Solve by Substitution 2x+2y=2 , 3x-y=1

$2 x + 2 y = 2$ , $3 x - y = 1$

Step 1

Solve for $x$ in $2 x + 2 y = 2$ .

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

$2 x = 2 - 2 y$

$3 x - y = 1$

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

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

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

$3 x - y = 1$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

$3 x - y = 1$

Divide $x$ by $1$ .

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

$3 x - y = 1$

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

$3 x - y = 1$

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

$3 x - y = 1$

Simplify the right side.

Simplify each term.

Divide $2$ by $2$ .

$x = 1 + \frac{- 2 y}{2}$

$3 x - y = 1$

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

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

$x = 1 + \frac{2 (- y)}{2}$

$3 x - y = 1$

Cancel the common factors.

Factor $2$ out of $2$ .

$x = 1 + \frac{2 (- y)}{2 (1)}$

$3 x - y = 1$

Cancel the common factor.

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

$3 x - y = 1$

Rewrite the expression.

$x = 1 + \frac{- y}{1}$

$3 x - y = 1$

Divide $- y$ by $1$ .

$x = 1 - y$

$3 x - y = 1$

$x = 1 - y$

$3 x - y = 1$

$x = 1 - y$

$3 x - y = 1$

$x = 1 - y$

$3 x - y = 1$

$x = 1 - y$

$3 x - y = 1$

$x = 1 - y$

$3 x - y = 1$

$x = 1 - y$

$3 x - y = 1$

Step 2

Replace all occurrences of $x$ with $1 - y$ in each equation.

Replace all occurrences of $x$ in $3 x - y = 1$ with $1 - y$ .

$3 (1 - y) - y = 1$

$x = 1 - y$

Simplify the left side.

Simplify $3 (1 - y) - y$ .

Simplify each term.

Apply the distributive property.

$3 \cdot 1 + 3 (- y) - y = 1$

$x = 1 - y$

Multiply $3$ by $1$ .

$3 + 3 (- y) - y = 1$

$x = 1 - y$

Multiply $- 1$ by $3$ .

$3 - 3 y - y = 1$

$x = 1 - y$

$3 - 3 y - y = 1$

$x = 1 - y$

Subtract $y$ from $- 3 y$ .

$3 - 4 y = 1$

$x = 1 - y$

$3 - 4 y = 1$

$x = 1 - y$

$3 - 4 y = 1$

$x = 1 - y$

$3 - 4 y = 1$

$x = 1 - y$

Step 3

Solve for $y$ in $3 - 4 y = 1$ .

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

Subtract $3$ from both sides of the equation.

$- 4 y = 1 - 3$

$x = 1 - y$

Subtract $3$ from $1$ .

$- 4 y = - 2$

$x = 1 - y$

$- 4 y = - 2$

$x = 1 - y$

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

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

$\frac{- 4 y}{- 4} = \frac{- 2}{- 4}$

$x = 1 - y$

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

$\frac{- 4 y}{- 4} = \frac{- 2}{- 4}$

$x = 1 - y$

Divide $y$ by $1$ .

$y = \frac{- 2}{- 4}$

$x = 1 - y$

$y = \frac{- 2}{- 4}$

$x = 1 - y$

$y = \frac{- 2}{- 4}$

$x = 1 - y$

Simplify the right side.

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

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

$y = \frac{- 2 \cdot 1}{- 4}$

$x = 1 - y$

Cancel the common factors.

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

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

$x = 1 - y$

Cancel the common factor.

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

$x = 1 - y$

Rewrite the expression.

$y = \frac{1}{2}$

$x = 1 - y$

$y = \frac{1}{2}$

$x = 1 - y$

$y = \frac{1}{2}$

$x = 1 - y$

$y = \frac{1}{2}$

$x = 1 - y$

$y = \frac{1}{2}$

$x = 1 - y$

$y = \frac{1}{2}$

$x = 1 - y$

Step 4

Replace all occurrences of $y$ with $\frac{1}{2}$ in each equation.

Replace all occurrences of $y$ in $x = 1 - y$ with $\frac{1}{2}$ .

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

$y = \frac{1}{2}$

Simplify the right side.

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

Write $1$ as a fraction with a common denominator.

$x = \frac{2}{2} - \frac{1}{2}$

$y = \frac{1}{2}$

Combine the numerators over the common denominator.

$x = \frac{2 - 1}{2}$

$y = \frac{1}{2}$

Subtract $1$ from $2$ .

$x = \frac{1}{2}$

$y = \frac{1}{2}$

$x = \frac{1}{2}$

$y = \frac{1}{2}$

$x = \frac{1}{2}$

$y = \frac{1}{2}$

$x = \frac{1}{2}$

$y = \frac{1}{2}$

Step 5

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

$(\frac{1}{2}, \frac{1}{2})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(\frac{1}{2}, \frac{1}{2})$

Equation Form:

$x = \frac{1}{2}, y = \frac{1}{2}$

Step 7

Solve by Substitution 2x+2y=2 , 3x-y=1

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