Algebra Examples

$\begin{array}{c} x & q (x) \\ 1 & 2 \\ 9 & 162 \\ 2 & 8 \\ 8 & 128 \\ 3 & 18 \end{array}$

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

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

$2 = a (1) + b 162 = a (9) + b 8 = a (2) + b 128 = a (8) + b 18 = a (3) + b$

Step 1.3

Calculate the values of

$a$ and

$b$ .

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Step 1.3.1

Solve for

$a$ in

$2 = a + b$ .

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Step 1.3.1.1

Rewrite the equation as

$a + b = 2$ .

$a + b = 2$

$162 = a (9) + b$

$8 = a (2) + b$

$128 = a (8) + b$

$18 = a (3) + b$

Step 1.3.1.2

Subtract

$b$ from both sides of the equation.

$a = 2 - b$

$162 = a (9) + b$

$8 = a (2) + b$

$128 = a (8) + b$

$18 = a (3) + b$

$a = 2 - b$

$162 = a (9) + b$

$8 = a (2) + b$

$128 = a (8) + b$

$18 = a (3) + b$

Step 1.3.2

Replace all occurrences of

$a$ with

$2 - b$ in each equation.

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Step 1.3.2.1

Replace all occurrences of

$a$ in

$162 = a (9) + b$ with

$2 - b$ .

$162 = (2 - b) (9) + b$

$a = 2 - b$

$8 = a (2) + b$

$128 = a (8) + b$

$18 = a (3) + b$

Step 1.3.2.2

Simplify the right side.

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Step 1.3.2.2.1

Simplify

$(2 - b) (9) + b$ .

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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.

$162 = 2 \cdot 9 - b \cdot 9 + b$

$a = 2 - b$

$8 = a (2) + b$

$128 = a (8) + b$

$18 = a (3) + b$

Step 1.3.2.2.1.1.2

Multiply

$2$ by

$9$ .

$162 = 18 - b \cdot 9 + b$

$a = 2 - b$

$8 = a (2) + b$

$128 = a (8) + b$

$18 = a (3) + b$

Step 1.3.2.2.1.1.3

Multiply

$9$ by

$- 1$ .

$162 = 18 - 9 b + b$

$a = 2 - b$

$8 = a (2) + b$

$128 = a (8) + b$

$18 = a (3) + b$

$162 = 18 - 9 b + b$

$a = 2 - b$

$8 = a (2) + b$

$128 = a (8) + b$

$18 = a (3) + b$

Step 1.3.2.2.1.2

Add

$- 9 b$ and

$b$ .

$162 = 18 - 8 b$

$a = 2 - b$

$8 = a (2) + b$

$128 = a (8) + b$

$18 = a (3) + b$

$162 = 18 - 8 b$

$a = 2 - b$

$8 = a (2) + b$

$128 = a (8) + b$

$18 = a (3) + b$

$162 = 18 - 8 b$

$a = 2 - b$

$8 = a (2) + b$

$128 = a (8) + b$

$18 = a (3) + b$

Step 1.3.2.3

Replace all occurrences of

$a$ in

$8 = a (2) + b$ with

$2 - b$ .

$8 = (2 - b) (2) + b$

$162 = 18 - 8 b$

$a = 2 - b$

$128 = a (8) + b$

$18 = a (3) + b$

Step 1.3.2.4

Simplify the right side.

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Step 1.3.2.4.1

Simplify

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

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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.

$8 = 2 \cdot 2 - b \cdot 2 + b$

$162 = 18 - 8 b$

$a = 2 - b$

$128 = a (8) + b$

$18 = a (3) + b$

Step 1.3.2.4.1.1.2

Multiply

$2$ by

$2$ .

$8 = 4 - b \cdot 2 + b$

$162 = 18 - 8 b$

$a = 2 - b$

$128 = a (8) + b$

$18 = a (3) + b$

Step 1.3.2.4.1.1.3

Multiply

$2$ by

$- 1$ .

$8 = 4 - 2 b + b$

$162 = 18 - 8 b$

$a = 2 - b$

$128 = a (8) + b$

$18 = a (3) + b$

$8 = 4 - 2 b + b$

$162 = 18 - 8 b$

$a = 2 - b$

$128 = a (8) + b$

$18 = a (3) + b$

Step 1.3.2.4.1.2

Add

$- 2 b$ and

$b$ .

$8 = 4 - b$

$162 = 18 - 8 b$

$a = 2 - b$

$128 = a (8) + b$

$18 = a (3) + b$

$8 = 4 - b$

$162 = 18 - 8 b$

$a = 2 - b$

$128 = a (8) + b$

$18 = a (3) + b$

$8 = 4 - b$

$162 = 18 - 8 b$

$a = 2 - b$

$128 = a (8) + b$

$18 = a (3) + b$

Step 1.3.2.5

Replace all occurrences of

$a$ in

$128 = a (8) + b$ with

$2 - b$ .

$128 = (2 - b) (8) + b$

$8 = 4 - b$

$162 = 18 - 8 b$

$a = 2 - b$

$18 = a (3) + b$

Step 1.3.2.6

Simplify the right side.

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Step 1.3.2.6.1

Simplify

$(2 - b) (8) + b$ .

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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.

$128 = 2 \cdot 8 - b \cdot 8 + b$

$8 = 4 - b$

$162 = 18 - 8 b$

$a = 2 - b$

$18 = a (3) + b$

Step 1.3.2.6.1.1.2

Multiply

$2$ by

$8$ .

$128 = 16 - b \cdot 8 + b$

$8 = 4 - b$

$162 = 18 - 8 b$

$a = 2 - b$

$18 = a (3) + b$

Step 1.3.2.6.1.1.3

Multiply

$8$ by

$- 1$ .

$128 = 16 - 8 b + b$

$8 = 4 - b$

$162 = 18 - 8 b$

$a = 2 - b$

$18 = a (3) + b$

$128 = 16 - 8 b + b$

$8 = 4 - b$

$162 = 18 - 8 b$

$a = 2 - b$

$18 = a (3) + b$

Step 1.3.2.6.1.2

Add

$- 8 b$ and

$b$ .

$128 = 16 - 7 b$

$8 = 4 - b$

$162 = 18 - 8 b$

$a = 2 - b$

$18 = a (3) + b$

$128 = 16 - 7 b$

$8 = 4 - b$

$162 = 18 - 8 b$

$a = 2 - b$

$18 = a (3) + b$

$128 = 16 - 7 b$

$8 = 4 - b$

$162 = 18 - 8 b$

$a = 2 - b$

$18 = a (3) + b$

Step 1.3.2.7

Replace all occurrences of

$a$ in

$18 = a (3) + b$ with

$2 - b$ .

$18 = (2 - b) (3) + b$

$128 = 16 - 7 b$

$8 = 4 - b$

$162 = 18 - 8 b$

$a = 2 - b$

Step 1.3.2.8

Simplify the right side.

