Finite Math Examples

$4 x + 3 y = - 9$ , $2 x - 3 y = - 16$

Step 1

Solve for $x$ in $4 x + 3 y = - 9$ .

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Step 1.1

Subtract $3 y$ from both sides of the equation.

$4 x = - 9 - 3 y$

$2 x - 3 y = - 16$

Step 1.2

Divide each term in $4 x = - 9 - 3 y$ by $4$ and simplify.

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Step 1.2.1

Divide each term in $4 x = - 9 - 3 y$ by $4$ .

$\frac{4 x}{4} = \frac{- 9}{4} + \frac{- 3 y}{4}$

$2 x - 3 y = - 16$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $4$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 9}{4} + \frac{- 3 y}{4}$

$2 x - 3 y = - 16$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 9}{4} + \frac{- 3 y}{4}$

$2 x - 3 y = - 16$

$x = \frac{- 9}{4} + \frac{- 3 y}{4}$

$2 x - 3 y = - 16$

$x = \frac{- 9}{4} + \frac{- 3 y}{4}$

$2 x - 3 y = - 16$

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

Move the negative in front of the fraction.

$x = - \frac{9}{4} + \frac{- 3 y}{4}$

$2 x - 3 y = - 16$

Step 1.2.3.1.2

Move the negative in front of the fraction.

$x = - \frac{9}{4} - \frac{3 y}{4}$

$2 x - 3 y = - 16$

$x = - \frac{9}{4} - \frac{3 y}{4}$

$2 x - 3 y = - 16$

$x = - \frac{9}{4} - \frac{3 y}{4}$

$2 x - 3 y = - 16$

$x = - \frac{9}{4} - \frac{3 y}{4}$

$2 x - 3 y = - 16$

$x = - \frac{9}{4} - \frac{3 y}{4}$

$2 x - 3 y = - 16$

Step 2

Replace all occurrences of $x$ with $- \frac{9}{4} - \frac{3 y}{4}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $2 x - 3 y = - 16$ with $- \frac{9}{4} - \frac{3 y}{4}$ .

$2 (- \frac{9}{4} - \frac{3 y}{4}) - 3 y = - 16$

$x = - \frac{9}{4} - \frac{3 y}{4}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $2 (- \frac{9}{4} - \frac{3 y}{4}) - 3 y$ .

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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.

$2 (- \frac{9}{4}) + 2 (- \frac{3 y}{4}) - 3 y = - 16$

$x = - \frac{9}{4} - \frac{3 y}{4}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Move the leading negative in $- \frac{9}{4}$ into the numerator.

$2 (\frac{- 9}{4}) + 2 (- \frac{3 y}{4}) - 3 y = - 16$

$x = - \frac{9}{4} - \frac{3 y}{4}$

Step 2.2.1.1.2.2

Factor $2$ out of $4$ .

$2 (\frac{- 9}{2 (2)}) + 2 (- \frac{3 y}{4}) - 3 y = - 16$

$x = - \frac{9}{4} - \frac{3 y}{4}$

Step 2.2.1.1.2.3

Cancel the common factor.

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

$x = - \frac{9}{4} - \frac{3 y}{4}$

Step 2.2.1.1.2.4

Rewrite the expression.

$\frac{- 9}{2} + 2 (- \frac{3 y}{4}) - 3 y = - 16$

$x = - \frac{9}{4} - \frac{3 y}{4}$

$\frac{- 9}{2} + 2 (- \frac{3 y}{4}) - 3 y = - 16$

$x = - \frac{9}{4} - \frac{3 y}{4}$

Step 2.2.1.1.3

Cancel the common factor of $2$ .

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Step 2.2.1.1.3.1

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

$\frac{- 9}{2} + 2 (\frac{- 3 y}{4}) - 3 y = - 16$

$x = - \frac{9}{4} - \frac{3 y}{4}$

Step 2.2.1.1.3.2

Factor $2$ out of $4$ .

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

$x = - \frac{9}{4} - \frac{3 y}{4}$

Step 2.2.1.1.3.3

Cancel the common factor.

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

$x = - \frac{9}{4} - \frac{3 y}{4}$

Step 2.2.1.1.3.4

Rewrite the expression.

$\frac{- 9}{2} + \frac{- 3 y}{2} - 3 y = - 16$

$x = - \frac{9}{4} - \frac{3 y}{4}$

$\frac{- 9}{2} + \frac{- 3 y}{2} - 3 y = - 16$

$x = - \frac{9}{4} - \frac{3 y}{4}$

Step 2.2.1.1.4

Simplify each term.

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Step 2.2.1.1.4.1

Move the negative in front of the fraction.

$- \frac{9}{2} + \frac{- 3 y}{2} - 3 y = - 16$

$x = - \frac{9}{4} - \frac{3 y}{4}$

Step 2.2.1.1.4.2

Move the negative in front of the fraction.

