Solve the System of Equations 4x-5y=-6 and -6x+7y=12

$4 x - 5 y = - 6$ and $- 6 x + 7 y = 12$

Step 1

Solve for $x$ in $4 x - 5 y = - 6$ .

Add $5 y$ to both sides of the equation.

$4 x = - 6 + 5 y$

$- 6 x + 7 y = 12$

Divide each term in $4 x = - 6 + 5 y$ by $4$ and simplify.

Divide each term in $4 x = - 6 + 5 y$ by $4$ .

$\frac{4 x}{4} = \frac{- 6}{4} + \frac{5 y}{4}$

$- 6 x + 7 y = 12$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 6}{4} + \frac{5 y}{4}$

$- 6 x + 7 y = 12$

Divide $x$ by $1$ .

$x = \frac{- 6}{4} + \frac{5 y}{4}$

$- 6 x + 7 y = 12$

$x = \frac{- 6}{4} + \frac{5 y}{4}$

$- 6 x + 7 y = 12$

$x = \frac{- 6}{4} + \frac{5 y}{4}$

$- 6 x + 7 y = 12$

Simplify the right side.

Simplify each term.

Cancel the common factor of $- 6$ and $4$ .

Factor $2$ out of $- 6$ .

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

$- 6 x + 7 y = 12$

Cancel the common factors.

Factor $2$ out of $4$ .

$x = \frac{2 \cdot - 3}{2 \cdot 2} + \frac{5 y}{4}$

$- 6 x + 7 y = 12$

Cancel the common factor.

$x = \frac{2 \cdot - 3}{2 \cdot 2} + \frac{5 y}{4}$

$- 6 x + 7 y = 12$

Rewrite the expression.

$x = \frac{- 3}{2} + \frac{5 y}{4}$

$- 6 x + 7 y = 12$

$x = \frac{- 3}{2} + \frac{5 y}{4}$

$- 6 x + 7 y = 12$

$x = \frac{- 3}{2} + \frac{5 y}{4}$

$- 6 x + 7 y = 12$

Move the negative in front of the fraction.

$x = - \frac{3}{2} + \frac{5 y}{4}$

$- 6 x + 7 y = 12$

$x = - \frac{3}{2} + \frac{5 y}{4}$

$- 6 x + 7 y = 12$

$x = - \frac{3}{2} + \frac{5 y}{4}$

$- 6 x + 7 y = 12$

$x = - \frac{3}{2} + \frac{5 y}{4}$

$- 6 x + 7 y = 12$

$x = - \frac{3}{2} + \frac{5 y}{4}$

$- 6 x + 7 y = 12$

Step 2

Replace all occurrences of $x$ with $- \frac{3}{2} + \frac{5 y}{4}$ in each equation.

Replace all occurrences of $x$ in $- 6 x + 7 y = 12$ with $- \frac{3}{2} + \frac{5 y}{4}$ .

$- 6 (- \frac{3}{2} + \frac{5 y}{4}) + 7 y = 12$

$x = - \frac{3}{2} + \frac{5 y}{4}$

Simplify the left side.

Simplify $- 6 (- \frac{3}{2} + \frac{5 y}{4}) + 7 y$ .

Simplify each term.

Apply the distributive property.

$- 6 (- \frac{3}{2}) - 6 \frac{5 y}{4} + 7 y = 12$

$x = - \frac{3}{2} + \frac{5 y}{4}$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$- 6 (\frac{- 3}{2}) - 6 \frac{5 y}{4} + 7 y = 12$

$x = - \frac{3}{2} + \frac{5 y}{4}$

Factor $2$ out of $- 6$ .

$2 (- 3) (\frac{- 3}{2}) - 6 \frac{5 y}{4} + 7 y = 12$

$x = - \frac{3}{2} + \frac{5 y}{4}$

Cancel the common factor.

$2 \cdot (- 3 (\frac{- 3}{2})) - 6 \frac{5 y}{4} + 7 y = 12$

$x = - \frac{3}{2} + \frac{5 y}{4}$

Rewrite the expression.

$- 3 \cdot - 3 - 6 \frac{5 y}{4} + 7 y = 12$

$x = - \frac{3}{2} + \frac{5 y}{4}$

$- 3 \cdot - 3 - 6 \frac{5 y}{4} + 7 y = 12$

$x = - \frac{3}{2} + \frac{5 y}{4}$

Multiply $- 3$ by $- 3$ .

$9 - 6 \frac{5 y}{4} + 7 y = 12$

$x = - \frac{3}{2} + \frac{5 y}{4}$

Cancel the common factor of $2$ .

