Solve the System of Equations -2x-3y=6 and -x+6y=18

$- 2 x - 3 y = 6$ and $- x + 6 y = 18$

Step 1

Solve for $x$ in $- 2 x - 3 y = 6$ .

Add $3 y$ to both sides of the equation.

$- 2 x = 6 + 3 y$

$- x + 6 y = 18$

Divide each term in $- 2 x = 6 + 3 y$ by $- 2$ and simplify.

Divide each term in $- 2 x = 6 + 3 y$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{6}{- 2} + \frac{3 y}{- 2}$

$- x + 6 y = 18$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{6}{- 2} + \frac{3 y}{- 2}$

$- x + 6 y = 18$

Divide $x$ by $1$ .

$x = \frac{6}{- 2} + \frac{3 y}{- 2}$

$- x + 6 y = 18$

$x = \frac{6}{- 2} + \frac{3 y}{- 2}$

$- x + 6 y = 18$

$x = \frac{6}{- 2} + \frac{3 y}{- 2}$

$- x + 6 y = 18$

Simplify the right side.

Simplify each term.

Divide $6$ by $- 2$ .

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

$- x + 6 y = 18$

Move the negative in front of the fraction.

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

$- x + 6 y = 18$

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

$- x + 6 y = 18$

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

$- x + 6 y = 18$

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

$- x + 6 y = 18$

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

$- x + 6 y = 18$

Step 2

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

Replace all occurrences of $x$ in $- x + 6 y = 18$ with $- 3 - \frac{3 y}{2}$ .

$- (- 3 - \frac{3 y}{2}) + 6 y = 18$

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

Simplify the left side.

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

Simplify each term.

Apply the distributive property.

$3 + \frac{3 y}{2} + 6 y = 18$

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

Multiply $- 1$ by $- 3$ .

$3 + \frac{3 y}{2} + 6 y = 18$

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

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

Multiply $- 1$ by $- 1$ .

$3 + 1 (\frac{3 y}{2}) + 6 y = 18$

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

Multiply $\frac{3 y}{2}$ by $1$ .

$3 + \frac{3 y}{2} + 6 y = 18$

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

$3 + \frac{3 y}{2} + 6 y = 18$

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

$3 + \frac{3 y}{2} + 6 y = 18$

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

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

$3 + \frac{3 y}{2} + 6 y \cdot \frac{2}{2} = 18$

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

Simplify terms.

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

$3 + \frac{3 y}{2} + \frac{6 y \cdot 2}{2} = 18$

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

Combine the numerators over the common denominator.

$3 + \frac{3 y + 6 y \cdot 2}{2} = 18$

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

$3 + \frac{3 y + 6 y \cdot 2}{2} = 18$

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

Simplify the numerator.

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

Factor $3 y$ out of $3 y$ .

$3 + \frac{3 y (1) + 6 y \cdot 2}{2} = 18$

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

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

$3 + \frac{3 y (1) + 3 y (2 \cdot 2)}{2} = 18$

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

Factor $3 y$ out of $3 y (1) + 3 y (2 \cdot 2)$ .

$3 + \frac{3 y (1 + 2 \cdot 2)}{2} = 18$

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

$3 + \frac{3 y (1 + 2 \cdot 2)}{2} = 18$

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

Multiply $2$ by $2$ .

$3 + \frac{3 y (1 + 4)}{2} = 18$

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

Add $1$ and $4$ .

$3 + \frac{3 y \cdot 5}{2} = 18$

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

Multiply $5$ by $3$ .

$3 + \frac{15 y}{2} = 18$

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

$3 + \frac{15 y}{2} = 18$

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

$3 + \frac{15 y}{2} = 18$

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

$3 + \frac{15 y}{2} = 18$

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

$3 + \frac{15 y}{2} = 18$

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

Step 3

Solve for $y$ in $3 + \frac{15 y}{2} = 18$ .

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

Subtract $3$ from both sides of the equation.

$\frac{15 y}{2} = 18 - 3$

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

Subtract $3$ from $18$ .

$\frac{15 y}{2} = 15$

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

$\frac{15 y}{2} = 15$

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

Multiply both sides of the equation by $\frac{2}{15}$ .

$\frac{2}{15} \cdot \frac{15 y}{2} = \frac{2}{15} \cdot 15$

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

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{2}{15} \cdot \frac{15 y}{2}$ .

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2}{15} \cdot \frac{15 y}{2} = \frac{2}{15} \cdot 15$

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

Rewrite the expression.

$\frac{1}{15} \cdot (15 y) = \frac{2}{15} \cdot 15$

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

$\frac{1}{15} \cdot (15 y) = \frac{2}{15} \cdot 15$

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

Cancel the common factor of $15$ .

Factor $15$ out of $15 y$ .

$\frac{1}{15} \cdot (15 (y)) = \frac{2}{15} \cdot 15$

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

Cancel the common factor.

$\frac{1}{15} \cdot (15 y) = \frac{2}{15} \cdot 15$

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

Rewrite the expression.

$y = \frac{2}{15} \cdot 15$

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

$y = \frac{2}{15} \cdot 15$

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

$y = \frac{2}{15} \cdot 15$

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

$y = \frac{2}{15} \cdot 15$

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

Simplify the right side.

Cancel the common factor of $15$ .

Cancel the common factor.

$y = \frac{2}{15} \cdot 15$

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

Rewrite the expression.

$y = 2$

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

$y = 2$

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

$y = 2$

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

$y = 2$

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

$y = 2$

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

Step 4

Replace all occurrences of $y$ with $2$ in each equation.

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

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

$y = 2$

Simplify the right side.

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

Simplify each term.

Cancel the common factor of $2$ .

Cancel the common factor.

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

$y = 2$

Divide $3$ by $1$ .

$x = - 3 - 1 \cdot 3$

$y = 2$

$x = - 3 - 1 \cdot 3$

$y = 2$

Multiply $- 1$ by $3$ .

$x = - 3 - 3$

$y = 2$

$x = - 3 - 3$

$y = 2$

Subtract $3$ from $- 3$ .

$x = - 6$

$y = 2$

$x = - 6$

$y = 2$

$x = - 6$

$y = 2$

$x = - 6$

$y = 2$

Step 5

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

$(- 6, 2)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(- 6, 2)$

Equation Form:

$x = - 6, y = 2$

Step 7

Solve the System of Equations -2x-3y=6 and -x+6y=18

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