$- \frac{9}{2} - \frac{3 y}{2} - 3 y = - 16$

$x = - \frac{9}{4} - \frac{3 y}{4}$

$- \frac{9}{2} - \frac{3 y}{2} - 3 y = - 16$

$x = - \frac{9}{4} - \frac{3 y}{4}$

$- \frac{9}{2} - \frac{3 y}{2} - 3 y = - 16$

$x = - \frac{9}{4} - \frac{3 y}{4}$

Step 2.2.1.2

To write $- 3 y$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{9}{2} - \frac{3 y}{2} - 3 y \cdot \frac{2}{2} = - 16$

$x = - \frac{9}{4} - \frac{3 y}{4}$

Step 2.2.1.3

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

$- \frac{9}{2} - \frac{3 y}{2} + \frac{- 3 y \cdot 2}{2} = - 16$

$x = - \frac{9}{4} - \frac{3 y}{4}$

Step 2.2.1.4

Combine the numerators over the common denominator.

$- \frac{9}{2} + \frac{- 3 y - 3 y \cdot 2}{2} = - 16$

$x = - \frac{9}{4} - \frac{3 y}{4}$

Step 2.2.1.5

Combine the numerators over the common denominator.

$\frac{- 9 - 3 y - 3 y \cdot 2}{2} = - 16$

$x = - \frac{9}{4} - \frac{3 y}{4}$

Step 2.2.1.6

Multiply $2$ by $- 3$ .

$\frac{- 9 - 3 y - 6 y}{2} = - 16$

$x = - \frac{9}{4} - \frac{3 y}{4}$

Step 2.2.1.7

Subtract $6 y$ from $- 3 y$ .

$\frac{- 9 - 9 y}{2} = - 16$

$x = - \frac{9}{4} - \frac{3 y}{4}$

Step 2.2.1.8

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

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Step 2.2.1.8.1

Factor $9$ out of $- 9$ .

$\frac{9 (- 1) - 9 y}{2} = - 16$

$x = - \frac{9}{4} - \frac{3 y}{4}$

Step 2.2.1.8.2

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

$\frac{9 (- 1) + 9 (- y)}{2} = - 16$

$x = - \frac{9}{4} - \frac{3 y}{4}$

Step 2.2.1.8.3

Factor $9$ out of $9 (- 1) + 9 (- y)$ .

$\frac{9 (- 1 - y)}{2} = - 16$

$x = - \frac{9}{4} - \frac{3 y}{4}$

$\frac{9 (- 1 - y)}{2} = - 16$

$x = - \frac{9}{4} - \frac{3 y}{4}$

Step 2.2.1.9

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

$\frac{9 (- 1 \cdot 1 - y)}{2} = - 16$

$x = - \frac{9}{4} - \frac{3 y}{4}$

Step 2.2.1.10

Factor $- 1$ out of $- y$ .

$\frac{9 (- 1 \cdot 1 - (y))}{2} = - 16$

$x = - \frac{9}{4} - \frac{3 y}{4}$

Step 2.2.1.11

Factor $- 1$ out of $- 1 (1) - (y)$ .

$\frac{9 (- 1 (1 + y))}{2} = - 16$

$x = - \frac{9}{4} - \frac{3 y}{4}$

Step 2.2.1.12

Move the negative in front of the fraction.

$- \frac{9 (1 + y)}{2} = - 16$

$x = - \frac{9}{4} - \frac{3 y}{4}$

$- \frac{9 (1 + y)}{2} = - 16$

$x = - \frac{9}{4} - \frac{3 y}{4}$

$- \frac{9 (1 + y)}{2} = - 16$

$x = - \frac{9}{4} - \frac{3 y}{4}$

$- \frac{9 (1 + y)}{2} = - 16$

$x = - \frac{9}{4} - \frac{3 y}{4}$

Step 3

Solve for $y$ in $- \frac{9 (1 + y)}{2} = - 16$ .

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Step 3.1

Multiply both sides of the equation by $- \frac{2}{9}$ .

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

$x = - \frac{9}{4} - \frac{3 y}{4}$

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

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

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Step 3.2.1.1.1

Cancel the common factor of $2$ .

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Step 3.2.1.1.1.1

Move the leading negative in $- \frac{2}{9}$ into the numerator.

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

$x = - \frac{9}{4} - \frac{3 y}{4}$

Step 3.2.1.1.1.2

Move the leading negative in $- \frac{9 (1 + y)}{2}$ into the numerator.

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

$x = - \frac{9}{4} - \frac{3 y}{4}$

Step 3.2.1.1.1.3

Factor $2$ out of $- 2$ .

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

$x = - \frac{9}{4} - \frac{3 y}{4}$

Step 3.2.1.1.1.4

Cancel the common factor.

