - (- 4 - \frac{4 c}{3}) - c$

$159 = 207 + 16 c$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

Step 2.3.4.8

Simplify the right side.

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Step 2.3.4.8.1

Simplify $- 1 - (- 4 - \frac{4 c}{3}) - c$ .

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Step 2.3.4.8.1.1

Simplify each term.

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Step 2.3.4.8.1.1.1

Apply the distributive property.

$a = - 1 + 4 + \frac{4 c}{3} - c$

$159 = 207 + 16 c$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

Step 2.3.4.8.1.1.2

Multiply $- 1$ by $- 4$ .

$a = - 1 + 4 + \frac{4 c}{3} - c$

$159 = 207 + 16 c$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

Step 2.3.4.8.1.1.3

Multiply $- - \frac{4 c}{3}$ .

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Step 2.3.4.8.1.1.3.1

Multiply $- 1$ by $- 1$ .

$a = - 1 + 4 + 1 (\frac{4 c}{3}) - c$

$159 = 207 + 16 c$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

Step 2.3.4.8.1.1.3.2

Multiply $\frac{4 c}{3}$ by $1$ .

$a = - 1 + 4 + \frac{4 c}{3} - c$

$159 = 207 + 16 c$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$a = - 1 + 4 + \frac{4 c}{3} - c$

$159 = 207 + 16 c$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$a = - 1 + 4 + \frac{4 c}{3} - c$

$159 = 207 + 16 c$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

Step 2.3.4.8.1.2

Add $- 1$ and $4$ .

$a = 3 + \frac{4 c}{3} - c$

$159 = 207 + 16 c$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

Step 2.3.4.8.1.3

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

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

$159 = 207 + 16 c$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

Step 2.3.4.8.1.4

Simplify terms.

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Step 2.3.4.8.1.4.1

Combine $- c$ and $\frac{3}{3}$ .

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

$159 = 207 + 16 c$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

Step 2.3.4.8.1.4.2

Combine the numerators over the common denominator.

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

$159 = 207 + 16 c$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

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

$159 = 207 + 16 c$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

Step 2.3.4.8.1.5

Simplify each term.

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Step 2.3.4.8.1.5.1

Simplify the numerator.

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Step 2.3.4.8.1.5.1.1

Factor $c$ out of $4 c - c \cdot 3$ .

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Step 2.3.4.8.1.5.1.1.1

Factor $c$ out of $4 c$ .

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

$159 = 207 + 16 c$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

Step 2.3.4.8.1.5.1.1.2

Factor $c$ out of $- c \cdot 3$ .

$a = 3 + \frac{c \cdot 4 + c (- 1 \cdot 3)}{3}$

$159 = 207 + 16 c$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

Step 2.3.4.8.1.5.1.1.3

Factor $c$ out of $c \cdot 4 + c (- 1 \cdot 3)$ .

$a = 3 + \frac{c (4 - 1 \cdot 3)}{3}$

$159 = 207 + 16 c$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$a = 3 + \frac{c (4 - 1 \cdot 3)}{3}$

$159 = 207 + 16 c$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

Step 2.3.4.8.1.5.1.2

Multiply $- 1$ by $3$ .

$a = 3 + \frac{c (4 - 3)}{3}$

$159 = 207 + 16 c$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

Step 2.3.4.8.1.5.1.3

Subtract $3$ from $4$ .

$a = 3 + \frac{c \cdot 1}{3}$

$159 = 207 + 16 c$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$a = 3 + \frac{c \cdot 1}{3}$

$159 = 207 + 16 c$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

Step 2.3.4.8.1.5.2

Multiply $c$ by $1$ .

$a = 3 + \frac{c}{3}$

$159 = 207 + 16 c$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$a = 3 + \frac{c}{3}$

$159 = 207 + 16 c$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$a = 3 + \frac{c}{3}$

$159 = 207 + 16 c$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$a = 3 + \frac{c}{3}$

$159 = 207 + 16 c$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$a = 3 + \frac{c}{3}$

$159 = 207 + 16 c$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

Step 2.3.5

Solve for $c$ in $159 = 207 + 16 c$ .

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Step 2.3.5.1

Rewrite the equation as $207 + 16 c = 159$ .

$207 + 16 c = 159$

$a = 3 + \frac{c}{3}$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

Step 2.3.5.2

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

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Step 2.3.5.2.1

Subtract $207$ from both sides of the equation.

$16 c = 159 - 207$

$a = 3 + \frac{c}{3}$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

Step 2.3.5.2.2

Subtract $207$ from $159$ .