Factor $2$ out of $- 6$ .

$9 + 2 (- 3) (\frac{5 y}{4}) + 7 y = 12$

$x = - \frac{3}{2} + \frac{5 y}{4}$

Factor $2$ out of $4$ .

$9 + 2 \cdot (- 3 \frac{5 y}{2 \cdot 2}) + 7 y = 12$

$x = - \frac{3}{2} + \frac{5 y}{4}$

Cancel the common factor.

$9 + 2 \cdot (- 3 \frac{5 y}{2 \cdot 2}) + 7 y = 12$

$x = - \frac{3}{2} + \frac{5 y}{4}$

Rewrite the expression.

$9 - 3 \frac{5 y}{2} + 7 y = 12$

$x = - \frac{3}{2} + \frac{5 y}{4}$

$9 - 3 \frac{5 y}{2} + 7 y = 12$

$x = - \frac{3}{2} + \frac{5 y}{4}$

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

$9 + \frac{- 3 (5 y)}{2} + 7 y = 12$

$x = - \frac{3}{2} + \frac{5 y}{4}$

Multiply $5$ by $- 3$ .

$9 + \frac{- 15 y}{2} + 7 y = 12$

$x = - \frac{3}{2} + \frac{5 y}{4}$

Move the negative in front of the fraction.

$9 - \frac{15 y}{2} + 7 y = 12$

$x = - \frac{3}{2} + \frac{5 y}{4}$

$9 - \frac{15 y}{2} + 7 y = 12$

$x = - \frac{3}{2} + \frac{5 y}{4}$

To write $7 y$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$9 - \frac{15 y}{2} + 7 y \cdot \frac{2}{2} = 12$

$x = - \frac{3}{2} + \frac{5 y}{4}$

Simplify terms.

Combine $7 y$ and $\frac{2}{2}$ .

$9 - \frac{15 y}{2} + \frac{7 y \cdot 2}{2} = 12$

$x = - \frac{3}{2} + \frac{5 y}{4}$

Combine the numerators over the common denominator.

$9 + \frac{- 15 y + 7 y \cdot 2}{2} = 12$

$x = - \frac{3}{2} + \frac{5 y}{4}$

$9 + \frac{- 15 y + 7 y \cdot 2}{2} = 12$

$x = - \frac{3}{2} + \frac{5 y}{4}$

Simplify each term.

Simplify the numerator.

Factor $y$ out of $- 15 y + 7 y \cdot 2$ .

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

$9 + \frac{y \cdot - 15 + 7 y \cdot 2}{2} = 12$

$x = - \frac{3}{2} + \frac{5 y}{4}$

Factor $y$ out of $7 y \cdot 2$ .

$9 + \frac{y \cdot - 15 + y (7 \cdot 2)}{2} = 12$

$x = - \frac{3}{2} + \frac{5 y}{4}$

Factor $y$ out of $y \cdot - 15 + y (7 \cdot 2)$ .

$9 + \frac{y (- 15 + 7 \cdot 2)}{2} = 12$

$x = - \frac{3}{2} + \frac{5 y}{4}$

$9 + \frac{y (- 15 + 7 \cdot 2)}{2} = 12$

$x = - \frac{3}{2} + \frac{5 y}{4}$

Multiply $7$ by $2$ .

$9 + \frac{y (- 15 + 14)}{2} = 12$

$x = - \frac{3}{2} + \frac{5 y}{4}$

Add $- 15$ and $14$ .

$9 + \frac{y \cdot - 1}{2} = 12$

$x = - \frac{3}{2} + \frac{5 y}{4}$

$9 + \frac{y \cdot - 1}{2} = 12$

$x = - \frac{3}{2} + \frac{5 y}{4}$

Move $- 1$ to the left of $y$ .

$9 + \frac{- 1 \cdot y}{2} = 12$

$x = - \frac{3}{2} + \frac{5 y}{4}$

Move the negative in front of the fraction.

$9 - \frac{y}{2} = 12$

$x = - \frac{3}{2} + \frac{5 y}{4}$

$9 - \frac{y}{2} = 12$

$x = - \frac{3}{2} + \frac{5 y}{4}$

$9 - \frac{y}{2} = 12$

$x = - \frac{3}{2} + \frac{5 y}{4}$

$9 - \frac{y}{2} = 12$

$x = - \frac{3}{2} + \frac{5 y}{4}$

$9 - \frac{y}{2} = 12$

$x = - \frac{3}{2} + \frac{5 y}{4}$

Step 3

Solve for $y$ in $9 - \frac{y}{2} = 12$ .