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Step 1.3.2.8.1

Simplify

$(2 - b) (3) + b$ .

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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.

$18 = 2 \cdot 3 - b \cdot 3 + b$

$128 = 16 - 7 b$

$8 = 4 - b$

$162 = 18 - 8 b$

$a = 2 - b$

Step 1.3.2.8.1.1.2

Multiply

$2$ by

$3$ .

$18 = 6 - b \cdot 3 + b$

$128 = 16 - 7 b$

$8 = 4 - b$

$162 = 18 - 8 b$

$a = 2 - b$

Step 1.3.2.8.1.1.3

Multiply

$3$ by

$- 1$ .

$18 = 6 - 3 b + b$

$128 = 16 - 7 b$

$8 = 4 - b$

$162 = 18 - 8 b$

$a = 2 - b$

$18 = 6 - 3 b + b$

$128 = 16 - 7 b$

$8 = 4 - b$

$162 = 18 - 8 b$

$a = 2 - b$

Step 1.3.2.8.1.2

Add

$- 3 b$ and

$b$ .

$18 = 6 - 2 b$

$128 = 16 - 7 b$

$8 = 4 - b$

$162 = 18 - 8 b$

$a = 2 - b$

$18 = 6 - 2 b$

$128 = 16 - 7 b$

$8 = 4 - b$

$162 = 18 - 8 b$

$a = 2 - b$

$18 = 6 - 2 b$

$128 = 16 - 7 b$

$8 = 4 - b$

$162 = 18 - 8 b$

$a = 2 - b$

$18 = 6 - 2 b$

$128 = 16 - 7 b$

$8 = 4 - b$

$162 = 18 - 8 b$

$a = 2 - b$

Step 1.3.3

Solve for

$b$ in

$18 = 6 - 2 b$ .

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Step 1.3.3.1

Rewrite the equation as

$6 - 2 b = 18$ .

$6 - 2 b = 18$

$128 = 16 - 7 b$

$8 = 4 - b$

$162 = 18 - 8 b$

$a = 2 - b$

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

Subtract

$6$ from both sides of the equation.

$- 2 b = 18 - 6$

$128 = 16 - 7 b$

$8 = 4 - b$

$162 = 18 - 8 b$

$a = 2 - b$

Step 1.3.3.2.2

Subtract

$6$ from

$18$ .

$- 2 b = 12$

$128 = 16 - 7 b$

$8 = 4 - b$

$162 = 18 - 8 b$

$a = 2 - b$

$- 2 b = 12$

$128 = 16 - 7 b$

$8 = 4 - b$

$162 = 18 - 8 b$

$a = 2 - b$

Step 1.3.3.3

Divide each term in

$- 2 b = 12$ by

$- 2$ and simplify.

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Step 1.3.3.3.1

Divide each term in

$- 2 b = 12$ by

$- 2$ .

$\frac{- 2 b}{- 2} = \frac{12}{- 2}$

$128 = 16 - 7 b$

$8 = 4 - b$

$162 = 18 - 8 b$

$a = 2 - b$

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.

$\frac{- 2 b}{- 2} = \frac{12}{- 2}$

$128 = 16 - 7 b$

$8 = 4 - b$

$162 = 18 - 8 b$

$a = 2 - b$

Step 1.3.3.3.2.1.2

Divide

$b$ by

$1$ .

$b = \frac{12}{- 2}$

$128 = 16 - 7 b$

$8 = 4 - b$

$162 = 18 - 8 b$

$a = 2 - b$

$b = \frac{12}{- 2}$

$128 = 16 - 7 b$

$8 = 4 - b$

$162 = 18 - 8 b$

$a = 2 - b$

$b = \frac{12}{- 2}$

$128 = 16 - 7 b$

$8 = 4 - b$

$162 = 18 - 8 b$

$a = 2 - b$

Step 1.3.3.3.3

Simplify the right side.

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Step 1.3.3.3.3.1

Divide

$12$ by

$- 2$ .

$b = - 6$

$128 = 16 - 7 b$

$8 = 4 - b$

$162 = 18 - 8 b$

$a = 2 - b$

$b = - 6$

$128 = 16 - 7 b$

$8 = 4 - b$

$162 = 18 - 8 b$

$a = 2 - b$

$b = - 6$

$128 = 16 - 7 b$

$8 = 4 - b$

$162 = 18 - 8 b$

$a = 2 - b$

$b = - 6$

$128 = 16 - 7 b$

$8 = 4 - b$

$162 = 18 - 8 b$

$a = 2 - b$

Step 1.3.4

Replace all occurrences of

$b$ with

$- 6$ in each equation.

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Step 1.3.4.1

Replace all occurrences of

$b$ in

$128 = 16 - 7 b$ with

$- 6$ .

$128 = 16 - 7 \cdot - 6$

$b = - 6$

$8 = 4 - b$

$162 = 18 - 8 b$

$a = 2 - b$

Step 1.3.4.2

Simplify the right side.

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Step 1.3.4.2.1

Simplify

$16 - 7 \cdot - 6$ .

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Step 1.3.4.2.1.1

Multiply

$- 7$ by

$- 6$ .

$128 = 16 + 42$

$b = - 6$

$8 = 4 - b$

$162 = 18 - 8 b$

$a = 2 - b$

Step 1.3.4.2.1.2

Add

$16$ and

$42$ .

$128 = 58$

$b = - 6$

$8 = 4 - b$

$162 = 18 - 8 b$

$a = 2 - b$

$128 = 58$

$b = - 6$

$8 = 4 - b$

$162 = 18 - 8 b$

$a = 2 - b$

$128 = 58$

$b = - 6$

$8 = 4 - b$

$162 = 18 - 8 b$

$a = 2 - b$

Step 1.3.4.3

Replace all occurrences of

$b$ in

$8 = 4 - b$ with

$- 6$ .

$8 = 4 - (- 6)$

$128 = 58$

$b = - 6$

$162 = 18 - 8 b$

$a = 2 - b$

Step 1.3.4.4

Simplify the right side.

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Step 1.3.4.4.1

Simplify

$4 - (- 6)$ .

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Step 1.3.4.4.1.1

Multiply

$- 1$ by

$- 6$ .

$8 = 4 + 6$

$128 = 58$

$b = - 6$

$162 = 18 - 8 b$

$a = 2 - b$

Step 1.3.4.4.1.2

Add

$4$ and

$6$ .

$8 = 10$

$128 = 58$

$b = - 6$

$162 = 18 - 8 b$

$a = 2 - b$

$8 = 10$

$128 = 58$

$b = - 6$

$162 = 18 - 8 b$

$a = 2 - b$

$8 = 10$

$128 = 58$

$b = - 6$

$162 = 18 - 8 b$

$a = 2 - b$

Step 1.3.4.5

Replace all occurrences of

$b$ in

$162 = 18 - 8 b$ with

$- 6$ .

