Solve by Substitution 2x+3y=-18 , 5x-6y=9

$2 x + 3 y = - 18$ , $5 x - 6 y = 9$

Step 1

Solve for $x$ in $2 x + 3 y = - 18$ .

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

$2 x = - 18 - 3 y$

$5 x - 6 y = 9$

Divide each term in $2 x = - 18 - 3 y$ by $2$ and simplify.

Divide each term in $2 x = - 18 - 3 y$ by $2$ .

$\frac{2 x}{2} = \frac{- 18}{2} + \frac{- 3 y}{2}$

$5 x - 6 y = 9$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 18}{2} + \frac{- 3 y}{2}$

$5 x - 6 y = 9$

Divide $x$ by $1$ .

$x = \frac{- 18}{2} + \frac{- 3 y}{2}$

$5 x - 6 y = 9$

$x = \frac{- 18}{2} + \frac{- 3 y}{2}$

$5 x - 6 y = 9$

$x = \frac{- 18}{2} + \frac{- 3 y}{2}$

$5 x - 6 y = 9$

Simplify the right side.

Simplify each term.

Divide $- 18$ by $2$ .

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

$5 x - 6 y = 9$

Move the negative in front of the fraction.

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

$5 x - 6 y = 9$

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

$5 x - 6 y = 9$

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

$5 x - 6 y = 9$

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

$5 x - 6 y = 9$

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

$5 x - 6 y = 9$

Step 2

Replace all occurrences of $x$ with $- 9 - \frac{3 y}{2}$ in each equation.

Replace all occurrences of $x$ in $5 x - 6 y = 9$ with $- 9 - \frac{3 y}{2}$ .

$5 (- 9 - \frac{3 y}{2}) - 6 y = 9$

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

Simplify the left side.

Simplify $5 (- 9 - \frac{3 y}{2}) - 6 y$ .

Simplify each term.

Apply the distributive property.

$5 \cdot - 9 + 5 (- \frac{3 y}{2}) - 6 y = 9$

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

Multiply $5$ by $- 9$ .

$- 45 + 5 (- \frac{3 y}{2}) - 6 y = 9$

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

Multiply $5 (- \frac{3 y}{2})$ .

Multiply $- 1$ by $5$ .

$- 45 - 5 \frac{3 y}{2} - 6 y = 9$

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

Combine $- 5$ and $\frac{3 y}{2}$ .

$- 45 + \frac{- 5 (3 y)}{2} - 6 y = 9$

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

Multiply $3$ by $- 5$ .

$- 45 + \frac{- 15 y}{2} - 6 y = 9$

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

$- 45 + \frac{- 15 y}{2} - 6 y = 9$

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

Move the negative in front of the fraction.

$- 45 - \frac{15 y}{2} - 6 y = 9$

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

$- 45 - \frac{15 y}{2} - 6 y = 9$

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

To write $- 6 y$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- 45 - \frac{15 y}{2} - 6 y \cdot \frac{2}{2} = 9$

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

Simplify terms.

Combine $- 6 y$ and $\frac{2}{2}$ .

$- 45 - \frac{15 y}{2} + \frac{- 6 y \cdot 2}{2} = 9$

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

Combine the numerators over the common denominator.

$- 45 + \frac{- 15 y - 6 y \cdot 2}{2} = 9$

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

$- 45 + \frac{- 15 y - 6 y \cdot 2}{2} = 9$

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

Simplify each term.

Simplify the numerator.

Factor $3 y$ out of $- 15 y - 6 y \cdot 2$ .

Factor $3 y$ out of $- 15 y$ .

$- 45 + \frac{3 y (- 5) - 6 y \cdot 2}{2} = 9$

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

Factor $3 y$ out of $- 6 y \cdot 2$ .

$- 45 + \frac{3 y (- 5) + 3 y (- 2 \cdot 2)}{2} = 9$

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

Factor $3 y$ out of $3 y (- 5) + 3 y (- 2 \cdot 2)$ .

$- 45 + \frac{3 y (- 5 - 2 \cdot 2)}{2} = 9$

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

$- 45 + \frac{3 y (- 5 - 2 \cdot 2)}{2} = 9$

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

Multiply $- 2$ by $2$ .

$- 45 + \frac{3 y (- 5 - 4)}{2} = 9$

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

Subtract $4$ from $- 5$ .

$- 45 + \frac{3 y \cdot - 9}{2} = 9$

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

Multiply $- 9$ by $3$ .

$- 45 + \frac{- 27 y}{2} = 9$

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

$- 45 + \frac{- 27 y}{2} = 9$

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

Move the negative in front of the fraction.

$- 45 - \frac{27 y}{2} = 9$

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

$- 45 - \frac{27 y}{2} = 9$

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

$- 45 - \frac{27 y}{2} = 9$

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

$- 45 - \frac{27 y}{2} = 9$

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

$- 45 - \frac{27 y}{2} = 9$

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

Step 3

Solve for $y$ in $- 45 - \frac{27 y}{2} = 9$ .

