Solve the System of Equations -4x+2y=-6 and -x+8y=21

$- 4 x + 2 y = - 6$ and $- x + 8 y = 21$

Step 1

Solve for $x$ in $- 4 x + 2 y = - 6$ .

Subtract $2 y$ from both sides of the equation.

$- 4 x = - 6 - 2 y$

$- x + 8 y = 21$

Divide each term in $- 4 x = - 6 - 2 y$ by $- 4$ and simplify.

Divide each term in $- 4 x = - 6 - 2 y$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{- 6}{- 4} + \frac{- 2 y}{- 4}$

$- x + 8 y = 21$

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{- 6}{- 4} + \frac{- 2 y}{- 4}$

$- x + 8 y = 21$

Divide $x$ by $1$ .

$x = \frac{- 6}{- 4} + \frac{- 2 y}{- 4}$

$- x + 8 y = 21$

$x = \frac{- 6}{- 4} + \frac{- 2 y}{- 4}$

$- x + 8 y = 21$

$x = \frac{- 6}{- 4} + \frac{- 2 y}{- 4}$

$- x + 8 y = 21$

Simplify the right side.

Simplify each term.

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

Factor $- 2$ out of $- 6$ .

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

$- x + 8 y = 21$

Cancel the common factors.

Factor $- 2$ out of $- 4$ .

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

$- x + 8 y = 21$

Cancel the common factor.

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

$- x + 8 y = 21$

Rewrite the expression.

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

$- x + 8 y = 21$

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

$- x + 8 y = 21$

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

$- x + 8 y = 21$

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

Factor $- 2$ out of $- 2 y$ .

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

$- x + 8 y = 21$

Cancel the common factors.

Factor $- 2$ out of $- 4$ .

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

$- x + 8 y = 21$

Cancel the common factor.

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

$- x + 8 y = 21$

Rewrite the expression.

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

$- x + 8 y = 21$

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

$- x + 8 y = 21$

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

$- x + 8 y = 21$

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

$- x + 8 y = 21$

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

$- x + 8 y = 21$

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

$- x + 8 y = 21$

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

$- x + 8 y = 21$

Step 2

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

Replace all occurrences of $x$ in $- x + 8 y = 21$ with $\frac{3}{2} + \frac{y}{2}$ .

$- (\frac{3}{2} + \frac{y}{2}) + 8 y = 21$

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

Simplify the left side.

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

Apply the distributive property.

$- \frac{3}{2} - \frac{y}{2} + 8 y = 21$

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

To write $8 y$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{3}{2} - \frac{y}{2} + 8 y \cdot \frac{2}{2} = 21$

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

Combine $8 y$ and $\frac{2}{2}$ .

$- \frac{3}{2} - \frac{y}{2} + \frac{8 y \cdot 2}{2} = 21$

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

Combine the numerators over the common denominator.

$- \frac{3}{2} + \frac{- y + 8 y \cdot 2}{2} = 21$

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

Combine the numerators over the common denominator.

$\frac{- 3 - y + 8 y \cdot 2}{2} = 21$

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

Multiply $2$ by $8$ .

$\frac{- 3 - y + 16 y}{2} = 21$

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

Add $- y$ and $16 y$ .

$\frac{- 3 + 15 y}{2} = 21$

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

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

Factor $3$ out of $- 3$ .

$\frac{3 (- 1) + 15 y}{2} = 21$

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

Factor $3$ out of $15 y$ .

$\frac{3 (- 1) + 3 (5 y)}{2} = 21$

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

Factor $3$ out of $3 (- 1) + 3 (5 y)$ .

$\frac{3 (- 1 + 5 y)}{2} = 21$

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

$\frac{3 (- 1 + 5 y)}{2} = 21$

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

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

$\frac{3 (- 1 \cdot 1 + 5 y)}{2} = 21$

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

Factor $- 1$ out of $5 y$ .

$\frac{3 (- 1 \cdot 1 - (- 5 y))}{2} = 21$

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

Factor $- 1$ out of $- 1 (1) - (- 5 y)$ .

$\frac{3 (- 1 (1 - 5 y))}{2} = 21$

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

Move the negative in front of the fraction.

$- \frac{3 (1 - 5 y)}{2} = 21$

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

$- \frac{3 (1 - 5 y)}{2} = 21$

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

$- \frac{3 (1 - 5 y)}{2} = 21$

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

$- \frac{3 (1 - 5 y)}{2} = 21$

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

Step 3

Solve for $y$ in $- \frac{3 (1 - 5 y)}{2} = 21$ .

Multiply both sides of the equation by $- \frac{2}{3}$ .

$- \frac{2}{3} \cdot (- \frac{3 (1 - 5 y)}{2}) = - \frac{2}{3} \cdot 21$

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

Simplify both sides of the equation.

