Algebra Examples

$12 y - 8 x y - 3 y^{2} = 0$ $12 x - 6 x y - 4 x^{2} = 0$

Step 1

Solve for $x$ in $12 y - 8 x y - 3 y^{2} = 0$ .

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Step 1.1

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

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Step 1.1.1

Subtract $12 y$ from both sides of the equation.

$- 8 x y - 3 y^{2} = - 12 y$

$12 x - 6 x y - 4 x^{2} = 0$

Step 1.1.2

Add $3 y^{2}$ to both sides of the equation.

$- 8 x y = - 12 y + 3 y^{2}$

$12 x - 6 x y - 4 x^{2} = 0$

$- 8 x y = - 12 y + 3 y^{2}$

$12 x - 6 x y - 4 x^{2} = 0$

Step 1.2

Divide each term in $- 8 x y = - 12 y + 3 y^{2}$ by $- 8 y$ and simplify.

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Step 1.2.1

Divide each term in $- 8 x y = - 12 y + 3 y^{2}$ by $- 8 y$ .

$\frac{- 8 x y}{- 8 y} = \frac{- 12 y}{- 8 y} + \frac{3 y^{2}}{- 8 y}$

$12 x - 6 x y - 4 x^{2} = 0$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $- 8$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{- 8 x y}{- 8 y} = \frac{- 12 y}{- 8 y} + \frac{3 y^{2}}{- 8 y}$

$12 x - 6 x y - 4 x^{2} = 0$

Step 1.2.2.1.2

Rewrite the expression.

$\frac{x y}{y} = \frac{- 12 y}{- 8 y} + \frac{3 y^{2}}{- 8 y}$

$12 x - 6 x y - 4 x^{2} = 0$

$\frac{x y}{y} = \frac{- 12 y}{- 8 y} + \frac{3 y^{2}}{- 8 y}$

$12 x - 6 x y - 4 x^{2} = 0$

Step 1.2.2.2

Cancel the common factor of $y$ .

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Step 1.2.2.2.1

Cancel the common factor.

$\frac{x y}{y} = \frac{- 12 y}{- 8 y} + \frac{3 y^{2}}{- 8 y}$

$12 x - 6 x y - 4 x^{2} = 0$

Step 1.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 12 y}{- 8 y} + \frac{3 y^{2}}{- 8 y}$

$12 x - 6 x y - 4 x^{2} = 0$

$x = \frac{- 12 y}{- 8 y} + \frac{3 y^{2}}{- 8 y}$

$12 x - 6 x y - 4 x^{2} = 0$

$x = \frac{- 12 y}{- 8 y} + \frac{3 y^{2}}{- 8 y}$

$12 x - 6 x y - 4 x^{2} = 0$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Cancel the common factor of $- 12$ and $- 8$ .

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Step 1.2.3.1.1.1

Factor $- 4$ out of $- 12 y$ .

$x = \frac{- 4 (3 y)}{- 8 y} + \frac{3 y^{2}}{- 8 y}$

$12 x - 6 x y - 4 x^{2} = 0$

Step 1.2.3.1.1.2

Cancel the common factors.

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Step 1.2.3.1.1.2.1

Factor $- 4$ out of $- 8 y$ .

$x = \frac{- 4 (3 y)}{- 4 (2 y)} + \frac{3 y^{2}}{- 8 y}$

$12 x - 6 x y - 4 x^{2} = 0$

Step 1.2.3.1.1.2.2

Cancel the common factor.

$x = \frac{- 4 (3 y)}{- 4 (2 y)} + \frac{3 y^{2}}{- 8 y}$

$12 x - 6 x y - 4 x^{2} = 0$

Step 1.2.3.1.1.2.3

Rewrite the expression.

$x = \frac{3 y}{2 y} + \frac{3 y^{2}}{- 8 y}$

$12 x - 6 x y - 4 x^{2} = 0$

$x = \frac{3 y}{2 y} + \frac{3 y^{2}}{- 8 y}$

$12 x - 6 x y - 4 x^{2} = 0$

$x = \frac{3 y}{2 y} + \frac{3 y^{2}}{- 8 y}$

$12 x - 6 x y - 4 x^{2} = 0$

Step 1.2.3.1.2

Cancel the common factor of $y$ .

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Step 1.2.3.1.2.1

Cancel the common factor.

$x = \frac{3 y}{2 y} + \frac{3 y^{2}}{- 8 y}$

$12 x - 6 x y - 4 x^{2} = 0$

Step 1.2.3.1.2.2

Rewrite the expression.

$x = \frac{3}{2} + \frac{3 y^{2}}{- 8 y}$

$12 x - 6 x y - 4 x^{2} = 0$

$x = \frac{3}{2} + \frac{3 y^{2}}{- 8 y}$

$12 x - 6 x y - 4 x^{2} = 0$

Step 1.2.3.1.3

Cancel the common factor of $y^{2}$ and $y$ .

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Step 1.2.3.1.3.1

Factor $y$ out of $3 y^{2}$ .

$x = \frac{3}{2} + \frac{y (3 y)}{- 8 y}$

$12 x - 6 x y - 4 x^{2} = 0$

Step 1.2.3.1.3.2

Cancel the common factors.

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Step 1.2.3.1.3.2.1

Factor $y$ out of $- 8 y$ .

$x = \frac{3}{2} + \frac{y (3 y)}{y \cdot - 8}$

$12 x - 6 x y - 4 x^{2} = 0$

Step 1.2.3.1.3.2.2

Cancel the common factor.

$x = \frac{3}{2} + \frac{y (3 y)}{y \cdot - 8}$

$12 x - 6 x y - 4 x^{2} = 0$

Step 1.2.3.1.3.2.3

Rewrite the expression.