$162 = 18 - 8 \cdot - 6$

$8 = 10$

$128 = 58$

$b = - 6$

$a = 2 - b$

Step 1.3.4.6

Simplify the right side.

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Step 1.3.4.6.1

Simplify

$18 - 8 \cdot - 6$ .

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Step 1.3.4.6.1.1

Multiply

$- 8$ by

$- 6$ .

$162 = 18 + 48$

$8 = 10$

$128 = 58$

$b = - 6$

$a = 2 - b$

Step 1.3.4.6.1.2

Add

$18$ and

$48$ .

$162 = 66$

$8 = 10$

$128 = 58$

$b = - 6$

$a = 2 - b$

$162 = 66$

$8 = 10$

$128 = 58$

$b = - 6$

$a = 2 - b$

$162 = 66$

$8 = 10$

$128 = 58$

$b = - 6$

$a = 2 - b$

Step 1.3.4.7

Replace all occurrences of

$b$ in

$a = 2 - b$ with

$- 6$ .

$a = 2 - (- 6)$

$162 = 66$

$8 = 10$

$128 = 58$

$b = - 6$

Step 1.3.4.8

Simplify the right side.

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Step 1.3.4.8.1

Simplify

$2 - (- 6)$ .

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Step 1.3.4.8.1.1

Multiply

$- 1$ by

$- 6$ .

$a = 2 + 6$

$162 = 66$

$8 = 10$

$128 = 58$

$b = - 6$

Step 1.3.4.8.1.2

Add

$2$ and

$6$ .

$a = 8$

$162 = 66$

$8 = 10$

$128 = 58$

$b = - 6$

$a = 8$

$162 = 66$

$8 = 10$

$128 = 58$

$b = - 6$

$a = 8$

$162 = 66$

$8 = 10$

$128 = 58$

$b = - 6$

$a = 8$

$162 = 66$

$8 = 10$

$128 = 58$

$b = - 6$

Step 1.3.5

Since

$162 = 66$ 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

$2 = a + b + c$ .

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Step 2.3.1.1

Rewrite the equation as

$a + b + c = 2$ .

$a + b + c = 2$

$162 = a \cdot 9^{2} + b (9) + c$

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

$128 = a \cdot 8^{2} + b (8) + c$

$18 = a \cdot 3^{2} + b (3) + c$

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.

$a + c = 2 - b$

$162 = a \cdot 9^{2} + b (9) + c$

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

$128 = a \cdot 8^{2} + b (8) + c$

$18 = a \cdot 3^{2} + b (3) + c$

Step 2.3.1.2.2

Subtract

$c$ from both sides of the equation.

$a = 2 - b - c$

$162 = a \cdot 9^{2} + b (9) + c$

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

$128 = a \cdot 8^{2} + b (8) + c$

$18 = a \cdot 3^{2} + b (3) + c$

$a = 2 - b - c$

$162 = a \cdot 9^{2} + b (9) + c$

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

$128 = a \cdot 8^{2} + b (8) + c$

$18 = a \cdot 3^{2} + b (3) + c$

$a = 2 - b - c$

$162 = a \cdot 9^{2} + b (9) + c$

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

$128 = a \cdot 8^{2} + b (8) + c$

$18 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2

Replace all occurrences of

$a$ with

$2 - b - c$ in each equation.

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Step 2.3.2.1

Replace all occurrences of

$a$ in

$162 = a \cdot 9^{2} + b (9) + c$ with

$2 - b - c$ .

$162 = (2 - b - c) \cdot 9^{2} + b (9) + c$

$a = 2 - b - c$

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

$128 = a \cdot 8^{2} + b (8) + c$

$18 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.2

Simplify the right side.

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Step 2.3.2.2.1

Simplify

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

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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

$2$ .

$162 = (2 - b - c) \cdot 81 + b (9) + c$

$a = 2 - b - c$

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

$128 = a \cdot 8^{2} + b (8) + c$

$18 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.2.1.1.2

Apply the distributive property.

$162 = 2 \cdot 81 - b \cdot 81 - c \cdot 81 + b (9) + c$

$a = 2 - b - c$

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

$128 = a \cdot 8^{2} + b (8) + c$

$18 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.2.1.1.3

Simplify.

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Step 2.3.2.2.1.1.3.1

Multiply

$2$ by

$81$ .

$162 = 162 - b \cdot 81 - c \cdot 81 + b (9) + c$

$a = 2 - b - c$

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

$128 = a \cdot 8^{2} + b (8) + c$

$18 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.2.1.1.3.2

Multiply

$81$ by

$- 1$ .

$162 = 162 - 81 b - c \cdot 81 + b (9) + c$

$a = 2 - b - c$

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

$128 = a \cdot 8^{2} + b (8) + c$

$18 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.2.1.1.3.3

Multiply

$81$ by

$- 1$ .

$162 = 162 - 81 b - 81 c + b (9) + c$

$a = 2 - b - c$

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

$128 = a \cdot 8^{2} + b (8) + c$

$18 = a \cdot 3^{2} + b (3) + c$

$162 = 162 - 81 b - 81 c + b (9) + c$

$a = 2 - b - c$

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

$128 = a \cdot 8^{2} + b (8) + c$

$18 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.2.1.1.4

Move

$9$ to the left of

$b$ .

$162 = 162 - 81 b - 81 c + 9 b + c$

$a = 2 - b - c$

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

$128 = a \cdot 8^{2} + b (8) + c$

$18 = a \cdot 3^{2} + b (3) + c$

$162 = 162 - 81 b - 81 c + 9 b + c$

$a = 2 - b - c$

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

$128 = a \cdot 8^{2} + b (8) + c$

$18 = a \cdot 3^{2} + b (3) + c$

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

$9 b$ .

$162 = 162 - 72 b - 81 c + c$

$a = 2 - b - c$

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

$128 = a \cdot 8^{2} + b (8) + c$

$18 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.2.1.2.2

Add

$- 81 c$ and

$c$ .

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

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

$128 = a \cdot 8^{2} + b (8) + c$

$18 = a \cdot 3^{2} + b (3) + c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

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

$128 = a \cdot 8^{2} + b (8) + c$

$18 = a \cdot 3^{2} + b (3) + c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

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

$128 = a \cdot 8^{2} + b (8) + c$

$18 = a \cdot 3^{2} + b (3) + c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

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

$128 = a \cdot 8^{2} + b (8) + c$

$18 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.3

Replace all occurrences of

$a$ in

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

$2 - b - c$ .

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

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

$128 = a \cdot 8^{2} + b (8) + c$

$18 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.4

Simplify the right side.

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Step 2.3.2.4.1

Simplify

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

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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

$2$ .

$8 = (2 - b - c) \cdot 4 + b (2) + c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

$128 = a \cdot 8^{2} + b (8) + c$

$18 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.4.1.1.2

Apply the distributive property.