$16 c = - 48$

$a = 3 + \frac{c}{3}$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$16 c = - 48$

$a = 3 + \frac{c}{3}$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

Step 2.3.5.3

Divide each term in $16 c = - 48$ by $16$ and simplify.

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Step 2.3.5.3.1

Divide each term in $16 c = - 48$ by $16$ .

$\frac{16 c}{16} = \frac{- 48}{16}$

$a = 3 + \frac{c}{3}$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

Step 2.3.5.3.2

Simplify the left side.

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Step 2.3.5.3.2.1

Cancel the common factor of $16$ .

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Step 2.3.5.3.2.1.1

Cancel the common factor.

$\frac{16 c}{16} = \frac{- 48}{16}$

$a = 3 + \frac{c}{3}$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

Step 2.3.5.3.2.1.2

Divide $c$ by $1$ .

$c = \frac{- 48}{16}$

$a = 3 + \frac{c}{3}$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$c = \frac{- 48}{16}$

$a = 3 + \frac{c}{3}$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$c = \frac{- 48}{16}$

$a = 3 + \frac{c}{3}$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

Step 2.3.5.3.3

Simplify the right side.

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Step 2.3.5.3.3.1

Divide $- 48$ by $16$ .

$c = - 3$

$a = 3 + \frac{c}{3}$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$c = - 3$

$a = 3 + \frac{c}{3}$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$c = - 3$

$a = 3 + \frac{c}{3}$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$c = - 3$

$a = 3 + \frac{c}{3}$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

Step 2.3.6

Replace all occurrences of $c$ with $- 3$ in each equation.

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Step 2.3.6.1

Replace all occurrences of $c$ in $a = 3 + \frac{c}{3}$ with $- 3$ .

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

$c = - 3$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

Step 2.3.6.2

Simplify the right side.

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Step 2.3.6.2.1

Simplify $3 + \frac{- 3}{3}$ .

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Step 2.3.6.2.1.1

Divide $- 3$ by $3$ .

$a = 3 - 1$

$c = - 3$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

Step 2.3.6.2.1.2

Subtract $1$ from $3$ .

$a = 2$

$c = - 3$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$a = 2$

$c = - 3$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$a = 2$

$c = - 3$

$5 = 4 - \frac{c}{3}$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

Step 2.3.6.3

Replace all occurrences of $c$ in $5 = 4 - \frac{c}{3}$ with $- 3$ .

$5 = 4 - \frac{- 3}{3}$

$a = 2$

$c = - 3$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

Step 2.3.6.4

Simplify the right side.

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Step 2.3.6.4.1

Simplify $4 - \frac{- 3}{3}$ .

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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

Divide $- 3$ by $3$ .

$5 = 4 + 1$

$a = 2$

$c = - 3$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

Step 2.3.6.4.1.1.2

Multiply $- 1$ by $- 1$ .

$5 = 4 + 1$

$a = 2$

$c = - 3$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$5 = 4 + 1$

$a = 2$

$c = - 3$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

Step 2.3.6.4.1.2

Add $4$ and $1$ .

$5 = 5$

$a = 2$

$c = - 3$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$5 = 5$

$a = 2$

$c = - 3$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

$5 = 5$

$a = 2$

$c = - 3$

$125 = 160 + \frac{35 c}{3}$

$b = - 4 - \frac{4 c}{3}$

Step 2.3.6.5

Replace all occurrences of $c$ in $125 = 160 + \frac{35 c}{3}$ with $- 3$ .

$125 = 160 + \frac{35 (- 3)}{3}$

$5 = 5$

$a = 2$

$c = - 3$

$b = - 4 - \frac{4 c}{3}$

Step 2.3.6.6

Simplify the right side.

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Step 2.3.6.6.1

Simplify $160 + \frac{35 (- 3)}{3}$ .

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Step 2.3.6.6.1.1

Multiply $35$ by $- 3$ .

$125 = 160 + \frac{- 105}{3}$

$5 = 5$

$a = 2$

$c = - 3$

$b = - 4 - \frac{4 c}{3}$

Step 2.3.6.6.1.2

Divide $- 105$ by $3$ .

$125 = 160 - 35$

$5 = 5$

$a = 2$

$c = - 3$

$b = - 4 - \frac{4 c}{3}$

Step 2.3.6.6.1.3

Subtract $35$ from $160$ .

$125 = 125$

$5 = 5$

$a = 2$

$c = - 3$

$b = - 4 - \frac{4 c}{3}$

$125 = 125$

$5 = 5$

$a = 2$

$c = - 3$

$b = - 4 - \frac{4 c}{3}$

$125 = 125$

$5 = 5$

$a = 2$

$c = - 3$

$b = - 4 - \frac{4 c}{3}$

Step 2.3.6.7

Replace all occurrences of $c$ in $b = - 4 - \frac{4 c}{3}$ with $- 3$ .

