Solve by Substitution 2x+3y=3 , 3x-2y=11

$2 x + 3 y = 3$ , $3 x - 2 y = 11$

Step 1

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

Subtract $3 y$ from both sides of the equation.

$2 x = 3 - 3 y$

$3 x - 2 y = 11$

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

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

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

$3 x - 2 y = 11$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

$3 x - 2 y = 11$

Divide $x$ by $1$ .

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

$3 x - 2 y = 11$

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

$3 x - 2 y = 11$

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

$3 x - 2 y = 11$

Simplify the right side.

Move the negative in front of the fraction.

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

$3 x - 2 y = 11$

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

$3 x - 2 y = 11$

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

$3 x - 2 y = 11$

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

$3 x - 2 y = 11$

Step 2

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

Replace all occurrences of $x$ in $3 x - 2 y = 11$ with $\frac{3}{2} - \frac{3 y}{2}$ .

$3 (\frac{3}{2} - \frac{3 y}{2}) - 2 y = 11$

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

Simplify the left side.

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

Simplify each term.

Apply the distributive property.

$3 (\frac{3}{2}) + 3 (- \frac{3 y}{2}) - 2 y = 11$

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

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

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

$\frac{3 \cdot 3}{2} + 3 (- \frac{3 y}{2}) - 2 y = 11$

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

Multiply $3$ by $3$ .

$\frac{9}{2} + 3 (- \frac{3 y}{2}) - 2 y = 11$

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

$\frac{9}{2} + 3 (- \frac{3 y}{2}) - 2 y = 11$

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

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

Multiply $- 1$ by $3$ .

$\frac{9}{2} - 3 \frac{3 y}{2} - 2 y = 11$

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

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

$\frac{9}{2} + \frac{- 3 (3 y)}{2} - 2 y = 11$

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

Multiply $3$ by $- 3$ .

$\frac{9}{2} + \frac{- 9 y}{2} - 2 y = 11$

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

$\frac{9}{2} + \frac{- 9 y}{2} - 2 y = 11$

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

Move the negative in front of the fraction.

$\frac{9}{2} - \frac{9 y}{2} - 2 y = 11$

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

$\frac{9}{2} - \frac{9 y}{2} - 2 y = 11$

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

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

$\frac{9}{2} - \frac{9 y}{2} - 2 y \cdot \frac{2}{2} = 11$

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

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

$\frac{9}{2} - \frac{9 y}{2} + \frac{- 2 y \cdot 2}{2} = 11$

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

Combine the numerators over the common denominator.

$\frac{9}{2} + \frac{- 9 y - 2 y \cdot 2}{2} = 11$

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

Combine the numerators over the common denominator.

$\frac{9 - 9 y - 2 y \cdot 2}{2} = 11$

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

Multiply $2$ by $- 2$ .

$\frac{9 - 9 y - 4 y}{2} = 11$

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

Subtract $4 y$ from $- 9 y$ .

$\frac{9 - 13 y}{2} = 11$

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

$\frac{9 - 13 y}{2} = 11$

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

$\frac{9 - 13 y}{2} = 11$

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

$\frac{9 - 13 y}{2} = 11$

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

Step 3

Solve for $y$ in $\frac{9 - 13 y}{2} = 11$ .

Multiply both sides by $2$ .

$\frac{9 - 13 y}{2} \cdot 2 = 11 \cdot 2$

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

Simplify.

Simplify the left side.

Simplify $\frac{9 - 13 y}{2} \cdot 2$ .

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{9 - 13 y}{2} \cdot 2 = 11 \cdot 2$

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

Rewrite the expression.

$9 - 13 y = 11 \cdot 2$

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

$9 - 13 y = 11 \cdot 2$

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

Reorder $9$ and $- 13 y$ .

$- 13 y + 9 = 11 \cdot 2$

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

$- 13 y + 9 = 11 \cdot 2$

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

$- 13 y + 9 = 11 \cdot 2$

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

Simplify the right side.

Multiply $11$ by $2$ .

$- 13 y + 9 = 22$

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

$- 13 y + 9 = 22$

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

$- 13 y + 9 = 22$

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

Solve for $y$ .

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

Subtract $9$ from both sides of the equation.

$- 13 y = 22 - 9$

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

Subtract $9$ from $22$ .

$- 13 y = 13$

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

$- 13 y = 13$

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

Divide each term in $- 13 y = 13$ by $- 13$ and simplify.

Divide each term in $- 13 y = 13$ by $- 13$ .

$\frac{- 13 y}{- 13} = \frac{13}{- 13}$

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

Simplify the left side.

Cancel the common factor of $- 13$ .

Cancel the common factor.

$\frac{- 13 y}{- 13} = \frac{13}{- 13}$

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

Divide $y$ by $1$ .

$y = \frac{13}{- 13}$

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

$y = \frac{13}{- 13}$

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

$y = \frac{13}{- 13}$

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

Simplify the right side.

Divide $13$ by $- 13$ .

$y = - 1$

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

$y = - 1$

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

$y = - 1$

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

$y = - 1$

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

$y = - 1$

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

Step 4

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

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

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

$y = - 1$

Simplify the right side.

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

Combine the numerators over the common denominator.

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

$y = - 1$

Simplify the expression.

Multiply $- 3$ by $- 1$ .

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

$y = - 1$

Add $3$ and $3$ .

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

$y = - 1$

Divide $6$ by $2$ .

$x = 3$

$y = - 1$

$x = 3$

$y = - 1$

$x = 3$

$y = - 1$

$x = 3$

$y = - 1$

$x = 3$

$y = - 1$

Step 5

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

$(3, - 1)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(3, - 1)$

Equation Form:

$x = 3, y = - 1$

Step 7

Solve by Substitution 2x+3y=3 , 3x-2y=11

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