Solve by Substitution 2x+3y=18 , 5x-y=11

$2 x + 3 y = 18$ , $5 x - y = 11$

Step 1

Solve for $x$ in $2 x + 3 y = 18$ .

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

$2 x = 18 - 3 y$

$5 x - y = 11$

Divide each term in $2 x = 18 - 3 y$ by $2$ and simplify.

Divide each term in $2 x = 18 - 3 y$ by $2$ .

$\frac{2 x}{2} = \frac{18}{2} + \frac{- 3 y}{2}$

$5 x - y = 11$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{18}{2} + \frac{- 3 y}{2}$

$5 x - y = 11$

Divide $x$ by $1$ .

$x = \frac{18}{2} + \frac{- 3 y}{2}$

$5 x - y = 11$

$x = \frac{18}{2} + \frac{- 3 y}{2}$

$5 x - y = 11$

$x = \frac{18}{2} + \frac{- 3 y}{2}$

$5 x - y = 11$

Simplify the right side.

Simplify each term.

Divide $18$ by $2$ .

$x = 9 + \frac{- 3 y}{2}$

$5 x - y = 11$

Move the negative in front of the fraction.

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

$5 x - y = 11$

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

$5 x - y = 11$

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

$5 x - y = 11$

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

$5 x - y = 11$

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

$5 x - y = 11$

Step 2

Replace all occurrences of $x$ with $9 - \frac{3 y}{2}$ in each equation.

Replace all occurrences of $x$ in $5 x - y = 11$ with $9 - \frac{3 y}{2}$ .

$5 (9 - \frac{3 y}{2}) - y = 11$

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

Simplify the left side.

Simplify $5 (9 - \frac{3 y}{2}) - y$ .

Simplify each term.

Apply the distributive property.

$5 \cdot 9 + 5 (- \frac{3 y}{2}) - y = 11$

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

Multiply $5$ by $9$ .

$45 + 5 (- \frac{3 y}{2}) - y = 11$

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

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

Multiply $- 1$ by $5$ .

$45 - 5 \frac{3 y}{2} - y = 11$

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

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

$45 + \frac{- 5 (3 y)}{2} - y = 11$

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

Multiply $3$ by $- 5$ .

$45 + \frac{- 15 y}{2} - y = 11$

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

$45 + \frac{- 15 y}{2} - y = 11$

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

Move the negative in front of the fraction.

$45 - \frac{15 y}{2} - y = 11$

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

$45 - \frac{15 y}{2} - y = 11$

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

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

$45 - \frac{15 y}{2} - y \cdot \frac{2}{2} = 11$

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

Simplify terms.

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

$45 - \frac{15 y}{2} + \frac{- y \cdot 2}{2} = 11$

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

Combine the numerators over the common denominator.

$45 + \frac{- 15 y - y \cdot 2}{2} = 11$

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

$45 + \frac{- 15 y - y \cdot 2}{2} = 11$

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

Simplify each term.

Simplify the numerator.

Factor $y$ out of $- 15 y - y \cdot 2$ .

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

$45 + \frac{y \cdot - 15 - y \cdot 2}{2} = 11$

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

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

$45 + \frac{y \cdot - 15 + y (- 1 \cdot 2)}{2} = 11$

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

Factor $y$ out of $y \cdot - 15 + y (- 1 \cdot 2)$ .

$45 + \frac{y (- 15 - 1 \cdot 2)}{2} = 11$

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

$45 + \frac{y (- 15 - 1 \cdot 2)}{2} = 11$

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

Multiply $- 1$ by $2$ .

$45 + \frac{y (- 15 - 2)}{2} = 11$

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

Subtract $2$ from $- 15$ .

$45 + \frac{y \cdot - 17}{2} = 11$

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

$45 + \frac{y \cdot - 17}{2} = 11$

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

Move $- 17$ to the left of $y$ .

$45 + \frac{- 17 \cdot y}{2} = 11$

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

Move the negative in front of the fraction.

$45 - \frac{17 y}{2} = 11$

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

$45 - \frac{17 y}{2} = 11$

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

$45 - \frac{17 y}{2} = 11$

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

$45 - \frac{17 y}{2} = 11$

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

$45 - \frac{17 y}{2} = 11$

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

Step 3

Solve for $y$ in $45 - \frac{17 y}{2} = 11$ .

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

Subtract $45$ from both sides of the equation.

$- \frac{17 y}{2} = 11 - 45$

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

Subtract $45$ from $11$ .

$- \frac{17 y}{2} = - 34$

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

$- \frac{17 y}{2} = - 34$

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

Multiply both sides of the equation by $- \frac{2}{17}$ .