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

$125 = 125$

$5 = 5$

$a = 2$

$c = - 3$

Step 2.3.6.8

Simplify the right side.

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Step 2.3.6.8.1

Simplify $- 4 - \frac{4 (- 3)}{3}$ .

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Step 2.3.6.8.1.1

Simplify each term.

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Step 2.3.6.8.1.1.1

Cancel the common factor of $- 3$ and $3$ .

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Step 2.3.6.8.1.1.1.1

Factor $3$ out of $4 (- 3)$ .

$b = - 4 - \frac{3 (4 \cdot (- 1))}{3}$

$125 = 125$

$5 = 5$

$a = 2$

$c = - 3$

Step 2.3.6.8.1.1.1.2

Cancel the common factors.

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Step 2.3.6.8.1.1.1.2.1

Factor $3$ out of $3$ .

$b = - 4 - \frac{3 (4 \cdot (- 1))}{3 (1)}$

$125 = 125$

$5 = 5$

$a = 2$

$c = - 3$

Step 2.3.6.8.1.1.1.2.2

Cancel the common factor.

$b = - 4 - \frac{3 (4 \cdot (- 1))}{3 \cdot 1}$

$125 = 125$

$5 = 5$

$a = 2$

$c = - 3$

Step 2.3.6.8.1.1.1.2.3

Rewrite the expression.

$b = - 4 - \frac{4 \cdot (- 1)}{1}$

$125 = 125$

$5 = 5$

$a = 2$

$c = - 3$

Step 2.3.6.8.1.1.1.2.4

Divide $4 \cdot (- 1)$ by $1$ .

$b = - 4 - (4 \cdot (- 1))$

$125 = 125$

$5 = 5$

$a = 2$

$c = - 3$

$b = - 4 - (4 \cdot (- 1))$

$125 = 125$

$5 = 5$

$a = 2$

$c = - 3$

$b = - 4 - (4 \cdot (- 1))$

$125 = 125$

$5 = 5$

$a = 2$

$c = - 3$

Step 2.3.6.8.1.1.2

Multiply $- (4 \cdot (- 1))$ .

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Step 2.3.6.8.1.1.2.1

Multiply $4$ by $- 1$ .

$b = - 4 + 4$

$125 = 125$

$5 = 5$

$a = 2$

$c = - 3$

Step 2.3.6.8.1.1.2.2

Multiply $- 1$ by $- 4$ .

$b = - 4 + 4$

$125 = 125$

$5 = 5$

$a = 2$

$c = - 3$

$b = - 4 + 4$

$125 = 125$

$5 = 5$

$a = 2$

$c = - 3$

$b = - 4 + 4$

$125 = 125$

$5 = 5$

$a = 2$

$c = - 3$

Step 2.3.6.8.1.2

Add $- 4$ and $4$ .

$b = 0$

$125 = 125$

$5 = 5$

$a = 2$

$c = - 3$

$b = 0$

$125 = 125$

$5 = 5$

$a = 2$

$c = - 3$

$b = 0$

$125 = 125$

$5 = 5$

$a = 2$

$c = - 3$

$b = 0$

$125 = 125$

$5 = 5$

$a = 2$

$c = - 3$

Step 2.3.7

Remove any equations from the system that are always true.

$b = 0$

$a = 2$

$c = - 3$

Step 2.3.8

List all of the solutions.

$b = 0, a = 2, c = - 3$

Step 2.4

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

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Step 2.4.1

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = 2$ , $b = 0$ , $c = - 3$ , and $x = 1$ .

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Step 2.4.1.1

Simplify each term.

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Step 2.4.1.1.1

One to any power is one.

$y = 2 \cdot 1 + (0) \cdot (1) - 3$

Step 2.4.1.1.2

Multiply $2$ by $1$ .

$y = 2 + (0) \cdot (1) - 3$

Step 2.4.1.1.3

Multiply $0$ by $1$ .

$y = 2 + 0 - 3$

Step 2.4.1.2

Simplify by adding and subtracting.

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Step 2.4.1.2.1

Add $2$ and $0$ .

$y = 2 - 3$

Step 2.4.1.2.2

Subtract $3$ from $2$ .

$y = - 1$

Step 2.4.2

If the table has a quadratic function rule, $y = q (x)$ for the corresponding $x$ value, $x = 1$ . This check passes since $y = - 1$ and $q (x) = - 1$ .

$- 1 = - 1$

Step 2.4.3

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = 2$ , $b = 0$ , $c = - 3$ , and $x = 9$ .

