Solve by Substitution 2x-y=5 , x+3y=7

$2 x - y = 5$ , $x + 3 y = 7$

Step 1

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

$x = 7 - 3 y$

$2 x - y = 5$

Step 2

Replace all occurrences of $x$ with $7 - 3 y$ in each equation.

Replace all occurrences of $x$ in $2 x - y = 5$ with $7 - 3 y$ .

$2 (7 - 3 y) - y = 5$

$x = 7 - 3 y$

Simplify the left side.

Simplify $2 (7 - 3 y) - y$ .

Simplify each term.

Apply the distributive property.

$2 \cdot 7 + 2 (- 3 y) - y = 5$

$x = 7 - 3 y$

Multiply $2$ by $7$ .

$14 + 2 (- 3 y) - y = 5$

$x = 7 - 3 y$

Multiply $- 3$ by $2$ .

$14 - 6 y - y = 5$

$x = 7 - 3 y$

$14 - 6 y - y = 5$

$x = 7 - 3 y$

Subtract $y$ from $- 6 y$ .

$14 - 7 y = 5$

$x = 7 - 3 y$

$14 - 7 y = 5$

$x = 7 - 3 y$

$14 - 7 y = 5$

$x = 7 - 3 y$

$14 - 7 y = 5$

$x = 7 - 3 y$

Step 3

Solve for $y$ in $14 - 7 y = 5$ .

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

Subtract $14$ from both sides of the equation.

$- 7 y = 5 - 14$

$x = 7 - 3 y$

Subtract $14$ from $5$ .

$- 7 y = - 9$

$x = 7 - 3 y$

$- 7 y = - 9$

$x = 7 - 3 y$

Divide each term in $- 7 y = - 9$ by $- 7$ and simplify.

Divide each term in $- 7 y = - 9$ by $- 7$ .

$\frac{- 7 y}{- 7} = \frac{- 9}{- 7}$

$x = 7 - 3 y$

Simplify the left side.

Cancel the common factor of $- 7$ .

Cancel the common factor.

$\frac{- 7 y}{- 7} = \frac{- 9}{- 7}$

$x = 7 - 3 y$

Divide $y$ by $1$ .

$y = \frac{- 9}{- 7}$

$x = 7 - 3 y$

$y = \frac{- 9}{- 7}$

$x = 7 - 3 y$

$y = \frac{- 9}{- 7}$

$x = 7 - 3 y$

Simplify the right side.

Dividing two negative values results in a positive value.

$y = \frac{9}{7}$

$x = 7 - 3 y$

$y = \frac{9}{7}$

$x = 7 - 3 y$

$y = \frac{9}{7}$

$x = 7 - 3 y$

$y = \frac{9}{7}$

$x = 7 - 3 y$

Step 4

Replace all occurrences of $y$ with $\frac{9}{7}$ in each equation.

Replace all occurrences of $y$ in $x = 7 - 3 y$ with $\frac{9}{7}$ .

$x = 7 - 3 (\frac{9}{7})$

$y = \frac{9}{7}$

Simplify the right side.

Simplify $7 - 3 (\frac{9}{7})$ .

Simplify each term.

Multiply $- 3 (\frac{9}{7})$ .

Combine $- 3$ and $\frac{9}{7}$ .

$x = 7 + \frac{- 3 \cdot 9}{7}$

$y = \frac{9}{7}$

Multiply $- 3$ by $9$ .

$x = 7 + \frac{- 27}{7}$

$y = \frac{9}{7}$

$x = 7 + \frac{- 27}{7}$

$y = \frac{9}{7}$

Move the negative in front of the fraction.

$x = 7 - \frac{27}{7}$

$y = \frac{9}{7}$

$x = 7 - \frac{27}{7}$

$y = \frac{9}{7}$

To write $7$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$x = 7 \cdot \frac{7}{7} - \frac{27}{7}$

$y = \frac{9}{7}$

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

$x = \frac{7 \cdot 7}{7} - \frac{27}{7}$

$y = \frac{9}{7}$

Combine the numerators over the common denominator.

$x = \frac{7 \cdot 7 - 27}{7}$

$y = \frac{9}{7}$

Simplify the numerator.

Multiply $7$ by $7$ .

$x = \frac{49 - 27}{7}$

$y = \frac{9}{7}$

Subtract $27$ from $49$ .

$x = \frac{22}{7}$

$y = \frac{9}{7}$

$x = \frac{22}{7}$

$y = \frac{9}{7}$

$x = \frac{22}{7}$

$y = \frac{9}{7}$

$x = \frac{22}{7}$

$y = \frac{9}{7}$

$x = \frac{22}{7}$

$y = \frac{9}{7}$

Step 5

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

$(\frac{22}{7}, \frac{9}{7})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(\frac{22}{7}, \frac{9}{7})$

Equation Form:

$x = \frac{22}{7}, y = \frac{9}{7}$

Step 7

Solve by Substitution 2x-y=5 , x+3y=7

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