$x = \frac{3}{2} + \frac{3 y}{- 8}$

$12 x - 6 x y - 4 x^{2} = 0$

$x = \frac{3}{2} + \frac{3 y}{- 8}$

$12 x - 6 x y - 4 x^{2} = 0$

$x = \frac{3}{2} + \frac{3 y}{- 8}$

$12 x - 6 x y - 4 x^{2} = 0$

Step 1.2.3.1.4

Move the negative in front of the fraction.

$x = \frac{3}{2} - \frac{3 y}{8}$

$12 x - 6 x y - 4 x^{2} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

$12 x - 6 x y - 4 x^{2} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

$12 x - 6 x y - 4 x^{2} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

$12 x - 6 x y - 4 x^{2} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

$12 x - 6 x y - 4 x^{2} = 0$

Step 2

Replace all occurrences of $x$ with $\frac{3}{2} - \frac{3 y}{8}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $12 x - 6 x y - 4 x^{2} = 0$ with $\frac{3}{2} - \frac{3 y}{8}$ .

$12 (\frac{3}{2} - \frac{3 y}{8}) - 6 (\frac{3}{2} - \frac{3 y}{8}) \cdot y - 4 {(\frac{3}{2} - \frac{3 y}{8})}^{2} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $12 (\frac{3}{2} - \frac{3 y}{8}) - 6 (\frac{3}{2} - \frac{3 y}{8}) \cdot y - 4 {(\frac{3}{2} - \frac{3 y}{8})}^{2}$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Apply the distributive property.

$12 (\frac{3}{2}) + 12 (- \frac{3 y}{8}) - 6 (\frac{3}{2} - \frac{3 y}{8}) \cdot y - 4 {(\frac{3}{2} - \frac{3 y}{8})}^{2} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Factor $2$ out of $12$ .

$2 (6) (\frac{3}{2}) + 12 (- \frac{3 y}{8}) - 6 (\frac{3}{2} - \frac{3 y}{8}) \cdot y - 4 {(\frac{3}{2} - \frac{3 y}{8})}^{2} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.2.2

Cancel the common factor.

$2 \cdot (6 (\frac{3}{2})) + 12 (- \frac{3 y}{8}) - 6 (\frac{3}{2} - \frac{3 y}{8}) \cdot y - 4 {(\frac{3}{2} - \frac{3 y}{8})}^{2} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.2.3

Rewrite the expression.

$6 \cdot 3 + 12 (- \frac{3 y}{8}) - 6 (\frac{3}{2} - \frac{3 y}{8}) \cdot y - 4 {(\frac{3}{2} - \frac{3 y}{8})}^{2} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

$6 \cdot 3 + 12 (- \frac{3 y}{8}) - 6 (\frac{3}{2} - \frac{3 y}{8}) \cdot y - 4 {(\frac{3}{2} - \frac{3 y}{8})}^{2} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.3

Multiply $6$ by $3$ .

$18 + 12 (- \frac{3 y}{8}) - 6 (\frac{3}{2} - \frac{3 y}{8}) \cdot y - 4 {(\frac{3}{2} - \frac{3 y}{8})}^{2} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.4

Cancel the common factor of $4$ .

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Step 2.2.1.1.4.1

Move the leading negative in $- \frac{3 y}{8}$ into the numerator.

$18 + 12 (\frac{- 3 y}{8}) - 6 (\frac{3}{2} - \frac{3 y}{8}) \cdot y - 4 {(\frac{3}{2} - \frac{3 y}{8})}^{2} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.4.2

Factor $4$ out of $12$ .

$18 + 4 (3) (\frac{- 3 y}{8}) - 6 (\frac{3}{2} - \frac{3 y}{8}) \cdot y - 4 {(\frac{3}{2} - \frac{3 y}{8})}^{2} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.4.3

Factor $4$ out of $8$ .

$18 + 4 \cdot (3 (\frac{- 3 y}{4 \cdot 2})) - 6 (\frac{3}{2} - \frac{3 y}{8}) \cdot y - 4 {(\frac{3}{2} - \frac{3 y}{8})}^{2} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.4.4

Cancel the common factor.

$18 + 4 \cdot (3 (\frac{- 3 y}{4 \cdot 2})) - 6 (\frac{3}{2} - \frac{3 y}{8}) \cdot y - 4 {(\frac{3}{2} - \frac{3 y}{8})}^{2} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.4.5

Rewrite the expression.

$18 + 3 (\frac{- 3 y}{2}) - 6 (\frac{3}{2} - \frac{3 y}{8}) \cdot y - 4 {(\frac{3}{2} - \frac{3 y}{8})}^{2} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

$18 + 3 (\frac{- 3 y}{2}) - 6 (\frac{3}{2} - \frac{3 y}{8}) \cdot y - 4 {(\frac{3}{2} - \frac{3 y}{8})}^{2} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.5

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

$18 + \frac{3 (- 3 y)}{2} - 6 (\frac{3}{2} - \frac{3 y}{8}) \cdot y - 4 {(\frac{3}{2} - \frac{3 y}{8})}^{2} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.6

Multiply $- 3$ by $3$ .

$18 + \frac{- 9 y}{2} - 6 (\frac{3}{2} - \frac{3 y}{8}) \cdot y - 4 {(\frac{3}{2} - \frac{3 y}{8})}^{2} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.7

Move the negative in front of the fraction.

$18 - \frac{9 y}{2} - 6 (\frac{3}{2} - \frac{3 y}{8}) \cdot y - 4 {(\frac{3}{2} - \frac{3 y}{8})}^{2} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.8

Apply the distributive property.

