Solve by Substitution 3x+2y=4 , 2x-y=5

$3 x + 2 y = 4$ , $2 x - y = 5$

Step 1

Solve for $x$ in $3 x + 2 y = 4$ .

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

$3 x = 4 - 2 y$

$2 x - y = 5$

Divide each term in $3 x = 4 - 2 y$ by $3$ and simplify.

Divide each term in $3 x = 4 - 2 y$ by $3$ .

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

$2 x - y = 5$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

$2 x - y = 5$

Divide $x$ by $1$ .

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

$2 x - y = 5$

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

$2 x - y = 5$

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

$2 x - y = 5$

Simplify the right side.

Move the negative in front of the fraction.

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

$2 x - y = 5$

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

$2 x - y = 5$

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

$2 x - y = 5$

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

$2 x - y = 5$

Step 2

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

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

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

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

Simplify the left side.

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

Simplify each term.

Apply the distributive property.

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

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

Multiply $2 (\frac{4}{3})$ .

Combine $2$ and $\frac{4}{3}$ .

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

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

Multiply $2$ by $4$ .

$\frac{8}{3} + 2 (- \frac{2 y}{3}) - y = 5$

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

$\frac{8}{3} + 2 (- \frac{2 y}{3}) - y = 5$

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

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

Multiply $- 1$ by $2$ .

$\frac{8}{3} - 2 \frac{2 y}{3} - y = 5$

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

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

$\frac{8}{3} + \frac{- 2 (2 y)}{3} - y = 5$

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

Multiply $2$ by $- 2$ .

$\frac{8}{3} + \frac{- 4 y}{3} - y = 5$

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

$\frac{8}{3} + \frac{- 4 y}{3} - y = 5$

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

Move the negative in front of the fraction.

$\frac{8}{3} - \frac{4 y}{3} - y = 5$

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

$\frac{8}{3} - \frac{4 y}{3} - y = 5$

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

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

$\frac{8}{3} - \frac{4 y}{3} - y \cdot \frac{3}{3} = 5$

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

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

$\frac{8}{3} - \frac{4 y}{3} + \frac{- y \cdot 3}{3} = 5$

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

Combine the numerators over the common denominator.

$\frac{8}{3} + \frac{- 4 y - y \cdot 3}{3} = 5$

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

Combine the numerators over the common denominator.

$\frac{8 - 4 y - y \cdot 3}{3} = 5$

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

Multiply $3$ by $- 1$ .

$\frac{8 - 4 y - 3 y}{3} = 5$

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

Subtract $3 y$ from $- 4 y$ .

$\frac{8 - 7 y}{3} = 5$

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

$\frac{8 - 7 y}{3} = 5$

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

$\frac{8 - 7 y}{3} = 5$

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

$\frac{8 - 7 y}{3} = 5$

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

Step 3

Solve for $y$ in $\frac{8 - 7 y}{3} = 5$ .

Multiply both sides by $3$ .

$\frac{8 - 7 y}{3} \cdot 3 = 5 \cdot 3$

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

Simplify.

Simplify the left side.

Simplify $\frac{8 - 7 y}{3} \cdot 3$ .

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{8 - 7 y}{3} \cdot 3 = 5 \cdot 3$

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

Rewrite the expression.

$8 - 7 y = 5 \cdot 3$

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

$8 - 7 y = 5 \cdot 3$

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

Reorder $8$ and $- 7 y$ .

$- 7 y + 8 = 5 \cdot 3$

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

$- 7 y + 8 = 5 \cdot 3$

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

$- 7 y + 8 = 5 \cdot 3$

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

Simplify the right side.

Multiply $5$ by $3$ .

$- 7 y + 8 = 15$

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

$- 7 y + 8 = 15$

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

$- 7 y + 8 = 15$

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

Solve for $y$ .

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

Subtract $8$ from both sides of the equation.

$- 7 y = 15 - 8$

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

Subtract $8$ from $15$ .

$- 7 y = 7$

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

$- 7 y = 7$

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

Divide each term in $- 7 y = 7$ by $- 7$ and simplify.

Divide each term in $- 7 y = 7$ by $- 7$ .

$\frac{- 7 y}{- 7} = \frac{7}{- 7}$

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

Simplify the left side.

Cancel the common factor of $- 7$ .

Cancel the common factor.

$\frac{- 7 y}{- 7} = \frac{7}{- 7}$

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

Divide $y$ by $1$ .

$y = \frac{7}{- 7}$

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

$y = \frac{7}{- 7}$

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

$y = \frac{7}{- 7}$

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

Simplify the right side.

Divide $7$ by $- 7$ .

$y = - 1$

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

$y = - 1$

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

$y = - 1$

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

$y = - 1$

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

$y = - 1$

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

Step 4

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

Replace all occurrences of $y$ in $x = \frac{4}{3} - \frac{2 y}{3}$ with $- 1$ .

$x = \frac{4}{3} - \frac{2 (- 1)}{3}$

$y = - 1$

Simplify the right side.

Simplify $\frac{4}{3} - \frac{2 (- 1)}{3}$ .

Combine the numerators over the common denominator.

$x = \frac{4 - 2 \cdot - 1}{3}$

$y = - 1$

Simplify the expression.

Multiply $- 2$ by $- 1$ .

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

$y = - 1$

Add $4$ and $2$ .

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

$y = - 1$

Divide $6$ by $3$ .

$x = 2$

$y = - 1$

$x = 2$

$y = - 1$

$x = 2$

$y = - 1$

$x = 2$

$y = - 1$

$x = 2$

$y = - 1$

Step 5

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

$(2, - 1)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(2, - 1)$

Equation Form:

$x = 2, y = - 1$

Step 7

Solve by Substitution 3x+2y=4 , 2x-y=5

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