$8 = 2 \cdot 4 - b \cdot 4 - c \cdot 4 + b (2) + c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

$128 = a \cdot 8^{2} + b (8) + c$

$18 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.4.1.1.3

Simplify.

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Step 2.3.2.4.1.1.3.1

Multiply

$2$ by

$4$ .

$8 = 8 - b \cdot 4 - c \cdot 4 + b (2) + c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

$128 = a \cdot 8^{2} + b (8) + c$

$18 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.4.1.1.3.2

Multiply

$4$ by

$- 1$ .

$8 = 8 - 4 b - c \cdot 4 + b (2) + c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

$128 = a \cdot 8^{2} + b (8) + c$

$18 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.4.1.1.3.3

Multiply

$4$ by

$- 1$ .

$8 = 8 - 4 b - 4 c + b (2) + c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

$128 = a \cdot 8^{2} + b (8) + c$

$18 = a \cdot 3^{2} + b (3) + c$

$8 = 8 - 4 b - 4 c + b (2) + c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

$128 = a \cdot 8^{2} + b (8) + c$

$18 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.4.1.1.4

Move

$2$ to the left of

$b$ .

$8 = 8 - 4 b - 4 c + 2 b + c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

$128 = a \cdot 8^{2} + b (8) + c$

$18 = a \cdot 3^{2} + b (3) + c$

$8 = 8 - 4 b - 4 c + 2 b + c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

$128 = a \cdot 8^{2} + b (8) + c$

$18 = a \cdot 3^{2} + b (3) + c$

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

$2 b$ .

$8 = 8 - 2 b - 4 c + c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

$128 = a \cdot 8^{2} + b (8) + c$

$18 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.4.1.2.2

Add

$- 4 c$ and

$c$ .

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

$128 = a \cdot 8^{2} + b (8) + c$

$18 = a \cdot 3^{2} + b (3) + c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

$128 = a \cdot 8^{2} + b (8) + c$

$18 = a \cdot 3^{2} + b (3) + c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

$128 = a \cdot 8^{2} + b (8) + c$

$18 = a \cdot 3^{2} + b (3) + c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

$128 = a \cdot 8^{2} + b (8) + c$

$18 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.5

Replace all occurrences of

$a$ in

$128 = a \cdot 8^{2} + b (8) + c$ with

$2 - b - c$ .

$128 = (2 - b - c) \cdot 8^{2} + b (8) + c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

$18 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.6

Simplify the right side.

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Step 2.3.2.6.1

Simplify

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

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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

$2$ .

$128 = (2 - b - c) \cdot 64 + b (8) + c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

$18 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.6.1.1.2

Apply the distributive property.

$128 = 2 \cdot 64 - b \cdot 64 - c \cdot 64 + b (8) + c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

$18 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.6.1.1.3

Simplify.

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Step 2.3.2.6.1.1.3.1

Multiply

$2$ by

$64$ .

$128 = 128 - b \cdot 64 - c \cdot 64 + b (8) + c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

$18 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.6.1.1.3.2

Multiply

$64$ by

$- 1$ .

$128 = 128 - 64 b - c \cdot 64 + b (8) + c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

$18 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.6.1.1.3.3

Multiply

$64$ by

$- 1$ .

$128 = 128 - 64 b - 64 c + b (8) + c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

$18 = a \cdot 3^{2} + b (3) + c$

$128 = 128 - 64 b - 64 c + b (8) + c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

$18 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.6.1.1.4

Move

$8$ to the left of

$b$ .

$128 = 128 - 64 b - 64 c + 8 b + c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

$18 = a \cdot 3^{2} + b (3) + c$

$128 = 128 - 64 b - 64 c + 8 b + c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

$18 = a \cdot 3^{2} + b (3) + c$

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

$8 b$ .

$128 = 128 - 56 b - 64 c + c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

$18 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.6.1.2.2

Add

$- 64 c$ and

$c$ .

$128 = 128 - 56 b - 63 c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

$18 = a \cdot 3^{2} + b (3) + c$

$128 = 128 - 56 b - 63 c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

$18 = a \cdot 3^{2} + b (3) + c$

$128 = 128 - 56 b - 63 c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

$18 = a \cdot 3^{2} + b (3) + c$

$128 = 128 - 56 b - 63 c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

$18 = a \cdot 3^{2} + b (3) + c$

Step 2.3.2.7

Replace all occurrences of

$a$ in

$18 = a \cdot 3^{2} + b (3) + c$ with

$2 - b - c$ .

$18 = (2 - b - c) \cdot 3^{2} + b (3) + c$

$128 = 128 - 56 b - 63 c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

Step 2.3.2.8

Simplify the right side.

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Step 2.3.2.8.1

Simplify

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

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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

$2$ .

$18 = (2 - b - c) \cdot 9 + b (3) + c$

$128 = 128 - 56 b - 63 c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

Step 2.3.2.8.1.1.2

Apply the distributive property.

$18 = 2 \cdot 9 - b \cdot 9 - c \cdot 9 + b (3) + c$

$128 = 128 - 56 b - 63 c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

Step 2.3.2.8.1.1.3

Simplify.

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Step 2.3.2.8.1.1.3.1

Multiply

$2$ by

$9$ .

$18 = 18 - b \cdot 9 - c \cdot 9 + b (3) + c$

$128 = 128 - 56 b - 63 c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

Step 2.3.2.8.1.1.3.2

Multiply

$9$ by

$- 1$ .

$18 = 18 - 9 b - c \cdot 9 + b (3) + c$

$128 = 128 - 56 b - 63 c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

Step 2.3.2.8.1.1.3.3

Multiply

$9$ by

$- 1$ .

$18 = 18 - 9 b - 9 c + b (3) + c$

$128 = 128 - 56 b - 63 c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

$18 = 18 - 9 b - 9 c + b (3) + c$

$128 = 128 - 56 b - 63 c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

Step 2.3.2.8.1.1.4

Move

$3$ to the left of

$b$ .

$18 = 18 - 9 b - 9 c + 3 b + c$

$128 = 128 - 56 b - 63 c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

$18 = 18 - 9 b - 9 c + 3 b + c$

$128 = 128 - 56 b - 63 c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

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

$3 b$ .

$18 = 18 - 6 b - 9 c + c$

$128 = 128 - 56 b - 63 c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

Step 2.3.2.8.1.2.2

Add

$- 9 c$ and

$c$ .

$18 = 18 - 6 b - 8 c$

$128 = 128 - 56 b - 63 c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

$18 = 18 - 6 b - 8 c$

$128 = 128 - 56 b - 63 c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

$18 = 18 - 6 b - 8 c$

$128 = 128 - 56 b - 63 c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

$18 = 18 - 6 b - 8 c$

$128 = 128 - 56 b - 63 c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

$18 = 18 - 6 b - 8 c$

$128 = 128 - 56 b - 63 c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

Step 2.3.3

Solve for

$b$ in

$18 = 18 - 6 b - 8 c$ .