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

Add $45$ to both sides of the equation.

$- \frac{27 y}{2} = 9 + 45$

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

Add $9$ and $45$ .

$- \frac{27 y}{2} = 54$

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

$- \frac{27 y}{2} = 54$

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

Multiply both sides of the equation by $- \frac{2}{27}$ .

$- \frac{2}{27} \cdot (- \frac{27 y}{2}) = - \frac{2}{27} \cdot 54$

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

Simplify both sides of the equation.

Simplify the left side.

Simplify $- \frac{2}{27} (- \frac{27 y}{2})$ .

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{2}{27}$ into the numerator.

$\frac{- 2}{27} \cdot (- \frac{27 y}{2}) = - \frac{2}{27} \cdot 54$

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

Move the leading negative in $- \frac{27 y}{2}$ into the numerator.

$\frac{- 2}{27} \cdot \frac{- 27 y}{2} = - \frac{2}{27} \cdot 54$

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

Factor $2$ out of $- 2$ .

$\frac{2 (- 1)}{27} \cdot \frac{- 27 y}{2} = - \frac{2}{27} \cdot 54$

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

Cancel the common factor.

$\frac{2 \cdot - 1}{27} \cdot \frac{- 27 y}{2} = - \frac{2}{27} \cdot 54$

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

Rewrite the expression.

$\frac{- 1}{27} \cdot (- 27 y) = - \frac{2}{27} \cdot 54$

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

$\frac{- 1}{27} \cdot (- 27 y) = - \frac{2}{27} \cdot 54$

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

Cancel the common factor of $27$ .

Factor $27$ out of $- 27 y$ .

$\frac{- 1}{27} \cdot (27 (- y)) = - \frac{2}{27} \cdot 54$

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

Cancel the common factor.

$\frac{- 1}{27} \cdot (27 (- y)) = - \frac{2}{27} \cdot 54$

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

Rewrite the expression.

$y = - \frac{2}{27} \cdot 54$

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

$y = - \frac{2}{27} \cdot 54$

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

Multiply.

Multiply $- 1$ by $- 1$ .

$1 y = - \frac{2}{27} \cdot 54$

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

Multiply $y$ by $1$ .

$y = - \frac{2}{27} \cdot 54$

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

$y = - \frac{2}{27} \cdot 54$

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

$y = - \frac{2}{27} \cdot 54$

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

$y = - \frac{2}{27} \cdot 54$

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

Simplify the right side.

Simplify $- \frac{2}{27} \cdot 54$ .

Cancel the common factor of $27$ .

Move the leading negative in $- \frac{2}{27}$ into the numerator.

$y = \frac{- 2}{27} \cdot 54$

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

Factor $27$ out of $54$ .

$y = \frac{- 2}{27} \cdot (27 (2))$

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

Cancel the common factor.

$y = \frac{- 2}{27} \cdot (27 \cdot 2)$

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

Rewrite the expression.

$y = - 2 \cdot 2$

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

$y = - 2 \cdot 2$

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

Multiply $- 2$ by $2$ .

$y = - 4$

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

$y = - 4$

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

$y = - 4$

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

$y = - 4$

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

$y = - 4$

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

Step 4

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

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

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

$y = - 4$

Simplify the right side.

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

Simplify each term.

Cancel the common factor of $- 4$ and $2$ .

Factor $2$ out of $3 (- 4)$ .

$x = - 9 - \frac{2 (3 \cdot (- 2))}{2}$

$y = - 4$

Cancel the common factors.

Factor $2$ out of $2$ .

$x = - 9 - \frac{2 (3 \cdot (- 2))}{2 (1)}$

$y = - 4$

Cancel the common factor.

$x = - 9 - \frac{2 (3 \cdot (- 2))}{2 \cdot 1}$

$y = - 4$

Rewrite the expression.

$x = - 9 - \frac{3 \cdot (- 2)}{1}$

$y = - 4$

Divide $3 \cdot (- 2)$ by $1$ .

$x = - 9 - (3 \cdot (- 2))$

$y = - 4$

$x = - 9 - (3 \cdot (- 2))$

$y = - 4$

$x = - 9 - (3 \cdot (- 2))$

$y = - 4$

Multiply $- (3 \cdot (- 2))$ .

Multiply $3$ by $- 2$ .

$x = - 9 + 6$

$y = - 4$

Multiply $- 1$ by $- 6$ .

$x = - 9 + 6$

$y = - 4$

$x = - 9 + 6$

$y = - 4$

$x = - 9 + 6$

$y = - 4$

Add $- 9$ and $6$ .

$x = - 3$

$y = - 4$

$x = - 3$

$y = - 4$

$x = - 3$

$y = - 4$

$x = - 3$

$y = - 4$

Step 5

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

$(- 3, - 4)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(- 3, - 4)$

Equation Form:

$x = - 3, y = - 4$

Step 7

Solve by Substitution 2x+3y=-18 , 5x-6y=9

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