$18 - \frac{9 y}{2} + (- 6 (\frac{3}{2}) - 6 (- \frac{3 y}{8})) \cdot y - 4 {(\frac{3}{2} - \frac{3 y}{8})}^{2} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.9

Cancel the common factor of $2$ .

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Step 2.2.1.1.9.1

Factor $2$ out of $- 6$ .

$18 - \frac{9 y}{2} + (2 (- 3) (\frac{3}{2}) - 6 (- \frac{3 y}{8})) \cdot y - 4 {(\frac{3}{2} - \frac{3 y}{8})}^{2} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.9.2

Cancel the common factor.

$18 - \frac{9 y}{2} + (2 \cdot (- 3 (\frac{3}{2})) - 6 (- \frac{3 y}{8})) \cdot y - 4 {(\frac{3}{2} - \frac{3 y}{8})}^{2} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.9.3

Rewrite the expression.

$18 - \frac{9 y}{2} + (- 3 \cdot 3 - 6 (- \frac{3 y}{8})) \cdot y - 4 {(\frac{3}{2} - \frac{3 y}{8})}^{2} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

$18 - \frac{9 y}{2} + (- 3 \cdot 3 - 6 (- \frac{3 y}{8})) \cdot y - 4 {(\frac{3}{2} - \frac{3 y}{8})}^{2} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.10

Multiply $- 3$ by $3$ .

$18 - \frac{9 y}{2} + (- 9 - 6 (- \frac{3 y}{8})) \cdot y - 4 {(\frac{3}{2} - \frac{3 y}{8})}^{2} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.11

Cancel the common factor of $2$ .

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Step 2.2.1.1.11.1

Move the leading negative in $- \frac{3 y}{8}$ into the numerator.

$18 - \frac{9 y}{2} + (- 9 - 6 \frac{- 3 y}{8}) \cdot y - 4 {(\frac{3}{2} - \frac{3 y}{8})}^{2} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.11.2

Factor $2$ out of $- 6$ .

$18 - \frac{9 y}{2} + (- 9 + 2 (- 3) (\frac{- 3 y}{8})) \cdot y - 4 {(\frac{3}{2} - \frac{3 y}{8})}^{2} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.11.3

Factor $2$ out of $8$ .

$18 - \frac{9 y}{2} + (- 9 + 2 \cdot (- 3 \frac{- 3 y}{2 \cdot 4})) \cdot y - 4 {(\frac{3}{2} - \frac{3 y}{8})}^{2} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.11.4

Cancel the common factor.

$18 - \frac{9 y}{2} + (- 9 + 2 \cdot (- 3 \frac{- 3 y}{2 \cdot 4})) \cdot y - 4 {(\frac{3}{2} - \frac{3 y}{8})}^{2} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.11.5

Rewrite the expression.

$18 - \frac{9 y}{2} + (- 9 - 3 \frac{- 3 y}{4}) \cdot y - 4 {(\frac{3}{2} - \frac{3 y}{8})}^{2} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

$18 - \frac{9 y}{2} + (- 9 - 3 \frac{- 3 y}{4}) \cdot y - 4 {(\frac{3}{2} - \frac{3 y}{8})}^{2} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.12

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

$18 - \frac{9 y}{2} + (- 9 + \frac{- 3 (- 3 y)}{4}) \cdot y - 4 {(\frac{3}{2} - \frac{3 y}{8})}^{2} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.13

Multiply $- 3$ by $- 3$ .

$18 - \frac{9 y}{2} + (- 9 + \frac{9 y}{4}) \cdot y - 4 {(\frac{3}{2} - \frac{3 y}{8})}^{2} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.14

Apply the distributive property.

$18 - \frac{9 y}{2} - 9 y + \frac{9 y}{4} \cdot y - 4 {(\frac{3}{2} - \frac{3 y}{8})}^{2} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.15

Multiply $\frac{9 y}{4} y$ .

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Step 2.2.1.1.15.1

Combine $\frac{9 y}{4}$ and $y$ .

$18 - \frac{9 y}{2} - 9 y + \frac{9 y \cdot y}{4} - 4 {(\frac{3}{2} - \frac{3 y}{8})}^{2} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.15.2

Raise $y$ to the power of $1$ .

$18 - \frac{9 y}{2} - 9 y + \frac{9 (y \cdot y)}{4} - 4 {(\frac{3}{2} - \frac{3 y}{8})}^{2} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.15.3

Raise $y$ to the power of $1$ .

$18 - \frac{9 y}{2} - 9 y + \frac{9 (y \cdot y)}{4} - 4 {(\frac{3}{2} - \frac{3 y}{8})}^{2} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.15.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{1 + 1}}{4} - 4 {(\frac{3}{2} - \frac{3 y}{8})}^{2} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.15.5

Add $1$ and $1$ .

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 4 {(\frac{3}{2} - \frac{3 y}{8})}^{2} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 4 {(\frac{3}{2} - \frac{3 y}{8})}^{2} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.16

Rewrite ${(\frac{3}{2} - \frac{3 y}{8})}^{2}$ as $(\frac{3}{2} - \frac{3 y}{8}) (\frac{3}{2} - \frac{3 y}{8})$ .

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 4 ((\frac{3}{2} - \frac{3 y}{8}) (\frac{3}{2} - \frac{3 y}{8})) = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.17

Expand $(\frac{3}{2} - \frac{3 y}{8}) (\frac{3}{2} - \frac{3 y}{8})$ using the FOIL Method.

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Step 2.2.1.1.17.1

Apply the distributive property.

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 4 (\frac{3}{2} \cdot (\frac{3}{2} - \frac{3 y}{8}) - \frac{3 y}{8} \cdot (\frac{3}{2} - \frac{3 y}{8})) = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.17.2

Apply the distributive property.