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Step 2.3.3.1

Rewrite the equation as

$18 - 6 b - 8 c = 18$ .

$18 - 6 b - 8 c = 18$

$128 = 128 - 56 b - 63 c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

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

Subtract

$18$ from both sides of the equation.

$- 6 b - 8 c = 18 - 18$

$128 = 128 - 56 b - 63 c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

Step 2.3.3.2.2

Add

$8 c$ to both sides of the equation.

$- 6 b = 18 - 18 + 8 c$

$128 = 128 - 56 b - 63 c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

Step 2.3.3.2.3

Subtract

$18$ from

$18$ .

$- 6 b = 0 + 8 c$

$128 = 128 - 56 b - 63 c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

Step 2.3.3.2.4

Add

$0$ and

$8 c$ .

$- 6 b = 8 c$

$128 = 128 - 56 b - 63 c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

$- 6 b = 8 c$

$128 = 128 - 56 b - 63 c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

Step 2.3.3.3

Divide each term in

$- 6 b = 8 c$ by

$- 6$ and simplify.

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Step 2.3.3.3.1

Divide each term in

$- 6 b = 8 c$ by

$- 6$ .

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

$128 = 128 - 56 b - 63 c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

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.

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

$128 = 128 - 56 b - 63 c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

Step 2.3.3.3.2.1.2

Divide

$b$ by

$1$ .

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

$128 = 128 - 56 b - 63 c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

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

$128 = 128 - 56 b - 63 c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

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

$128 = 128 - 56 b - 63 c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

Step 2.3.3.3.3

Simplify the right side.

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Step 2.3.3.3.3.1

Cancel the common factor of

$8$ and

$- 6$ .

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Step 2.3.3.3.3.1.1

Factor

$2$ out of

$8 c$ .

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

$128 = 128 - 56 b - 63 c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

Step 2.3.3.3.3.1.2

Cancel the common factors.

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Step 2.3.3.3.3.1.2.1

Factor

$2$ out of

$- 6$ .

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

$128 = 128 - 56 b - 63 c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

Step 2.3.3.3.3.1.2.2

Cancel the common factor.

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

$128 = 128 - 56 b - 63 c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

Step 2.3.3.3.3.1.2.3

Rewrite the expression.

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

$128 = 128 - 56 b - 63 c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

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

$128 = 128 - 56 b - 63 c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

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

$128 = 128 - 56 b - 63 c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

Step 2.3.3.3.3.2

Move the negative in front of the fraction.

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

$128 = 128 - 56 b - 63 c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

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

$128 = 128 - 56 b - 63 c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

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

$128 = 128 - 56 b - 63 c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

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

$128 = 128 - 56 b - 63 c$

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

Step 2.3.4

Replace all occurrences of

$b$ with

$- \frac{4 c}{3}$ in each equation.

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Step 2.3.4.1

Replace all occurrences of

$b$ in

$128 = 128 - 56 b - 63 c$ with

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

$128 = 128 - 56 (- \frac{4 c}{3}) - 63 c$

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

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

Step 2.3.4.2

Simplify the right side.

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Step 2.3.4.2.1

Simplify

$128 - 56 (- \frac{4 c}{3}) - 63 c$ .

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Step 2.3.4.2.1.1

Multiply

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

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Step 2.3.4.2.1.1.1

Multiply

$- 1$ by

$- 56$ .

$128 = 128 + 56 (\frac{4 c}{3}) - 63 c$

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

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

Step 2.3.4.2.1.1.2

Combine

$56$ and

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

$128 = 128 + \frac{56 (4 c)}{3} - 63 c$

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

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

Step 2.3.4.2.1.1.3

Multiply

$4$ by

$56$ .

$128 = 128 + \frac{224 c}{3} - 63 c$

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

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

$128 = 128 + \frac{224 c}{3} - 63 c$

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

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

Step 2.3.4.2.1.2

To write

$- 63 c$ as a fraction with a common denominator, multiply by

$\frac{3}{3}$ .

$128 = 128 + \frac{224 c}{3} - 63 c \cdot \frac{3}{3}$

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

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

Step 2.3.4.2.1.3

Simplify terms.

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Step 2.3.4.2.1.3.1

Combine

$- 63 c$ and

$\frac{3}{3}$ .

$128 = 128 + \frac{224 c}{3} + \frac{- 63 c \cdot 3}{3}$

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

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

Step 2.3.4.2.1.3.2

Combine the numerators over the common denominator.

$128 = 128 + \frac{224 c - 63 c \cdot 3}{3}$

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

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

$128 = 128 + \frac{224 c - 63 c \cdot 3}{3}$

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

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

Step 2.3.4.2.1.4

Simplify the numerator.

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Step 2.3.4.2.1.4.1

Factor

$7 c$ out of

$224 c - 63 c \cdot 3$ .

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Step 2.3.4.2.1.4.1.1

Factor

$7 c$ out of

$224 c$ .

$128 = 128 + \frac{7 c (32) - 63 c \cdot 3}{3}$

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

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

Step 2.3.4.2.1.4.1.2

Factor

$7 c$ out of

$- 63 c \cdot 3$ .

$128 = 128 + \frac{7 c (32) + 7 c (- 9 \cdot 3)}{3}$

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

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

Step 2.3.4.2.1.4.1.3

Factor

$7 c$ out of

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

$128 = 128 + \frac{7 c (32 - 9 \cdot 3)}{3}$

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

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

$128 = 128 + \frac{7 c (32 - 9 \cdot 3)}{3}$

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

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

Step 2.3.4.2.1.4.2

Multiply

$- 9$ by

$3$ .

$128 = 128 + \frac{7 c (32 - 27)}{3}$

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

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

Step 2.3.4.2.1.4.3

Subtract

$27$ from

$32$ .

$128 = 128 + \frac{7 c \cdot 5}{3}$

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

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

Step 2.3.4.2.1.4.4

Multiply

$5$ by

$7$ .

$128 = 128 + \frac{35 c}{3}$

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

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

$128 = 128 + \frac{35 c}{3}$

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

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

$128 = 128 + \frac{35 c}{3}$

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

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

$128 = 128 + \frac{35 c}{3}$

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

$8 = 8 - 2 b - 3 c$

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

Step 2.3.4.3

Replace all occurrences of

$b$ in

$8 = 8 - 2 b - 3 c$ with

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

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

$128 = 128 + \frac{35 c}{3}$

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

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

Step 2.3.4.4

Simplify the right side.

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Step 2.3.4.4.1

Simplify

$8 - 2 (- \frac{4 c}{3}) - 3 c$ .

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Step 2.3.4.4.1.1

Multiply

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

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Step 2.3.4.4.1.1.1

Multiply

$- 1$ by

$- 2$ .

