Solve by Substitution 3x-2y=2 , 5x-5y=10

$3 x - 2 y = 2$ , $5 x - 5 y = 10$

Step 1

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

Add $2 y$ to both sides of the equation.

$3 x = 2 + 2 y$

$5 x - 5 y = 10$

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

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

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

$5 x - 5 y = 10$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

$5 x - 5 y = 10$

Divide $x$ by $1$ .

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

$5 x - 5 y = 10$

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

$5 x - 5 y = 10$

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

$5 x - 5 y = 10$

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

$5 x - 5 y = 10$

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

$5 x - 5 y = 10$

Step 2

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

Replace all occurrences of $x$ in $5 x - 5 y = 10$ with $\frac{2}{3} + \frac{2 y}{3}$ .

$5 (\frac{2}{3} + \frac{2 y}{3}) - 5 y = 10$

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

Simplify the left side.

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

Simplify each term.

Apply the distributive property.

$5 (\frac{2}{3}) + 5 (\frac{2 y}{3}) - 5 y = 10$

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

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

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

$\frac{5 \cdot 2}{3} + 5 (\frac{2 y}{3}) - 5 y = 10$

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

Multiply $5$ by $2$ .

$\frac{10}{3} + 5 (\frac{2 y}{3}) - 5 y = 10$

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

$\frac{10}{3} + 5 (\frac{2 y}{3}) - 5 y = 10$

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

Multiply $5 \frac{2 y}{3}$ .

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

$\frac{10}{3} + \frac{5 (2 y)}{3} - 5 y = 10$

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

Multiply $2$ by $5$ .

$\frac{10}{3} + \frac{10 y}{3} - 5 y = 10$

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

$\frac{10}{3} + \frac{10 y}{3} - 5 y = 10$

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

$\frac{10}{3} + \frac{10 y}{3} - 5 y = 10$

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

To write $- 5 y$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{10}{3} + \frac{10 y}{3} - 5 y \cdot \frac{3}{3} = 10$

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

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

$\frac{10}{3} + \frac{10 y}{3} + \frac{- 5 y \cdot 3}{3} = 10$

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

Combine the numerators over the common denominator.

$\frac{10}{3} + \frac{10 y - 5 y \cdot 3}{3} = 10$

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

Combine the numerators over the common denominator.

$\frac{10 + 10 y - 5 y \cdot 3}{3} = 10$

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

Multiply $3$ by $- 5$ .

$\frac{10 + 10 y - 15 y}{3} = 10$

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

Subtract $15 y$ from $10 y$ .

$\frac{10 - 5 y}{3} = 10$

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

Factor $5$ out of $10 - 5 y$ .

Factor $5$ out of $10$ .

$\frac{5 (2) - 5 y}{3} = 10$

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

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

$\frac{5 (2) + 5 (- y)}{3} = 10$

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

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

$\frac{5 (2 - y)}{3} = 10$

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

$\frac{5 (2 - y)}{3} = 10$

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

$\frac{5 (2 - y)}{3} = 10$

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

$\frac{5 (2 - y)}{3} = 10$

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

$\frac{5 (2 - y)}{3} = 10$

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

Step 3

Solve for $y$ in $\frac{5 (2 - y)}{3} = 10$ .

Multiply both sides by $3$ .

$\frac{5 (2 - y)}{3} \cdot 3 = 10 \cdot 3$

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

Simplify.

Simplify the left side.

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

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{5 (2 - y)}{3} \cdot 3 = 10 \cdot 3$

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

Rewrite the expression.

$5 (2 - y) = 10 \cdot 3$

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

$5 (2 - y) = 10 \cdot 3$

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

Apply the distributive property.

$5 \cdot 2 + 5 (- y) = 10 \cdot 3$

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

Simplify the expression.

Multiply $5$ by $2$ .

$10 + 5 (- y) = 10 \cdot 3$

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

Multiply $- 1$ by $5$ .

$10 - 5 y = 10 \cdot 3$

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

Reorder $10$ and $- 5 y$ .

$- 5 y + 10 = 10 \cdot 3$

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

$- 5 y + 10 = 10 \cdot 3$

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

$- 5 y + 10 = 10 \cdot 3$

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

$- 5 y + 10 = 10 \cdot 3$

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

Simplify the right side.

Multiply $10$ by $3$ .

$- 5 y + 10 = 30$

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

$- 5 y + 10 = 30$

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

$- 5 y + 10 = 30$

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

Solve for $y$ .

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

Subtract $10$ from both sides of the equation.

$- 5 y = 30 - 10$

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

Subtract $10$ from $30$ .

$- 5 y = 20$

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

$- 5 y = 20$

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

Divide each term in $- 5 y = 20$ by $- 5$ and simplify.

Divide each term in $- 5 y = 20$ by $- 5$ .

$\frac{- 5 y}{- 5} = \frac{20}{- 5}$

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

Simplify the left side.

Cancel the common factor of $- 5$ .

Cancel the common factor.

$\frac{- 5 y}{- 5} = \frac{20}{- 5}$

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

Divide $y$ by $1$ .

$y = \frac{20}{- 5}$

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

$y = \frac{20}{- 5}$

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

$y = \frac{20}{- 5}$

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

Simplify the right side.

Divide $20$ by $- 5$ .

$y = - 4$

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

$y = - 4$

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

$y = - 4$

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

$y = - 4$

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

$y = - 4$

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

Step 4

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

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

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

$y = - 4$

Simplify the right side.

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

Combine the numerators over the common denominator.

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

$y = - 4$

Simplify the expression.

Multiply $2$ by $- 4$ .

$x = \frac{2 - 8}{3}$

$y = - 4$

Subtract $8$ from $2$ .

$x = \frac{- 6}{3}$

$y = - 4$

Divide $- 6$ by $3$ .

$x = - 2$

$y = - 4$

$x = - 2$

$y = - 4$

$x = - 2$

$y = - 4$

$x = - 2$

$y = - 4$

$x = - 2$

$y = - 4$

Step 5

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

$(- 2, - 4)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(- 2, - 4)$

Equation Form:

$x = - 2, y = - 4$

Step 7

Solve by Substitution 3x-2y=2 , 5x-5y=10

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