Solve by Substitution 2x-y=7 , 3x-2y=10

$2 x - y = 7$ , $3 x - 2 y = 10$

Step 1

Solve for $x$ in $2 x - y = 7$ .

Add $y$ to both sides of the equation.

$2 x = 7 + y$

$3 x - 2 y = 10$

Divide each term in $2 x = 7 + y$ by $2$ and simplify.

Divide each term in $2 x = 7 + y$ by $2$ .

$\frac{2 x}{2} = \frac{7}{2} + \frac{y}{2}$

$3 x - 2 y = 10$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{7}{2} + \frac{y}{2}$

$3 x - 2 y = 10$

Divide $x$ by $1$ .

$x = \frac{7}{2} + \frac{y}{2}$

$3 x - 2 y = 10$

$x = \frac{7}{2} + \frac{y}{2}$

$3 x - 2 y = 10$

$x = \frac{7}{2} + \frac{y}{2}$

$3 x - 2 y = 10$

$x = \frac{7}{2} + \frac{y}{2}$

$3 x - 2 y = 10$

$x = \frac{7}{2} + \frac{y}{2}$

$3 x - 2 y = 10$

Step 2

Replace all occurrences of $x$ with $\frac{7}{2} + \frac{y}{2}$ in each equation.

Replace all occurrences of $x$ in $3 x - 2 y = 10$ with $\frac{7}{2} + \frac{y}{2}$ .

$3 (\frac{7}{2} + \frac{y}{2}) - 2 y = 10$

$x = \frac{7}{2} + \frac{y}{2}$

Simplify the left side.

Simplify $3 (\frac{7}{2} + \frac{y}{2}) - 2 y$ .

Simplify each term.

Apply the distributive property.

$3 (\frac{7}{2}) + 3 (\frac{y}{2}) - 2 y = 10$

$x = \frac{7}{2} + \frac{y}{2}$

Multiply $3 (\frac{7}{2})$ .

Combine $3$ and $\frac{7}{2}$ .

$\frac{3 \cdot 7}{2} + 3 (\frac{y}{2}) - 2 y = 10$

$x = \frac{7}{2} + \frac{y}{2}$

Multiply $3$ by $7$ .

$\frac{21}{2} + 3 (\frac{y}{2}) - 2 y = 10$

$x = \frac{7}{2} + \frac{y}{2}$

$\frac{21}{2} + 3 (\frac{y}{2}) - 2 y = 10$

$x = \frac{7}{2} + \frac{y}{2}$

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

$\frac{21}{2} + \frac{3 y}{2} - 2 y = 10$

$x = \frac{7}{2} + \frac{y}{2}$

$\frac{21}{2} + \frac{3 y}{2} - 2 y = 10$

$x = \frac{7}{2} + \frac{y}{2}$

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

$\frac{21}{2} + \frac{3 y}{2} - 2 y \cdot \frac{2}{2} = 10$

$x = \frac{7}{2} + \frac{y}{2}$

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

$\frac{21}{2} + \frac{3 y}{2} + \frac{- 2 y \cdot 2}{2} = 10$

$x = \frac{7}{2} + \frac{y}{2}$

Combine the numerators over the common denominator.

$\frac{21}{2} + \frac{3 y - 2 y \cdot 2}{2} = 10$

$x = \frac{7}{2} + \frac{y}{2}$

Combine the numerators over the common denominator.

$\frac{21 + 3 y - 2 y \cdot 2}{2} = 10$

$x = \frac{7}{2} + \frac{y}{2}$

Multiply $2$ by $- 2$ .

$\frac{21 + 3 y - 4 y}{2} = 10$

$x = \frac{7}{2} + \frac{y}{2}$

Subtract $4 y$ from $3 y$ .