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

$128 = 128 + \frac{35 c}{3}$

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

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

Step 2.3.4.4.1.1.2

Combine

$2$ and

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

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

$128 = 128 + \frac{35 c}{3}$

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

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

Step 2.3.4.4.1.1.3

Multiply

$4$ by

$2$ .

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

$128 = 128 + \frac{35 c}{3}$

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

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

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

$128 = 128 + \frac{35 c}{3}$

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

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

Step 2.3.4.4.1.2

To write

$- 3 c$ as a fraction with a common denominator, multiply by

$\frac{3}{3}$ .

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

$128 = 128 + \frac{35 c}{3}$

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

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

Step 2.3.4.4.1.3

Simplify terms.

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Step 2.3.4.4.1.3.1

Combine

$- 3 c$ and

$\frac{3}{3}$ .

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

$128 = 128 + \frac{35 c}{3}$

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

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

Step 2.3.4.4.1.3.2

Combine the numerators over the common denominator.

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

$128 = 128 + \frac{35 c}{3}$

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

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

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

$128 = 128 + \frac{35 c}{3}$

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

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

Step 2.3.4.4.1.4

Simplify each term.

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Step 2.3.4.4.1.4.1

Simplify the numerator.

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Step 2.3.4.4.1.4.1.1

Factor

$c$ out of

$8 c - 3 c \cdot 3$ .

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Step 2.3.4.4.1.4.1.1.1

Factor

$c$ out of

$8 c$ .

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

$128 = 128 + \frac{35 c}{3}$

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

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

Step 2.3.4.4.1.4.1.1.2

Factor

$c$ out of

$- 3 c \cdot 3$ .

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

$128 = 128 + \frac{35 c}{3}$

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

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

Step 2.3.4.4.1.4.1.1.3

Factor

$c$ out of

$c \cdot 8 + c (- 3 \cdot 3)$ .

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

$128 = 128 + \frac{35 c}{3}$

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

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

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

$128 = 128 + \frac{35 c}{3}$

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

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

Step 2.3.4.4.1.4.1.2

Multiply

$- 3$ by

$3$ .

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

$128 = 128 + \frac{35 c}{3}$

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

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

Step 2.3.4.4.1.4.1.3

Subtract

$9$ from

$8$ .

$8 = 8 + \frac{c \cdot - 1}{3}$

$128 = 128 + \frac{35 c}{3}$

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

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

$8 = 8 + \frac{c \cdot - 1}{3}$

$128 = 128 + \frac{35 c}{3}$

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

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

Step 2.3.4.4.1.4.2

Move

$- 1$ to the left of

$c$ .

$8 = 8 + \frac{- 1 \cdot c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

Step 2.3.4.4.1.4.3

Move the negative in front of the fraction.

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

$162 = 162 - 72 b - 80 c$

$a = 2 - b - c$

Step 2.3.4.5

Replace all occurrences of

$b$ in

$162 = 162 - 72 b - 80 c$ with

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

$162 = 162 - 72 (- \frac{4 c}{3}) - 80 c$

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

$a = 2 - b - c$

Step 2.3.4.6

Simplify the right side.

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Step 2.3.4.6.1

Simplify

$162 - 72 (- \frac{4 c}{3}) - 80 c$ .

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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

Cancel the common factor of

$3$ .

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Step 2.3.4.6.1.1.1.1

Move the leading negative in

$- \frac{4 c}{3}$ into the numerator.

$162 = 162 - 72 \frac{- 4 c}{3} - 80 c$

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

$a = 2 - b - c$

Step 2.3.4.6.1.1.1.2

Factor

$3$ out of

$- 72$ .

$162 = 162 + 3 (- 24) (\frac{- 4 c}{3}) - 80 c$

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

$a = 2 - b - c$

Step 2.3.4.6.1.1.1.3

Cancel the common factor.

$162 = 162 + 3 \cdot (- 24 \frac{- 4 c}{3}) - 80 c$

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

$a = 2 - b - c$

Step 2.3.4.6.1.1.1.4

Rewrite the expression.

$162 = 162 - 24 (- 4 c) - 80 c$

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

$a = 2 - b - c$

$162 = 162 - 24 (- 4 c) - 80 c$

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

$a = 2 - b - c$

Step 2.3.4.6.1.1.2

Multiply

$- 4$ by

$- 24$ .

$162 = 162 + 96 c - 80 c$

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

$a = 2 - b - c$

$162 = 162 + 96 c - 80 c$

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

$a = 2 - b - c$

Step 2.3.4.6.1.2

Subtract

$80 c$ from

$96 c$ .

$162 = 162 + 16 c$

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

$a = 2 - b - c$

$162 = 162 + 16 c$

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

$a = 2 - b - c$

$162 = 162 + 16 c$

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

$a = 2 - b - c$

Step 2.3.4.7

Replace all occurrences of

$b$ in

$a = 2 - b - c$ with

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

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

$162 = 162 + 16 c$

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

Step 2.3.4.8

Simplify the right side.

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Step 2.3.4.8.1

Simplify

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

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Step 2.3.4.8.1.1

Multiply

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

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Step 2.3.4.8.1.1.1

Multiply

$- 1$ by

$- 1$ .

$a = 2 + 1 (\frac{4 c}{3}) - c$

$162 = 162 + 16 c$

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

Step 2.3.4.8.1.1.2

Multiply

$\frac{4 c}{3}$ by

$1$ .

$a = 2 + \frac{4 c}{3} - c$

$162 = 162 + 16 c$

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

$a = 2 + \frac{4 c}{3} - c$

$162 = 162 + 16 c$

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

Step 2.3.4.8.1.2

To write

$- c$ as a fraction with a common denominator, multiply by

$\frac{3}{3}$ .

$a = 2 + \frac{4 c}{3} - c \cdot \frac{3}{3}$

$162 = 162 + 16 c$

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

Step 2.3.4.8.1.3

Simplify terms.

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Step 2.3.4.8.1.3.1

Combine

$- c$ and

$\frac{3}{3}$ .

$a = 2 + \frac{4 c}{3} + \frac{- c \cdot 3}{3}$

$162 = 162 + 16 c$

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

Step 2.3.4.8.1.3.2

Combine the numerators over the common denominator.

$a = 2 + \frac{4 c - c \cdot 3}{3}$

$162 = 162 + 16 c$

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

$a = 2 + \frac{4 c - c \cdot 3}{3}$

$162 = 162 + 16 c$

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

Step 2.3.4.8.1.4

Simplify each term.

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Step 2.3.4.8.1.4.1

Simplify the numerator.

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Step 2.3.4.8.1.4.1.1

Factor

$c$ out of

$4 c - c \cdot 3$ .

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Step 2.3.4.8.1.4.1.1.1

Factor

$c$ out of

$4 c$ .