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 4 (\frac{3}{2} \cdot \frac{3}{2} + \frac{3}{2} \cdot (- \frac{3 y}{8}) - \frac{3 y}{8} \cdot (\frac{3}{2} - \frac{3 y}{8})) = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.17.3

Apply the distributive property.

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 4 (\frac{3}{2} \cdot \frac{3}{2} + \frac{3}{2} \cdot (- \frac{3 y}{8}) - \frac{3 y}{8} \cdot \frac{3}{2} - \frac{3 y}{8} \cdot (- \frac{3 y}{8})) = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 4 (\frac{3}{2} \cdot \frac{3}{2} + \frac{3}{2} \cdot (- \frac{3 y}{8}) - \frac{3 y}{8} \cdot \frac{3}{2} - \frac{3 y}{8} \cdot (- \frac{3 y}{8})) = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.18

Simplify and combine like terms.

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Step 2.2.1.1.18.1

Simplify each term.

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Step 2.2.1.1.18.1.1

Multiply $\frac{3}{2} \cdot \frac{3}{2}$ .

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Step 2.2.1.1.18.1.1.1

Multiply $\frac{3}{2}$ by $\frac{3}{2}$ .

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 4 (\frac{3 \cdot 3}{2 \cdot 2} + \frac{3}{2} \cdot (- \frac{3 y}{8}) - \frac{3 y}{8} \cdot \frac{3}{2} - \frac{3 y}{8} \cdot (- \frac{3 y}{8})) = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.18.1.1.2

Multiply $3$ by $3$ .

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 4 (\frac{9}{2 \cdot 2} + \frac{3}{2} \cdot (- \frac{3 y}{8}) - \frac{3 y}{8} \cdot \frac{3}{2} - \frac{3 y}{8} \cdot (- \frac{3 y}{8})) = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.18.1.1.3

Multiply $2$ by $2$ .

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 4 (\frac{9}{4} + \frac{3}{2} \cdot (- \frac{3 y}{8}) - \frac{3 y}{8} \cdot \frac{3}{2} - \frac{3 y}{8} \cdot (- \frac{3 y}{8})) = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 4 (\frac{9}{4} + \frac{3}{2} \cdot (- \frac{3 y}{8}) - \frac{3 y}{8} \cdot \frac{3}{2} - \frac{3 y}{8} \cdot (- \frac{3 y}{8})) = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.18.1.2

Multiply $\frac{3}{2} (- \frac{3 y}{8})$ .

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Step 2.2.1.1.18.1.2.1

Multiply $\frac{3}{2}$ by $\frac{3 y}{8}$ .

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 4 (\frac{9}{4} - \frac{3 (3 y)}{2 \cdot 8} - \frac{3 y}{8} \cdot \frac{3}{2} - \frac{3 y}{8} \cdot (- \frac{3 y}{8})) = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.18.1.2.2

Multiply $3$ by $3$ .

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 4 (\frac{9}{4} - \frac{9 y}{2 \cdot 8} - \frac{3 y}{8} \cdot \frac{3}{2} - \frac{3 y}{8} \cdot (- \frac{3 y}{8})) = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.18.1.2.3

Multiply $2$ by $8$ .

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 4 (\frac{9}{4} - \frac{9 y}{16} - \frac{3 y}{8} \cdot \frac{3}{2} - \frac{3 y}{8} \cdot (- \frac{3 y}{8})) = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 4 (\frac{9}{4} - \frac{9 y}{16} - \frac{3 y}{8} \cdot \frac{3}{2} - \frac{3 y}{8} \cdot (- \frac{3 y}{8})) = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.18.1.3

Multiply $- \frac{3 y}{8} \cdot \frac{3}{2}$ .

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Step 2.2.1.1.18.1.3.1

Multiply $\frac{3}{2}$ by $\frac{3 y}{8}$ .

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 4 (\frac{9}{4} - \frac{9 y}{16} - \frac{3 (3 y)}{2 \cdot 8} - \frac{3 y}{8} \cdot (- \frac{3 y}{8})) = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.18.1.3.2

Multiply $3$ by $3$ .

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 4 (\frac{9}{4} - \frac{9 y}{16} - \frac{9 y}{2 \cdot 8} - \frac{3 y}{8} \cdot (- \frac{3 y}{8})) = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.18.1.3.3

Multiply $2$ by $8$ .

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 4 (\frac{9}{4} - \frac{9 y}{16} - \frac{9 y}{16} - \frac{3 y}{8} \cdot (- \frac{3 y}{8})) = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 4 (\frac{9}{4} - \frac{9 y}{16} - \frac{9 y}{16} - \frac{3 y}{8} \cdot (- \frac{3 y}{8})) = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.18.1.4

Multiply $- \frac{3 y}{8} (- \frac{3 y}{8})$ .

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Step 2.2.1.1.18.1.4.1

Multiply $- 1$ by $- 1$ .

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 4 (\frac{9}{4} - \frac{9 y}{16} - \frac{9 y}{16} + 1 (\frac{3 y}{8}) \cdot \frac{3 y}{8}) = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.18.1.4.2

Multiply $\frac{3 y}{8}$ by $1$ .

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 4 (\frac{9}{4} - \frac{9 y}{16} - \frac{9 y}{16} + \frac{3 y}{8} \cdot \frac{3 y}{8}) = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.18.1.4.3

Multiply $\frac{3 y}{8}$ by $\frac{3 y}{8}$ .

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 4 (\frac{9}{4} - \frac{9 y}{16} - \frac{9 y}{16} + \frac{3 y (3 y)}{8 \cdot 8}) = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.18.1.4.4

Multiply $3$ by $3$ .