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Step 2.4.3.1

Simplify each term.

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Step 2.4.3.1.1

Raise $9$ to the power of $2$ .

$y = 2 \cdot 81 + (0) \cdot (9) - 3$

Step 2.4.3.1.2

Multiply $2$ by $81$ .

$y = 162 + (0) \cdot (9) - 3$

Step 2.4.3.1.3

Multiply $0$ by $9$ .

$y = 162 + 0 - 3$

Step 2.4.3.2

Simplify by adding and subtracting.

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Step 2.4.3.2.1

Add $162$ and $0$ .

$y = 162 - 3$

Step 2.4.3.2.2

Subtract $3$ from $162$ .

$y = 159$

Step 2.4.4

If the table has a quadratic function rule, $y = q (x)$ for the corresponding $x$ value, $x = 9$ . This check passes since $y = 159$ and $q (x) = 159$ .

$159 = 159$

Step 2.4.5

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = 2$ , $b = 0$ , $c = - 3$ , and $x = 2$ .

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Step 2.4.5.1

Simplify each term.

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Step 2.4.5.1.1

Multiply $2$ by ${(2)}^{2}$ by adding the exponents.

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Step 2.4.5.1.1.1

Multiply $2$ by ${(2)}^{2}$ .

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Step 2.4.5.1.1.1.1

Raise $2$ to the power of $1$ .

$y = 2 \cdot {(2)}^{2} + (0) \cdot (2) - 3$

Step 2.4.5.1.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = 2^{1 + 2} + (0) \cdot (2) - 3$

Step 2.4.5.1.1.2

Add $1$ and $2$ .

$y = 2^{3} + (0) \cdot (2) - 3$

Step 2.4.5.1.2

Raise $2$ to the power of $3$ .

$y = 8 + (0) \cdot (2) - 3$

Step 2.4.5.1.3

Multiply $0$ by $2$ .

$y = 8 + 0 - 3$

Step 2.4.5.2

Simplify by adding and subtracting.

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Step 2.4.5.2.1

Add $8$ and $0$ .

$y = 8 - 3$

Step 2.4.5.2.2

Subtract $3$ from $8$ .

$y = 5$

Step 2.4.6

If the table has a quadratic function rule, $y = q (x)$ for the corresponding $x$ value, $x = 2$ . This check passes since $y = 5$ and $q (x) = 5$ .

$5 = 5$

Step 2.4.7

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = 2$ , $b = 0$ , $c = - 3$ , and $x = 8$ .

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Step 2.4.7.1

Simplify each term.

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Step 2.4.7.1.1

Raise $8$ to the power of $2$ .

$y = 2 \cdot 64 + (0) \cdot (8) - 3$

Step 2.4.7.1.2

Multiply $2$ by $64$ .

$y = 128 + (0) \cdot (8) - 3$

Step 2.4.7.1.3

Multiply $0$ by $8$ .

$y = 128 + 0 - 3$

Step 2.4.7.2

Simplify by adding and subtracting.

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Step 2.4.7.2.1

Add $128$ and $0$ .

$y = 128 - 3$

Step 2.4.7.2.2

Subtract $3$ from $128$ .

$y = 125$

Step 2.4.8

If the table has a quadratic function rule, $y = q (x)$ for the corresponding $x$ value, $x = 8$ . This check passes since $y = 125$ and $q (x) = 125$ .

$125 = 125$

Step 2.4.9

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = 2$ , $b = 0$ , $c = - 3$ , and $x = 3$ .

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Step 2.4.9.1

Simplify each term.

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Step 2.4.9.1.1

Raise $3$ to the power of $2$ .

$y = 2 \cdot 9 + (0) \cdot (3) - 3$

Step 2.4.9.1.2

Multiply $2$ by $9$ .

$y = 18 + (0) \cdot (3) - 3$

Step 2.4.9.1.3

Multiply $0$ by $3$ .

$y = 18 + 0 - 3$

Step 2.4.9.2

Simplify by adding and subtracting.

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Step 2.4.9.2.1

Add $18$ and $0$ .

$y = 18 - 3$

Step 2.4.9.2.2

Subtract $3$ from $18$ .

$y = 15$

Step 2.4.10

If the table has a quadratic function rule, $y = q (x)$ for the corresponding $x$ value, $x = 3$ . This check passes since $y = 15$ and $q (x) = 15$ .

$15 = 15$

Step 2.4.11

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

The function is quadratic

Step 3

Since all $y = q (x)$ , the function is quadratic and follows the form $y = 2 x^{2} - 3$ .

$y = 2 x^{2} - 3$

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