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

$x = - \frac{9}{4} - \frac{3 y}{4}$

Step 3.2.1.1.1.5

Rewrite the expression.

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

$x = - \frac{9}{4} - \frac{3 y}{4}$

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

$x = - \frac{9}{4} - \frac{3 y}{4}$

Step 3.2.1.1.2

Cancel the common factor of $9$ .

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Step 3.2.1.1.2.1

Factor $9$ out of $- 9 (1 + y)$ .

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

$x = - \frac{9}{4} - \frac{3 y}{4}$

Step 3.2.1.1.2.2

Cancel the common factor.

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

$x = - \frac{9}{4} - \frac{3 y}{4}$

Step 3.2.1.1.2.3

Rewrite the expression.

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

$x = - \frac{9}{4} - \frac{3 y}{4}$

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

$x = - \frac{9}{4} - \frac{3 y}{4}$

Step 3.2.1.1.3

Multiply.

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Step 3.2.1.1.3.1

Multiply $- 1$ by $- 1$ .

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

$x = - \frac{9}{4} - \frac{3 y}{4}$

Step 3.2.1.1.3.2

Multiply $1 + y$ by $1$ .

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

$x = - \frac{9}{4} - \frac{3 y}{4}$

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

$x = - \frac{9}{4} - \frac{3 y}{4}$

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

$x = - \frac{9}{4} - \frac{3 y}{4}$

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

$x = - \frac{9}{4} - \frac{3 y}{4}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Multiply $- \frac{2}{9} \cdot - 16$ .

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Step 3.2.2.1.1

Multiply $- 16$ by $- 1$ .

$1 + y = 16 (\frac{2}{9})$

$x = - \frac{9}{4} - \frac{3 y}{4}$

Step 3.2.2.1.2

Combine $16$ and $\frac{2}{9}$ .

$1 + y = \frac{16 \cdot 2}{9}$

$x = - \frac{9}{4} - \frac{3 y}{4}$

Step 3.2.2.1.3

Multiply $16$ by $2$ .

$1 + y = \frac{32}{9}$

$x = - \frac{9}{4} - \frac{3 y}{4}$

$1 + y = \frac{32}{9}$

$x = - \frac{9}{4} - \frac{3 y}{4}$

$1 + y = \frac{32}{9}$

$x = - \frac{9}{4} - \frac{3 y}{4}$

$1 + y = \frac{32}{9}$

$x = - \frac{9}{4} - \frac{3 y}{4}$

Step 3.3

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

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Step 3.3.1

Subtract $1$ from both sides of the equation.

$y = \frac{32}{9} - 1$

$x = - \frac{9}{4} - \frac{3 y}{4}$

Step 3.3.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$y = \frac{32}{9} - 1 \cdot \frac{9}{9}$

$x = - \frac{9}{4} - \frac{3 y}{4}$

Step 3.3.3

Combine $- 1$ and $\frac{9}{9}$ .

$y = \frac{32}{9} + \frac{- 1 \cdot 9}{9}$

$x = - \frac{9}{4} - \frac{3 y}{4}$

Step 3.3.4

Combine the numerators over the common denominator.

$y = \frac{32 - 1 \cdot 9}{9}$

$x = - \frac{9}{4} - \frac{3 y}{4}$

Step 3.3.5

Simplify the numerator.

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Step 3.3.5.1

Multiply $- 1$ by $9$ .

$y = \frac{32 - 9}{9}$

$x = - \frac{9}{4} - \frac{3 y}{4}$

Step 3.3.5.2

Subtract $9$ from $32$ .

$y = \frac{23}{9}$

$x = - \frac{9}{4} - \frac{3 y}{4}$

$y = \frac{23}{9}$

$x = - \frac{9}{4} - \frac{3 y}{4}$

$y = \frac{23}{9}$

$x = - \frac{9}{4} - \frac{3 y}{4}$

$y = \frac{23}{9}$

$x = - \frac{9}{4} - \frac{3 y}{4}$

Step 4

Replace all occurrences of $y$ with $\frac{23}{9}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = - \frac{9}{4} - \frac{3 y}{4}$ with $\frac{23}{9}$ .

$x = - \frac{9}{4} - \frac{3 (\frac{23}{9})}{4}$

$y = \frac{23}{9}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- \frac{9}{4} - \frac{3 (\frac{23}{9})}{4}$ .

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Step 4.2.1.1

Combine the numerators over the common denominator.

$x = \frac{- 9 - 3 (\frac{23}{9})}{4}$

$y = \frac{23}{9}$

Step 4.2.1.2

Simplify each term.

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Step 4.2.1.2.1

Cancel the common factor of $3$ .

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Step 4.2.1.2.1.1

Factor $3$ out of $- 3$ .