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 4 (\frac{9}{4} - \frac{9 y}{16} - \frac{9 y}{16} + \frac{9 y \cdot y}{8 \cdot 8}) = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.18.1.4.5

Raise $y$ to the power of $1$ .

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 4 (\frac{9}{4} - \frac{9 y}{16} - \frac{9 y}{16} + \frac{9 (y \cdot y)}{8 \cdot 8}) = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.18.1.4.6

Raise $y$ to the power of $1$ .

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 4 (\frac{9}{4} - \frac{9 y}{16} - \frac{9 y}{16} + \frac{9 (y \cdot y)}{8 \cdot 8}) = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.18.1.4.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 4 (\frac{9}{4} - \frac{9 y}{16} - \frac{9 y}{16} + \frac{9 y^{1 + 1}}{8 \cdot 8}) = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.18.1.4.8

Add $1$ and $1$ .

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 4 (\frac{9}{4} - \frac{9 y}{16} - \frac{9 y}{16} + \frac{9 y^{2}}{8 \cdot 8}) = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.18.1.4.9

Multiply $8$ by $8$ .

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 4 (\frac{9}{4} - \frac{9 y}{16} - \frac{9 y}{16} + \frac{9 y^{2}}{64}) = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 4 (\frac{9}{4} - \frac{9 y}{16} - \frac{9 y}{16} + \frac{9 y^{2}}{64}) = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 4 (\frac{9}{4} - \frac{9 y}{16} - \frac{9 y}{16} + \frac{9 y^{2}}{64}) = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.18.2

Subtract $\frac{9 y}{16}$ from $- \frac{9 y}{16}$ .

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 4 (\frac{9}{4} - 2 \frac{9 y}{16} + \frac{9 y^{2}}{64}) = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 4 (\frac{9}{4} - 2 \frac{9 y}{16} + \frac{9 y^{2}}{64}) = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.19

Simplify each term.

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Step 2.2.1.1.19.1

Cancel the common factor of $2$ .

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Step 2.2.1.1.19.1.1

Factor $2$ out of $- 2$ .

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 4 (\frac{9}{4} + 2 (- 1) (\frac{9 y}{16}) + \frac{9 y^{2}}{64}) = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.19.1.2

Factor $2$ out of $16$ .

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 4 (\frac{9}{4} + 2 \cdot (- 1 \frac{9 y}{2 \cdot 8}) + \frac{9 y^{2}}{64}) = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.19.1.3

Cancel the common factor.

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 4 (\frac{9}{4} + 2 \cdot (- 1 \frac{9 y}{2 \cdot 8}) + \frac{9 y^{2}}{64}) = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.19.1.4

Rewrite the expression.

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 4 (\frac{9}{4} - 1 \frac{9 y}{8} + \frac{9 y^{2}}{64}) = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 4 (\frac{9}{4} - 1 \frac{9 y}{8} + \frac{9 y^{2}}{64}) = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.19.2

Rewrite $- 1 \frac{9 y}{8}$ as $- \frac{9 y}{8}$ .

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 4 (\frac{9}{4} - \frac{9 y}{8} + \frac{9 y^{2}}{64}) = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 4 (\frac{9}{4} - \frac{9 y}{8} + \frac{9 y^{2}}{64}) = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.20

Apply the distributive property.

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 4 (\frac{9}{4}) - 4 (- \frac{9 y}{8}) - 4 \frac{9 y^{2}}{64} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.21

Simplify.

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Step 2.2.1.1.21.1

Cancel the common factor of $4$ .

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Step 2.2.1.1.21.1.1

Factor $4$ out of $- 4$ .

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} + 4 (- 1) (\frac{9}{4}) - 4 (- \frac{9 y}{8}) - 4 \frac{9 y^{2}}{64} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.21.1.2

Cancel the common factor.

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} + 4 \cdot (- 1 (\frac{9}{4})) - 4 (- \frac{9 y}{8}) - 4 \frac{9 y^{2}}{64} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.21.1.3

Rewrite the expression.

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 1 \cdot 9 - 4 (- \frac{9 y}{8}) - 4 \frac{9 y^{2}}{64} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 1 \cdot 9 - 4 (- \frac{9 y}{8}) - 4 \frac{9 y^{2}}{64} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.21.2

Multiply $- 1$ by $9$ .

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 9 - 4 (- \frac{9 y}{8}) - 4 \frac{9 y^{2}}{64} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.21.3

Cancel the common factor of $4$ .

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Step 2.2.1.1.21.3.1

Move the leading negative in $- \frac{9 y}{8}$ into the numerator.

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 9 - 4 \frac{- 9 y}{8} - 4 \frac{9 y^{2}}{64} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.21.3.2

Factor $4$ out of $- 4$ .

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 9 + 4 (- 1) (\frac{- 9 y}{8}) - 4 \frac{9 y^{2}}{64} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.21.3.3

Factor $4$ out of $8$ .

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 9 + 4 \cdot (- 1 \frac{- 9 y}{4 \cdot 2}) - 4 \frac{9 y^{2}}{64} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.21.3.4

Cancel the common factor.

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 9 + 4 \cdot (- 1 \frac{- 9 y}{4 \cdot 2}) - 4 \frac{9 y^{2}}{64} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.21.3.5

Rewrite the expression.

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 9 - 1 \frac{- 9 y}{2} - 4 \frac{9 y^{2}}{64} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 9 - 1 \frac{- 9 y}{2} - 4 \frac{9 y^{2}}{64} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.21.4

Cancel the common factor of $4$ .

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Step 2.2.1.1.21.4.1

Factor $4$ out of $- 4$ .

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 9 - 1 \frac{- 9 y}{2} + 4 (- 1) (\frac{9 y^{2}}{64}) = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.21.4.2

Factor $4$ out of $64$ .