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

Subtract $9$ from both sides of the equation.

$- \frac{y}{2} = 12 - 9$

$x = - \frac{3}{2} + \frac{5 y}{4}$

Subtract $9$ from $12$ .

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

$x = - \frac{3}{2} + \frac{5 y}{4}$

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

$x = - \frac{3}{2} + \frac{5 y}{4}$

Multiply both sides of the equation by $- 2$ .

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

$x = - \frac{3}{2} + \frac{5 y}{4}$

Simplify both sides of the equation.

Simplify the left side.

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

Cancel the common factor of $2$ .

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

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

$x = - \frac{3}{2} + \frac{5 y}{4}$

Factor $2$ out of $- 2$ .

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

$x = - \frac{3}{2} + \frac{5 y}{4}$

Cancel the common factor.

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

$x = - \frac{3}{2} + \frac{5 y}{4}$

Rewrite the expression.

$y = - 2 \cdot 3$

$x = - \frac{3}{2} + \frac{5 y}{4}$

$y = - 2 \cdot 3$

$x = - \frac{3}{2} + \frac{5 y}{4}$

Multiply.

Multiply $- 1$ by $- 1$ .

$1 y = - 2 \cdot 3$

$x = - \frac{3}{2} + \frac{5 y}{4}$

Multiply $y$ by $1$ .

$y = - 2 \cdot 3$

$x = - \frac{3}{2} + \frac{5 y}{4}$

$y = - 2 \cdot 3$

$x = - \frac{3}{2} + \frac{5 y}{4}$

$y = - 2 \cdot 3$

$x = - \frac{3}{2} + \frac{5 y}{4}$

$y = - 2 \cdot 3$

$x = - \frac{3}{2} + \frac{5 y}{4}$

Simplify the right side.

Multiply $- 2$ by $3$ .

$y = - 6$

$x = - \frac{3}{2} + \frac{5 y}{4}$

$y = - 6$

$x = - \frac{3}{2} + \frac{5 y}{4}$

$y = - 6$

$x = - \frac{3}{2} + \frac{5 y}{4}$

$y = - 6$

$x = - \frac{3}{2} + \frac{5 y}{4}$

Step 4

Replace all occurrences of $y$ with $- 6$ in each equation.

Replace all occurrences of $y$ in $x = - \frac{3}{2} + \frac{5 y}{4}$ with $- 6$ .

$x = - \frac{3}{2} + \frac{5 (- 6)}{4}$

$y = - 6$

Simplify the right side.

Simplify $- \frac{3}{2} + \frac{5 (- 6)}{4}$ .

Multiply $5$ by $- 6$ .

$x = - \frac{3}{2} + \frac{- 30}{4}$

$y = - 6$

Simplify each term.

Cancel the common factor of $- 30$ and $4$ .

Factor $2$ out of $- 30$ .

$x = - \frac{3}{2} + \frac{2 (- 15)}{4}$

$y = - 6$

Cancel the common factors.

Factor $2$ out of $4$ .

$x = - \frac{3}{2} + \frac{2 \cdot - 15}{2 \cdot 2}$

$y = - 6$

Cancel the common factor.

$x = - \frac{3}{2} + \frac{2 \cdot - 15}{2 \cdot 2}$

$y = - 6$

Rewrite the expression.

$x = - \frac{3}{2} + \frac{- 15}{2}$

$y = - 6$

$x = - \frac{3}{2} + \frac{- 15}{2}$

$y = - 6$

$x = - \frac{3}{2} + \frac{- 15}{2}$

$y = - 6$

Move the negative in front of the fraction.

$x = - \frac{3}{2} - \frac{15}{2}$

$y = - 6$

$x = - \frac{3}{2} - \frac{15}{2}$

$y = - 6$

Combine fractions.

Combine the numerators over the common denominator.

$x = \frac{- 3 - 15}{2}$

$y = - 6$

Simplify the expression.

Subtract $15$ from $- 3$ .

$x = \frac{- 18}{2}$

$y = - 6$

Divide $- 18$ by $2$ .

$x = - 9$

$y = - 6$

$x = - 9$

$y = - 6$

$x = - 9$

$y = - 6$

$x = - 9$

$y = - 6$

$x = - 9$

$y = - 6$

$x = - 9$

$y = - 6$

Step 5

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

$(- 9, - 6)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(- 9, - 6)$

Equation Form:

$x = - 9, y = - 6$

Step 7

Solve the System of Equations 4x-5y=-6 and -6x+7y=12

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