Simplify the left side.

Simplify $- \frac{2}{3} (- \frac{3 (1 - 5 y)}{2})$ .

Cancel the common factor of $2$ .

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

$\frac{- 2}{3} \cdot (- \frac{3 (1 - 5 y)}{2}) = - \frac{2}{3} \cdot 21$

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

Move the leading negative in $- \frac{3 (1 - 5 y)}{2}$ into the numerator.

$\frac{- 2}{3} \cdot \frac{- 3 (1 - 5 y)}{2} = - \frac{2}{3} \cdot 21$

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

Factor $2$ out of $- 2$ .

$\frac{2 (- 1)}{3} \cdot \frac{- 3 (1 - 5 y)}{2} = - \frac{2}{3} \cdot 21$

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

Cancel the common factor.

$\frac{2 \cdot - 1}{3} \cdot \frac{- 3 (1 - 5 y)}{2} = - \frac{2}{3} \cdot 21$

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

Rewrite the expression.

$\frac{- 1}{3} \cdot (- 3 (1 - 5 y)) = - \frac{2}{3} \cdot 21$

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

$\frac{- 1}{3} \cdot (- 3 (1 - 5 y)) = - \frac{2}{3} \cdot 21$

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

Cancel the common factor of $3$ .

Factor $3$ out of $- 3 (1 - 5 y)$ .

$\frac{- 1}{3} \cdot (3 (- (1 - 5 y))) = - \frac{2}{3} \cdot 21$

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

Cancel the common factor.

$\frac{- 1}{3} \cdot (3 (- (1 - 5 y))) = - \frac{2}{3} \cdot 21$

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

Rewrite the expression.

$1 - 5 y = - \frac{2}{3} \cdot 21$

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

$1 - 5 y = - \frac{2}{3} \cdot 21$

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

Multiply.

Multiply $- 1$ by $- 1$ .

$1 (1 - 5 y) = - \frac{2}{3} \cdot 21$

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

Multiply $1 - 5 y$ by $1$ .

$1 - 5 y = - \frac{2}{3} \cdot 21$

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

$1 - 5 y = - \frac{2}{3} \cdot 21$

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

$1 - 5 y = - \frac{2}{3} \cdot 21$

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

$1 - 5 y = - \frac{2}{3} \cdot 21$

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

Simplify the right side.

Simplify $- \frac{2}{3} \cdot 21$ .

Cancel the common factor of $3$ .

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

$1 - 5 y = \frac{- 2}{3} \cdot 21$

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

Factor $3$ out of $21$ .

$1 - 5 y = \frac{- 2}{3} \cdot (3 (7))$

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

Cancel the common factor.

$1 - 5 y = \frac{- 2}{3} \cdot (3 \cdot 7)$

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

Rewrite the expression.

$1 - 5 y = - 2 \cdot 7$

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

$1 - 5 y = - 2 \cdot 7$

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

Multiply $- 2$ by $7$ .

$1 - 5 y = - 14$

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

$1 - 5 y = - 14$

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

$1 - 5 y = - 14$

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

$1 - 5 y = - 14$

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

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

Subtract $1$ from both sides of the equation.

$- 5 y = - 14 - 1$

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

Subtract $1$ from $- 14$ .

$- 5 y = - 15$

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

$- 5 y = - 15$

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

Divide each term in $- 5 y = - 15$ by $- 5$ and simplify.

Divide each term in $- 5 y = - 15$ by $- 5$ .

$\frac{- 5 y}{- 5} = \frac{- 15}{- 5}$

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

Simplify the left side.

Cancel the common factor of $- 5$ .

Cancel the common factor.

$\frac{- 5 y}{- 5} = \frac{- 15}{- 5}$

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

Divide $y$ by $1$ .

$y = \frac{- 15}{- 5}$

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

$y = \frac{- 15}{- 5}$

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

$y = \frac{- 15}{- 5}$

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

Simplify the right side.

Divide $- 15$ by $- 5$ .

$y = 3$

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

$y = 3$

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

$y = 3$

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

$y = 3$

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

Step 4

Replace all occurrences of $y$ with $3$ in each equation.

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

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

$y = 3$

Simplify the right side.

Simplify $\frac{3}{2} + \frac{3}{2}$ .

Combine the numerators over the common denominator.

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

$y = 3$

Simplify the expression.

Add $3$ and $3$ .

$x = \frac{6}{2}$

$y = 3$

Divide $6$ by $2$ .

$x = 3$

$y = 3$

$x = 3$

$y = 3$

$x = 3$

$y = 3$

$x = 3$

$y = 3$

$x = 3$

$y = 3$

Step 5

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

$(3, 3)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(3, 3)$

Equation Form:

$x = 3, y = 3$

Step 7

Solve the System of Equations -4x+2y=-6 and -x+8y=21

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