$a = 2 + \frac{c \cdot 4 - c \cdot 3}{3}$

$162 = 162 + 16 c$

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

Step 2.3.4.8.1.4.1.1.2

Factor

$c$ out of

$- c \cdot 3$ .

$a = 2 + \frac{c \cdot 4 + c (- 1 \cdot 3)}{3}$

$162 = 162 + 16 c$

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

Step 2.3.4.8.1.4.1.1.3

Factor

$c$ out of

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

$a = 2 + \frac{c (4 - 1 \cdot 3)}{3}$

$162 = 162 + 16 c$

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

$a = 2 + \frac{c (4 - 1 \cdot 3)}{3}$

$162 = 162 + 16 c$

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

Step 2.3.4.8.1.4.1.2

Multiply

$- 1$ by

$3$ .

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

$162 = 162 + 16 c$

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

Step 2.3.4.8.1.4.1.3

Subtract

$3$ from

$4$ .

$a = 2 + \frac{c \cdot 1}{3}$

$162 = 162 + 16 c$

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

$a = 2 + \frac{c \cdot 1}{3}$

$162 = 162 + 16 c$

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

Step 2.3.4.8.1.4.2

Multiply

$c$ by

$1$ .

$a = 2 + \frac{c}{3}$

$162 = 162 + 16 c$

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

$a = 2 + \frac{c}{3}$

$162 = 162 + 16 c$

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

$a = 2 + \frac{c}{3}$

$162 = 162 + 16 c$

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

$a = 2 + \frac{c}{3}$

$162 = 162 + 16 c$

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

$a = 2 + \frac{c}{3}$

$162 = 162 + 16 c$

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

Step 2.3.5

Solve for

$c$ in

$162 = 162 + 16 c$ .

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Step 2.3.5.1

Rewrite the equation as

$162 + 16 c = 162$ .

$162 + 16 c = 162$

$a = 2 + \frac{c}{3}$

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

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

$162$ from both sides of the equation.

$16 c = 162 - 162$

$a = 2 + \frac{c}{3}$

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

Step 2.3.5.2.2

Subtract

$162$ from

$162$ .

$16 c = 0$

$a = 2 + \frac{c}{3}$

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

$16 c = 0$

$a = 2 + \frac{c}{3}$

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

Step 2.3.5.3

Divide each term in

$16 c = 0$ by

$16$ and simplify.

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Step 2.3.5.3.1

Divide each term in

$16 c = 0$ by

$16$ .

$\frac{16 c}{16} = \frac{0}{16}$

$a = 2 + \frac{c}{3}$

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

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.

$\frac{16 c}{16} = \frac{0}{16}$

$a = 2 + \frac{c}{3}$

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

Step 2.3.5.3.2.1.2

Divide

$c$ by

$1$ .

$c = \frac{0}{16}$

$a = 2 + \frac{c}{3}$

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

$c = \frac{0}{16}$

$a = 2 + \frac{c}{3}$

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

$c = \frac{0}{16}$

$a = 2 + \frac{c}{3}$

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

Step 2.3.5.3.3

Simplify the right side.

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Step 2.3.5.3.3.1

Divide

$0$ by

$16$ .

$c = 0$

$a = 2 + \frac{c}{3}$

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

$c = 0$

$a = 2 + \frac{c}{3}$

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

$c = 0$

$a = 2 + \frac{c}{3}$

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

$c = 0$

$a = 2 + \frac{c}{3}$

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

Step 2.3.6

Replace all occurrences of

$c$ with

$0$ in each equation.

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Step 2.3.6.1

Replace all occurrences of

$c$ in

$a = 2 + \frac{c}{3}$ with

$0$ .

$a = 2 + \frac{0}{3}$

$c = 0$

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

Step 2.3.6.2

Simplify the right side.

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Step 2.3.6.2.1

Simplify

$2 + \frac{0}{3}$ .

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Step 2.3.6.2.1.1

Divide

$0$ by

$3$ .

$a = 2 + 0$

$c = 0$

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

Step 2.3.6.2.1.2

Add

$2$ and

$0$ .

$a = 2$

$c = 0$

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

$a = 2$

$c = 0$

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

$a = 2$

$c = 0$

$8 = 8 - \frac{c}{3}$

$128 = 128 + \frac{35 c}{3}$

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

Step 2.3.6.3

Replace all occurrences of

$c$ in

$8 = 8 - \frac{c}{3}$ with

$0$ .

$8 = 8 - \frac{0}{3}$

$a = 2$

$c = 0$

$128 = 128 + \frac{35 c}{3}$

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

Step 2.3.6.4

Simplify the right side.

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Step 2.3.6.4.1

Simplify

$8 - \frac{0}{3}$ .

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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

$0$ by

$3$ .

$8 = 8 - 0$

$a = 2$

$c = 0$

$128 = 128 + \frac{35 c}{3}$

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

Step 2.3.6.4.1.1.2

Multiply

$- 1$ by

$0$ .

$8 = 8 + 0$

$a = 2$

$c = 0$

$128 = 128 + \frac{35 c}{3}$

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

$8 = 8 + 0$

$a = 2$

$c = 0$

$128 = 128 + \frac{35 c}{3}$

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

Step 2.3.6.4.1.2

Add

$8$ and

$0$ .

$8 = 8$

$a = 2$

$c = 0$

$128 = 128 + \frac{35 c}{3}$

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

$8 = 8$

$a = 2$

$c = 0$

$128 = 128 + \frac{35 c}{3}$

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

$8 = 8$

$a = 2$

$c = 0$

$128 = 128 + \frac{35 c}{3}$

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

Step 2.3.6.5

Replace all occurrences of

$c$ in

$128 = 128 + \frac{35 c}{3}$ with

$0$ .

$128 = 128 + \frac{35 (0)}{3}$

$8 = 8$

$a = 2$

$c = 0$

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

Step 2.3.6.6

Simplify the right side.

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Step 2.3.6.6.1

Simplify

$128 + \frac{35 (0)}{3}$ .

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Step 2.3.6.6.1.1

Multiply

$35$ by

$0$ .

$128 = 128 + \frac{0}{3}$

$8 = 8$

$a = 2$

$c = 0$

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

Step 2.3.6.6.1.2

Divide

$0$ by

$3$ .

$128 = 128 + 0$

$8 = 8$

$a = 2$

$c = 0$

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

Step 2.3.6.6.1.3

Add

$128$ and

$0$ .

$128 = 128$

$8 = 8$

$a = 2$

$c = 0$

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

$128 = 128$

$8 = 8$

$a = 2$

$c = 0$

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

$128 = 128$

$8 = 8$

$a = 2$

$c = 0$

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

Step 2.3.6.7

Replace all occurrences of

$c$ in

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

$0$ .

$b = - \frac{4 (0)}{3}$

$128 = 128$

$8 = 8$

$a = 2$

$c = 0$

Step 2.3.6.8

Simplify the right side.