$\frac{21 - y}{2} = 10$

$x = \frac{7}{2} + \frac{y}{2}$

$\frac{21 - y}{2} = 10$

$x = \frac{7}{2} + \frac{y}{2}$

$\frac{21 - y}{2} = 10$

$x = \frac{7}{2} + \frac{y}{2}$

$\frac{21 - y}{2} = 10$

$x = \frac{7}{2} + \frac{y}{2}$

Step 3

Solve for $y$ in $\frac{21 - y}{2} = 10$ .

Multiply both sides by $2$ .

$\frac{21 - y}{2} \cdot 2 = 10 \cdot 2$

$x = \frac{7}{2} + \frac{y}{2}$

Simplify.

Simplify the left side.

Simplify $\frac{21 - y}{2} \cdot 2$ .

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{21 - y}{2} \cdot 2 = 10 \cdot 2$

$x = \frac{7}{2} + \frac{y}{2}$

Rewrite the expression.

$21 - y = 10 \cdot 2$

$x = \frac{7}{2} + \frac{y}{2}$

$21 - y = 10 \cdot 2$

$x = \frac{7}{2} + \frac{y}{2}$

Reorder $21$ and $- y$ .

$- y + 21 = 10 \cdot 2$

$x = \frac{7}{2} + \frac{y}{2}$

$- y + 21 = 10 \cdot 2$

$x = \frac{7}{2} + \frac{y}{2}$

$- y + 21 = 10 \cdot 2$

$x = \frac{7}{2} + \frac{y}{2}$

Simplify the right side.

Multiply $10$ by $2$ .

$- y + 21 = 20$

$x = \frac{7}{2} + \frac{y}{2}$

$- y + 21 = 20$

$x = \frac{7}{2} + \frac{y}{2}$

$- y + 21 = 20$

$x = \frac{7}{2} + \frac{y}{2}$

Solve for $y$ .

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

Subtract $21$ from both sides of the equation.

$- y = 20 - 21$

$x = \frac{7}{2} + \frac{y}{2}$

Subtract $21$ from $20$ .

$- y = - 1$

$x = \frac{7}{2} + \frac{y}{2}$

$- y = - 1$

$x = \frac{7}{2} + \frac{y}{2}$

Divide each term in $- y = - 1$ by $- 1$ and simplify.

Divide each term in $- y = - 1$ by $- 1$ .

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

$x = \frac{7}{2} + \frac{y}{2}$

Simplify the left side.

Dividing two negative values results in a positive value.

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

$x = \frac{7}{2} + \frac{y}{2}$

Divide $y$ by $1$ .

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

$x = \frac{7}{2} + \frac{y}{2}$

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

$x = \frac{7}{2} + \frac{y}{2}$

Simplify the right side.

Divide $- 1$ by $- 1$ .

$y = 1$

$x = \frac{7}{2} + \frac{y}{2}$

$y = 1$

$x = \frac{7}{2} + \frac{y}{2}$

$y = 1$

$x = \frac{7}{2} + \frac{y}{2}$

$y = 1$

$x = \frac{7}{2} + \frac{y}{2}$

$y = 1$

$x = \frac{7}{2} + \frac{y}{2}$

Step 4

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

Replace all occurrences of $y$ in $x = \frac{7}{2} + \frac{y}{2}$ with $1$ .

$x = \frac{7}{2} + \frac{1}{2}$

$y = 1$

Simplify the right side.

Simplify $\frac{7}{2} + \frac{1}{2}$ .

Combine the numerators over the common denominator.

$x = \frac{7 + 1}{2}$

$y = 1$

Simplify the expression.

Add $7$ and $1$ .

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

$y = 1$

Divide $8$ by $2$ .

$x = 4$

$y = 1$

$x = 4$

$y = 1$

$x = 4$

$y = 1$

$x = 4$

$y = 1$

$x = 4$

$y = 1$

Step 5

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

$(4, 1)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(4, 1)$

Equation Form:

$x = 4, y = 1$

Step 7

Solve by Substitution 2x-y=7 , 3x-2y=10

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