$- \frac{2}{17} \cdot (- \frac{17 y}{2}) = - \frac{2}{17} \cdot - 34$

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

Simplify both sides of the equation.

Simplify the left side.

Simplify $- \frac{2}{17} (- \frac{17 y}{2})$ .

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{2}{17}$ into the numerator.

$\frac{- 2}{17} \cdot (- \frac{17 y}{2}) = - \frac{2}{17} \cdot - 34$

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

Move the leading negative in $- \frac{17 y}{2}$ into the numerator.

$\frac{- 2}{17} \cdot \frac{- 17 y}{2} = - \frac{2}{17} \cdot - 34$

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

Factor $2$ out of $- 2$ .

$\frac{2 (- 1)}{17} \cdot \frac{- 17 y}{2} = - \frac{2}{17} \cdot - 34$

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

Cancel the common factor.

$\frac{2 \cdot - 1}{17} \cdot \frac{- 17 y}{2} = - \frac{2}{17} \cdot - 34$

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

Rewrite the expression.

$\frac{- 1}{17} \cdot (- 17 y) = - \frac{2}{17} \cdot - 34$

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

$\frac{- 1}{17} \cdot (- 17 y) = - \frac{2}{17} \cdot - 34$

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

Cancel the common factor of $17$ .

Factor $17$ out of $- 17 y$ .

$\frac{- 1}{17} \cdot (17 (- y)) = - \frac{2}{17} \cdot - 34$

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

Cancel the common factor.

$\frac{- 1}{17} \cdot (17 (- y)) = - \frac{2}{17} \cdot - 34$

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

Rewrite the expression.

$y = - \frac{2}{17} \cdot - 34$

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

$y = - \frac{2}{17} \cdot - 34$

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

Multiply.

Multiply $- 1$ by $- 1$ .

$1 y = - \frac{2}{17} \cdot - 34$

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

Multiply $y$ by $1$ .

$y = - \frac{2}{17} \cdot - 34$

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

$y = - \frac{2}{17} \cdot - 34$

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

$y = - \frac{2}{17} \cdot - 34$

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

$y = - \frac{2}{17} \cdot - 34$

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

Simplify the right side.

Simplify $- \frac{2}{17} \cdot - 34$ .

Cancel the common factor of $17$ .

Move the leading negative in $- \frac{2}{17}$ into the numerator.

$y = \frac{- 2}{17} \cdot - 34$

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

Factor $17$ out of $- 34$ .

$y = \frac{- 2}{17} \cdot (17 (- 2))$

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

Cancel the common factor.

$y = \frac{- 2}{17} \cdot (17 \cdot - 2)$

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

Rewrite the expression.

$y = - 2 \cdot - 2$

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

$y = - 2 \cdot - 2$

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

Multiply $- 2$ by $- 2$ .

$y = 4$

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

$y = 4$

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

$y = 4$

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

$y = 4$

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

$y = 4$

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

Step 4

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

Replace all occurrences of $y$ in $x = 9 - \frac{3 y}{2}$ with $4$ .

$x = 9 - \frac{3 (4)}{2}$

$y = 4$

Simplify the right side.

Simplify $9 - \frac{3 (4)}{2}$ .

Simplify each term.

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

Factor $2$ out of $3 (4)$ .

$x = 9 - \frac{2 (3 \cdot (2))}{2}$

$y = 4$

Cancel the common factors.

Factor $2$ out of $2$ .

$x = 9 - \frac{2 (3 \cdot (2))}{2 (1)}$

$y = 4$

Cancel the common factor.

$x = 9 - \frac{2 (3 \cdot (2))}{2 \cdot 1}$

$y = 4$

Rewrite the expression.

$x = 9 - \frac{3 \cdot (2)}{1}$

$y = 4$

Divide $3 \cdot (2)$ by $1$ .

$x = 9 - (3 \cdot (2))$

$y = 4$

$x = 9 - (3 \cdot (2))$

$y = 4$

$x = 9 - (3 \cdot (2))$

$y = 4$

Multiply $- (3 \cdot (2))$ .

Multiply $3$ by $2$ .

$x = 9 - 1 \cdot 6$

$y = 4$

Multiply $- 1$ by $6$ .

$x = 9 - 6$

$y = 4$

$x = 9 - 6$

$y = 4$

$x = 9 - 6$

$y = 4$

Subtract $6$ from $9$ .

$x = 3$

$y = 4$

$x = 3$

$y = 4$

$x = 3$

$y = 4$

$x = 3$

$y = 4$

Step 5

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

$(3, 4)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(3, 4)$

Equation Form:

$x = 3, y = 4$

Step 7

Solve by Substitution 2x+3y=18 , 5x-y=11

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