$x = \frac{- 9 + 3 (- 1) (\frac{23}{9})}{4}$

$y = \frac{23}{9}$

Step 4.2.1.2.1.2

Factor $3$ out of $9$ .

$x = \frac{- 9 + 3 \cdot (- 1 \frac{23}{3 \cdot 3})}{4}$

$y = \frac{23}{9}$

Step 4.2.1.2.1.3

Cancel the common factor.

$x = \frac{- 9 + 3 \cdot (- 1 \frac{23}{3 \cdot 3})}{4}$

$y = \frac{23}{9}$

Step 4.2.1.2.1.4

Rewrite the expression.

$x = \frac{- 9 - 1 (\frac{23}{3})}{4}$

$y = \frac{23}{9}$

$x = \frac{- 9 - 1 (\frac{23}{3})}{4}$

$y = \frac{23}{9}$

Step 4.2.1.2.2

Rewrite $- 1 (\frac{23}{3})$ as $- (\frac{23}{3})$ .

$x = \frac{- 9 - \frac{23}{3}}{4}$

$y = \frac{23}{9}$

$x = \frac{- 9 - \frac{23}{3}}{4}$

$y = \frac{23}{9}$

Step 4.2.1.3

To write $- 9$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = \frac{- 9 \cdot \frac{3}{3} - \frac{23}{3}}{4}$

$y = \frac{23}{9}$

Step 4.2.1.4

Combine $- 9$ and $\frac{3}{3}$ .

$x = \frac{\frac{- 9 \cdot 3}{3} - \frac{23}{3}}{4}$

$y = \frac{23}{9}$

Step 4.2.1.5

Combine the numerators over the common denominator.

$x = \frac{\frac{- 9 \cdot 3 - 23}{3}}{4}$

$y = \frac{23}{9}$

Step 4.2.1.6

Simplify the numerator.

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Step 4.2.1.6.1

Multiply $- 9$ by $3$ .

$x = \frac{\frac{- 27 - 23}{3}}{4}$

$y = \frac{23}{9}$

Step 4.2.1.6.2

Subtract $23$ from $- 27$ .

$x = \frac{\frac{- 50}{3}}{4}$

$y = \frac{23}{9}$

$x = \frac{\frac{- 50}{3}}{4}$

$y = \frac{23}{9}$

Step 4.2.1.7

Move the negative in front of the fraction.

$x = \frac{- \frac{50}{3}}{4}$

$y = \frac{23}{9}$

Step 4.2.1.8

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{50}{3} \cdot \frac{1}{4}$

$y = \frac{23}{9}$

Step 4.2.1.9

Cancel the common factor of $2$ .

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Step 4.2.1.9.1

Move the leading negative in $- \frac{50}{3}$ into the numerator.

$x = \frac{- 50}{3} \cdot \frac{1}{4}$

$y = \frac{23}{9}$

Step 4.2.1.9.2

Factor $2$ out of $- 50$ .

$x = \frac{2 (- 25)}{3} \cdot \frac{1}{4}$

$y = \frac{23}{9}$

Step 4.2.1.9.3

Factor $2$ out of $4$ .

$x = \frac{2 \cdot - 25}{3} \cdot \frac{1}{2 \cdot 2}$

$y = \frac{23}{9}$

Step 4.2.1.9.4

Cancel the common factor.

$x = \frac{2 \cdot - 25}{3} \cdot \frac{1}{2 \cdot 2}$

$y = \frac{23}{9}$

Step 4.2.1.9.5

Rewrite the expression.

$x = \frac{- 25}{3} \cdot \frac{1}{2}$

$y = \frac{23}{9}$

$x = \frac{- 25}{3} \cdot \frac{1}{2}$

$y = \frac{23}{9}$

Step 4.2.1.10

Multiply $\frac{- 25}{3}$ by $\frac{1}{2}$ .

$x = \frac{- 25}{3 \cdot 2}$

$y = \frac{23}{9}$

Step 4.2.1.11

Simplify the expression.

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Step 4.2.1.11.1

Multiply $3$ by $2$ .

$x = \frac{- 25}{6}$

$y = \frac{23}{9}$

Step 4.2.1.11.2

Move the negative in front of the fraction.

$x = - \frac{25}{6}$

$y = \frac{23}{9}$

$x = - \frac{25}{6}$

$y = \frac{23}{9}$

$x = - \frac{25}{6}$

$y = \frac{23}{9}$

$x = - \frac{25}{6}$

$y = \frac{23}{9}$

$x = - \frac{25}{6}$

$y = \frac{23}{9}$

Step 5

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

$(- \frac{25}{6}, \frac{23}{9})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(- \frac{25}{6}, \frac{23}{9})$

Equation Form:

$x = - \frac{25}{6}, y = \frac{23}{9}$

Step 7