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 9 - 1 \frac{- 9 y}{2} + 4 \cdot (- 1 \frac{9 y^{2}}{4 \cdot 16}) = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.21.4.3

Cancel the common factor.

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 9 - 1 \frac{- 9 y}{2} + 4 \cdot (- 1 \frac{9 y^{2}}{4 \cdot 16}) = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.21.4.4

Rewrite the expression.

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 9 - 1 \frac{- 9 y}{2} - 1 \frac{9 y^{2}}{16} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 9 - 1 \frac{- 9 y}{2} - 1 \frac{9 y^{2}}{16} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 9 - 1 \frac{- 9 y}{2} - 1 \frac{9 y^{2}}{16} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.22

Simplify each term.

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Step 2.2.1.1.22.1

Move the negative in front of the fraction.

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 9 - 1 (- \frac{9 y}{2}) - 1 \frac{9 y^{2}}{16} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.22.2

Multiply $- 1 (- \frac{9 y}{2})$ .

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Step 2.2.1.1.22.2.1

Multiply $- 1$ by $- 1$ .

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 9 + 1 (\frac{9 y}{2}) - 1 \frac{9 y^{2}}{16} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.22.2.2

Multiply $\frac{9 y}{2}$ by $1$ .

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 9 + \frac{9 y}{2} - 1 \frac{9 y^{2}}{16} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 9 + \frac{9 y}{2} - 1 \frac{9 y^{2}}{16} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.1.22.3

Rewrite $- 1 \frac{9 y^{2}}{16}$ as $- \frac{9 y^{2}}{16}$ .

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 9 + \frac{9 y}{2} - \frac{9 y^{2}}{16} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 9 + \frac{9 y}{2} - \frac{9 y^{2}}{16} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

$18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 9 + \frac{9 y}{2} - \frac{9 y^{2}}{16} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Combine the opposite terms in $18 - \frac{9 y}{2} - 9 y + \frac{9 y^{2}}{4} - 9 + \frac{9 y}{2} - \frac{9 y^{2}}{16}$ .

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Step 2.2.1.2.1.1

Add $- \frac{9 y}{2}$ and $\frac{9 y}{2}$ .

$18 - 9 y + \frac{9 y^{2}}{4} - 9 + 0 - \frac{9 y^{2}}{16} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.2.1.2

Add $18 - 9 y + \frac{9 y^{2}}{4} - 9$ and $0$ .

$18 - 9 y + \frac{9 y^{2}}{4} - 9 - \frac{9 y^{2}}{16} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

$18 - 9 y + \frac{9 y^{2}}{4} - 9 - \frac{9 y^{2}}{16} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.2.2

Subtract $9$ from $18$ .

$- 9 y + \frac{9 y^{2}}{4} + 9 - \frac{9 y^{2}}{16} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

$- 9 y + \frac{9 y^{2}}{4} + 9 - \frac{9 y^{2}}{16} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.3

To write $\frac{9 y^{2}}{4}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- 9 y + \frac{9 y^{2}}{4} \cdot \frac{4}{4} - \frac{9 y^{2}}{16} + 9 = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.4

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.2.1.4.1

Multiply $\frac{9 y^{2}}{4}$ by $\frac{4}{4}$ .

$- 9 y + \frac{9 y^{2} \cdot 4}{4 \cdot 4} - \frac{9 y^{2}}{16} + 9 = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.4.2

Multiply $4$ by $4$ .

$- 9 y + \frac{9 y^{2} \cdot 4}{16} - \frac{9 y^{2}}{16} + 9 = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

$- 9 y + \frac{9 y^{2} \cdot 4}{16} - \frac{9 y^{2}}{16} + 9 = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.5

Combine the numerators over the common denominator.

$- 9 y + \frac{9 y^{2} \cdot 4 - 9 y^{2}}{16} + 9 = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.6

Find the common denominator.

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Step 2.2.1.6.1

Write $- 9 y$ as a fraction with denominator $1$ .

$\frac{- 9 y}{1} + \frac{9 y^{2} \cdot 4 - 9 y^{2}}{16} + 9 = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.6.2

Multiply $\frac{- 9 y}{1}$ by $\frac{16}{16}$ .

$\frac{- 9 y}{1} \cdot \frac{16}{16} + \frac{9 y^{2} \cdot 4 - 9 y^{2}}{16} + 9 = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.6.3

Multiply $\frac{- 9 y}{1}$ by $\frac{16}{16}$ .

$\frac{- 9 y \cdot 16}{16} + \frac{9 y^{2} \cdot 4 - 9 y^{2}}{16} + 9 = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.6.4

Write $9$ as a fraction with denominator $1$ .

$\frac{- 9 y \cdot 16}{16} + \frac{9 y^{2} \cdot 4 - 9 y^{2}}{16} + \frac{9}{1} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.6.5

Multiply $\frac{9}{1}$ by $\frac{16}{16}$ .

$\frac{- 9 y \cdot 16}{16} + \frac{9 y^{2} \cdot 4 - 9 y^{2}}{16} + \frac{9}{1} \cdot \frac{16}{16} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.6.6

Multiply $\frac{9}{1}$ by $\frac{16}{16}$ .

$\frac{- 9 y \cdot 16}{16} + \frac{9 y^{2} \cdot 4 - 9 y^{2}}{16} + \frac{9 \cdot 16}{16} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

$\frac{- 9 y \cdot 16}{16} + \frac{9 y^{2} \cdot 4 - 9 y^{2}}{16} + \frac{9 \cdot 16}{16} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.7

Simplify terms.

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Step 2.2.1.7.1

Combine the numerators over the common denominator.

$\frac{- 9 y \cdot 16 + 9 y^{2} \cdot 4 - 9 y^{2} + 9 \cdot 16}{16} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.7.2

Simplify each term.