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Step 2.3.6.8.1

Simplify

$- \frac{4 (0)}{3}$ .

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Step 2.3.6.8.1.1

Cancel the common factor of

$0$ and

$3$ .

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Step 2.3.6.8.1.1.1

Factor

$3$ out of

$4 (0)$ .

$b = - \frac{3 (4 \cdot (0))}{3}$

$128 = 128$

$8 = 8$

$a = 2$

$c = 0$

Step 2.3.6.8.1.1.2

Cancel the common factors.

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Step 2.3.6.8.1.1.2.1

Factor

$3$ out of

$3$ .

$b = - \frac{3 (4 \cdot (0))}{3 (1)}$

$128 = 128$

$8 = 8$

$a = 2$

$c = 0$

Step 2.3.6.8.1.1.2.2

Cancel the common factor.

$b = - \frac{3 (4 \cdot (0))}{3 \cdot 1}$

$128 = 128$

$8 = 8$

$a = 2$

$c = 0$

Step 2.3.6.8.1.1.2.3

Rewrite the expression.

$b = - \frac{4 \cdot (0)}{1}$

$128 = 128$

$8 = 8$

$a = 2$

$c = 0$

Step 2.3.6.8.1.1.2.4

Divide

$4 \cdot (0)$ by

$1$ .

$b = - (4 \cdot (0))$

$128 = 128$

$8 = 8$

$a = 2$

$c = 0$

$b = - (4 \cdot (0))$

$128 = 128$

$8 = 8$

$a = 2$

$c = 0$

$b = - (4 \cdot (0))$

$128 = 128$

$8 = 8$

$a = 2$

$c = 0$

Step 2.3.6.8.1.2

Multiply

$- (4 \cdot (0))$ .

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Step 2.3.6.8.1.2.1

Multiply

$4$ by

$0$ .

$b = - 0$

$128 = 128$

$8 = 8$

$a = 2$

$c = 0$

Step 2.3.6.8.1.2.2

Multiply

$- 1$ by

$0$ .

$b = 0$

$128 = 128$

$8 = 8$

$a = 2$

$c = 0$

$b = 0$

$128 = 128$

$8 = 8$

$a = 2$

$c = 0$

$b = 0$

$128 = 128$

$8 = 8$

$a = 2$

$c = 0$

$b = 0$

$128 = 128$

$8 = 8$

$a = 2$

$c = 0$

$b = 0$

$128 = 128$

$8 = 8$

$a = 2$

$c = 0$

Step 2.3.7

Remove any equations from the system that are always true.

$b = 0$

$a = 2$

$c = 0$

Step 2.3.8

List all of the solutions.

$b = 0, a = 2, c = 0$

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

$a = 2$ ,

$b = 0$ ,

$c = 0$ , 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.

$y = 2 \cdot 1 + (0) \cdot (1) + 0$

Step 2.4.1.1.2

Multiply

$2$ by

$1$ .

$y = 2 + (0) \cdot (1) + 0$

Step 2.4.1.1.3

Multiply

$0$ by

$1$ .

$y = 2 + 0 + 0$

Step 2.4.1.2

Simplify by adding numbers.

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Step 2.4.1.2.1

Add

$2$ and

$0$ .

$y = 2 + 0$

Step 2.4.1.2.2

Add

$2$ and

$0$ .

$y = 2$

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

$y = 2$ and

$q (x) = 2$ .

$2 = 2$

Step 2.4.3

Calculate the value of

$y$ such that

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

$a = 2$ ,

$b = 0$ ,

$c = 0$ , 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

$2$ .

$y = 2 \cdot 81 + (0) \cdot (9) + 0$

Step 2.4.3.1.2

Multiply

$2$ by

$81$ .

$y = 162 + (0) \cdot (9) + 0$

Step 2.4.3.1.3

Multiply

$0$ by

$9$ .

$y = 162 + 0 + 0$

Step 2.4.3.2

Simplify by adding numbers.

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Step 2.4.3.2.1

Add

$162$ and

$0$ .

$y = 162 + 0$

Step 2.4.3.2.2

Add

$162$ and

$0$ .

$y = 162$

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

$y = 162$ and

$q (x) = 162$ .

$162 = 162$

Step 2.4.5

Calculate the value of

$y$ such that

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

$a = 2$ ,

$b = 0$ ,

$c = 0$ , 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

$1$ .

$y = 2 \cdot {(2)}^{2} + (0) \cdot (2) + 0$

Step 2.4.5.1.1.1.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = 2^{1 + 2} + (0) \cdot (2) + 0$

Step 2.4.5.1.1.2

Add

$1$ and

$2$ .

$y = 2^{3} + (0) \cdot (2) + 0$

Step 2.4.5.1.2

Raise

$2$ to the power of

$3$ .

$y = 8 + (0) \cdot (2) + 0$

Step 2.4.5.1.3

Multiply

$0$ by

$2$ .

$y = 8 + 0 + 0$

Step 2.4.5.2

Simplify by adding numbers.

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Step 2.4.5.2.1

Add

$8$ and

$0$ .

$y = 8 + 0$

Step 2.4.5.2.2

Add

$8$ and

$0$ .

$y = 8$

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

$y = 8$ and

$q (x) = 8$ .

$8 = 8$

Step 2.4.7

Calculate the value of

$y$ such that

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

$a = 2$ ,

$b = 0$ ,

$c = 0$ , 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

$2$ .

$y = 2 \cdot 64 + (0) \cdot (8) + 0$

Step 2.4.7.1.2

Multiply

$2$ by

$64$ .

$y = 128 + (0) \cdot (8) + 0$

Step 2.4.7.1.3

Multiply

$0$ by

$8$ .

$y = 128 + 0 + 0$

Step 2.4.7.2

Simplify by adding numbers.

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Step 2.4.7.2.1

Add

$128$ and

$0$ .

$y = 128 + 0$

Step 2.4.7.2.2

Add

$128$ and

$0$ .

$y = 128$

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

$y = 128$ and

$q (x) = 128$ .

$128 = 128$

Step 2.4.9

Calculate the value of

$y$ such that

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

$a = 2$ ,

$b = 0$ ,

$c = 0$ , 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

$2$ .

$y = 2 \cdot 9 + (0) \cdot (3) + 0$

Step 2.4.9.1.2

Multiply

$2$ by

$9$ .

$y = 18 + (0) \cdot (3) + 0$

Step 2.4.9.1.3

Multiply

$0$ by

$3$ .

$y = 18 + 0 + 0$

Step 2.4.9.2

Simplify by adding numbers.

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Step 2.4.9.2.1

Add

$18$ and

$0$ .

$y = 18 + 0$

Step 2.4.9.2.2

Add

$18$ and

$0$ .

$y = 18$

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

$y = 18$ and

$q (x) = 18$ .

$18 = 18$

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

$y = 2 x^{2}$ .

$y = 2 x^{2}$

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