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Step 2.2.1.7.2.1

Multiply $16$ by $- 9$ .

$\frac{- 144 y + 9 y^{2} \cdot 4 - 9 y^{2} + 9 \cdot 16}{16} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.7.2.2

Multiply $4$ by $9$ .

$\frac{- 144 y + 36 y^{2} - 9 y^{2} + 9 \cdot 16}{16} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.7.2.3

Multiply $9$ by $16$ .

$\frac{- 144 y + 36 y^{2} - 9 y^{2} + 144}{16} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

$\frac{- 144 y + 36 y^{2} - 9 y^{2} + 144}{16} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.7.3

Subtract $9 y^{2}$ from $36 y^{2}$ .

$\frac{- 144 y + 27 y^{2} + 144}{16} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

$\frac{- 144 y + 27 y^{2} + 144}{16} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.8

Simplify the numerator.

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Step 2.2.1.8.1

Factor $9$ out of $- 144 y + 27 y^{2} + 144$ .

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Step 2.2.1.8.1.1

Factor $9$ out of $- 144 y$ .

$\frac{9 (- 16 y) + 27 y^{2} + 144}{16} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.8.1.2

Factor $9$ out of $27 y^{2}$ .

$\frac{9 (- 16 y) + 9 (3 y^{2}) + 144}{16} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.8.1.3

Factor $9$ out of $144$ .

$\frac{9 (- 16 y) + 9 (3 y^{2}) + 9 (16)}{16} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.8.1.4

Factor $9$ out of $9 (- 16 y) + 9 (3 y^{2})$ .

$\frac{9 (- 16 y + 3 y^{2}) + 9 (16)}{16} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.8.1.5

Factor $9$ out of $9 (- 16 y + 3 y^{2}) + 9 (16)$ .

$\frac{9 (- 16 y + 3 y^{2} + 16)}{16} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

$\frac{9 (- 16 y + 3 y^{2} + 16)}{16} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.8.2

Let $u = y$ . Substitute $u$ for all occurrences of $y$ .

$\frac{9 (- 16 u + 3 u^{2} + 16)}{16} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.8.3

Factor by grouping.

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Step 2.2.1.8.3.1

Reorder terms.

$\frac{9 (3 u^{2} - 16 u + 16)}{16} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.8.3.2

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot 16 = 48$ and whose sum is $b = - 16$ .

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Step 2.2.1.8.3.2.1

Factor $- 16$ out of $- 16 u$ .

$\frac{9 (3 u^{2} - 16 u + 16)}{16} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.8.3.2.2

Rewrite $- 16$ as $- 4$ plus $- 12$

$\frac{9 (3 u^{2} + (- 4 - 12) u + 16)}{16} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.8.3.2.3

Apply the distributive property.

$\frac{9 (3 u^{2} - 4 u - 12 u + 16)}{16} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

$\frac{9 (3 u^{2} - 4 u - 12 u + 16)}{16} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.8.3.3

Factor out the greatest common factor from each group.

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Step 2.2.1.8.3.3.1

Group the first two terms and the last two terms.

$\frac{9 ((3 u^{2} - 4 u) - 12 u + 16)}{16} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.8.3.3.2

Factor out the greatest common factor (GCF) from each group.

$\frac{9 (u (3 u - 4) - 4 (3 u - 4))}{16} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

$\frac{9 (u (3 u - 4) - 4 (3 u - 4))}{16} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.8.3.4

Factor the polynomial by factoring out the greatest common factor, $3 u - 4$ .

$\frac{9 ((3 u - 4) (u - 4))}{16} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

$\frac{9 ((3 u - 4) (u - 4))}{16} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 2.2.1.8.4

Replace all occurrences of $u$ with $y$ .

$\frac{9 (3 y - 4) (y - 4)}{16} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

$\frac{9 (3 y - 4) (y - 4)}{16} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

$\frac{9 (3 y - 4) (y - 4)}{16} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

$\frac{9 (3 y - 4) (y - 4)}{16} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

$\frac{9 (3 y - 4) (y - 4)}{16} = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 3

Solve for $y$ in $\frac{9 (3 y - 4) (y - 4)}{16} = 0$ .

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Step 3.1

Set the numerator equal to zero.

$9 (3 y - 4) (y - 4) = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 3.2

Solve the equation for $y$ .

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Step 3.2.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$3 y - 4 = 0$

$y - 4 = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 3.2.2

Set $3 y - 4$ equal to $0$ and solve for $y$ .

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Step 3.2.2.1

Set $3 y - 4$ equal to $0$ .

$3 y - 4 = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 3.2.2.2

Solve $3 y - 4 = 0$ for $y$ .

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Step 3.2.2.2.1

Add $4$ to both sides of the equation.

$3 y = 4$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 3.2.2.2.2

Divide each term in $3 y = 4$ by $3$ and simplify.

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Step 3.2.2.2.2.1

Divide each term in $3 y = 4$ by $3$ .

$\frac{3 y}{3} = \frac{4}{3}$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 3.2.2.2.2.2

Simplify the left side.

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Step 3.2.2.2.2.2.1

Cancel the common factor of $3$ .

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Step 3.2.2.2.2.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{4}{3}$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 3.2.2.2.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{4}{3}$

$x = \frac{3}{2} - \frac{3 y}{8}$

$y = \frac{4}{3}$

$x = \frac{3}{2} - \frac{3 y}{8}$

$y = \frac{4}{3}$

$x = \frac{3}{2} - \frac{3 y}{8}$

$y = \frac{4}{3}$

$x = \frac{3}{2} - \frac{3 y}{8}$

$y = \frac{4}{3}$

$x = \frac{3}{2} - \frac{3 y}{8}$

$y = \frac{4}{3}$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 3.2.3

Set $y - 4$ equal to $0$ and solve for $y$ .

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Step 3.2.3.1

Set $y - 4$ equal to $0$ .

$y - 4 = 0$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 3.2.3.2

Add $4$ to both sides of the equation.

$y = 4$

$x = \frac{3}{2} - \frac{3 y}{8}$

$y = 4$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 3.2.4

The final solution is all the values that make $9 (3 y - 4) (y - 4) = 0$ true.

$y = \frac{4}{3}, 4$

$x = \frac{3}{2} - \frac{3 y}{8}$

$y = \frac{4}{3}, 4$

$x = \frac{3}{2} - \frac{3 y}{8}$

$y = \frac{4}{3}, 4$

$x = \frac{3}{2} - \frac{3 y}{8}$

Step 4

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

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Step 4.1

Replace all occurrences of $y$ in $x = \frac{3}{2} - \frac{3 y}{8}$ with $\frac{4}{3}$ .

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

$y = \frac{4}{3}$

Step 4.2

Simplify the right side.

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Step 4.2.1

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

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

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

$x = \frac{3}{2} - \frac{\frac{3 \cdot 4}{3}}{8}$

$y = \frac{4}{3}$

Step 4.2.1.1.2

Multiply $3$ by $4$ .

$x = \frac{3}{2} - \frac{\frac{12}{3}}{8}$

$y = \frac{4}{3}$

Step 4.2.1.1.3

Divide $12$ by $3$ .

$x = \frac{3}{2} - \frac{4}{8}$

$y = \frac{4}{3}$

Step 4.2.1.1.4

Cancel the common factor of $4$ and $8$ .

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Step 4.2.1.1.4.1

Factor $4$ out of $4$ .

$x = \frac{3}{2} - \frac{4 (1)}{8}$

$y = \frac{4}{3}$

Step 4.2.1.1.4.2

Cancel the common factors.

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Step 4.2.1.1.4.2.1

Factor $4$ out of $8$ .

$x = \frac{3}{2} - \frac{4 \cdot 1}{4 \cdot 2}$

$y = \frac{4}{3}$

Step 4.2.1.1.4.2.2

Cancel the common factor.

$x = \frac{3}{2} - \frac{4 \cdot 1}{4 \cdot 2}$

$y = \frac{4}{3}$

Step 4.2.1.1.4.2.3

Rewrite the expression.

$x = \frac{3}{2} - \frac{1}{2}$

$y = \frac{4}{3}$

$x = \frac{3}{2} - \frac{1}{2}$

$y = \frac{4}{3}$

$x = \frac{3}{2} - \frac{1}{2}$

$y = \frac{4}{3}$

$x = \frac{3}{2} - \frac{1}{2}$

$y = \frac{4}{3}$

Step 4.2.1.2

Combine fractions.

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Step 4.2.1.2.1

Combine the numerators over the common denominator.

$x = \frac{3 - 1}{2}$

$y = \frac{4}{3}$

Step 4.2.1.2.2

Simplify the expression.

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Step 4.2.1.2.2.1

Subtract $1$ from $3$ .

$x = \frac{2}{2}$

$y = \frac{4}{3}$

Step 4.2.1.2.2.2

Divide $2$ by $2$ .

$x = 1$

$y = \frac{4}{3}$

$x = 1$

$y = \frac{4}{3}$

$x = 1$

$y = \frac{4}{3}$

$x = 1$

$y = \frac{4}{3}$

$x = 1$

$y = \frac{4}{3}$

$x = 1$

$y = \frac{4}{3}$

Step 5

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

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Step 5.1

Replace all occurrences of $y$ in $x = \frac{3}{2} - \frac{3 y}{8}$ with $4$ .

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

$y = 4$

Step 5.2

Simplify the right side.

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Step 5.2.1

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

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Step 5.2.1.1

Cancel the common factor of $4$ and $8$ .

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Step 5.2.1.1.1

Factor $4$ out of $3 (4)$ .

$x = \frac{3}{2} - \frac{4 \cdot 3}{8}$

$y = 4$

Step 5.2.1.1.2

Cancel the common factors.

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Step 5.2.1.1.2.1

Factor $4$ out of $8$ .

$x = \frac{3}{2} - \frac{4 \cdot 3}{4 \cdot 2}$

$y = 4$

Step 5.2.1.1.2.2

Cancel the common factor.

$x = \frac{3}{2} - \frac{4 \cdot 3}{4 \cdot 2}$

$y = 4$

Step 5.2.1.1.2.3

Rewrite the expression.

$x = \frac{3}{2} - \frac{3}{2}$

$y = 4$

$x = \frac{3}{2} - \frac{3}{2}$

$y = 4$

$x = \frac{3}{2} - \frac{3}{2}$

$y = 4$

Step 5.2.1.2

Combine the numerators over the common denominator.

$x = \frac{3 - 3}{2}$

$y = 4$

Step 5.2.1.3

Simplify the expression.

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Step 5.2.1.3.1

Subtract $3$ from $3$ .

$x = \frac{0}{2}$

$y = 4$

Step 5.2.1.3.2

Divide $0$ by $2$ .

$x = 0$

$y = 4$

$x = 0$

$y = 4$

$x = 0$

$y = 4$

$x = 0$

$y = 4$

$x = 0$

$y = 4$

Step 6

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(1, \frac{4}{3})$

$(0, 4)$

Step 7

The result can be shown in multiple forms.

Point Form:

$(1, \frac{4}{3}), (0, 4)$

Equation Form:

$x = 1, y = \frac{4}{3}$

$x = 0, y = 4$

Step 8