Solve by Substitution 2y+5x=13 , 2y-3x=5

$2 y + 5 x = 13$ , $2 y - 3 x = 5$

Step 1

Solve for $y$ in $2 y + 5 x = 13$ .

Subtract $5 x$ from both sides of the equation.

$2 y = 13 - 5 x$

$2 y - 3 x = 5$

Divide each term in $2 y = 13 - 5 x$ by $2$ and simplify.

Divide each term in $2 y = 13 - 5 x$ by $2$ .

$\frac{2 y}{2} = \frac{13}{2} + \frac{- 5 x}{2}$

$2 y - 3 x = 5$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 y}{2} = \frac{13}{2} + \frac{- 5 x}{2}$

$2 y - 3 x = 5$

Divide $y$ by $1$ .

$y = \frac{13}{2} + \frac{- 5 x}{2}$

$2 y - 3 x = 5$

$y = \frac{13}{2} + \frac{- 5 x}{2}$

$2 y - 3 x = 5$

$y = \frac{13}{2} + \frac{- 5 x}{2}$

$2 y - 3 x = 5$

Simplify the right side.

Move the negative in front of the fraction.

$y = \frac{13}{2} - \frac{5 x}{2}$

$2 y - 3 x = 5$

$y = \frac{13}{2} - \frac{5 x}{2}$

$2 y - 3 x = 5$

$y = \frac{13}{2} - \frac{5 x}{2}$

$2 y - 3 x = 5$

$y = \frac{13}{2} - \frac{5 x}{2}$

$2 y - 3 x = 5$

Step 2

Replace all occurrences of $y$ with $\frac{13}{2} - \frac{5 x}{2}$ in each equation.

Replace all occurrences of $y$ in $2 y - 3 x = 5$ with $\frac{13}{2} - \frac{5 x}{2}$ .

$2 (\frac{13}{2} - \frac{5 x}{2}) - 3 x = 5$

$y = \frac{13}{2} - \frac{5 x}{2}$

Simplify the left side.

Simplify $2 (\frac{13}{2} - \frac{5 x}{2}) - 3 x$ .

Simplify each term.

Apply the distributive property.

$2 (\frac{13}{2}) + 2 (- \frac{5 x}{2}) - 3 x = 5$

$y = \frac{13}{2} - \frac{5 x}{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$2 (\frac{13}{2}) + 2 (- \frac{5 x}{2}) - 3 x = 5$

$y = \frac{13}{2} - \frac{5 x}{2}$

Rewrite the expression.

$13 + 2 (- \frac{5 x}{2}) - 3 x = 5$

$y = \frac{13}{2} - \frac{5 x}{2}$

$13 + 2 (- \frac{5 x}{2}) - 3 x = 5$

$y = \frac{13}{2} - \frac{5 x}{2}$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{5 x}{2}$ into the numerator.

$13 + 2 (\frac{- 5 x}{2}) - 3 x = 5$

$y = \frac{13}{2} - \frac{5 x}{2}$

Cancel the common factor.

$13 + 2 (\frac{- 5 x}{2}) - 3 x = 5$

$y = \frac{13}{2} - \frac{5 x}{2}$

Rewrite the expression.

$13 - 5 x - 3 x = 5$

$y = \frac{13}{2} - \frac{5 x}{2}$

$13 - 5 x - 3 x = 5$

$y = \frac{13}{2} - \frac{5 x}{2}$

$13 - 5 x - 3 x = 5$

$y = \frac{13}{2} - \frac{5 x}{2}$

Subtract $3 x$ from $- 5 x$ .

$13 - 8 x = 5$

$y = \frac{13}{2} - \frac{5 x}{2}$

$13 - 8 x = 5$

$y = \frac{13}{2} - \frac{5 x}{2}$

$13 - 8 x = 5$

$y = \frac{13}{2} - \frac{5 x}{2}$

$13 - 8 x = 5$

$y = \frac{13}{2} - \frac{5 x}{2}$

Step 3

Solve for $x$ in $13 - 8 x = 5$ .

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

Subtract $13$ from both sides of the equation.

$- 8 x = 5 - 13$

$y = \frac{13}{2} - \frac{5 x}{2}$

Subtract $13$ from $5$ .

$- 8 x = - 8$

$y = \frac{13}{2} - \frac{5 x}{2}$

$- 8 x = - 8$

$y = \frac{13}{2} - \frac{5 x}{2}$

Divide each term in $- 8 x = - 8$ by $- 8$ and simplify.

Divide each term in $- 8 x = - 8$ by $- 8$ .

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

$y = \frac{13}{2} - \frac{5 x}{2}$

Simplify the left side.

Cancel the common factor of $- 8$ .

Cancel the common factor.

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

$y = \frac{13}{2} - \frac{5 x}{2}$

Divide $x$ by $1$ .

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

$y = \frac{13}{2} - \frac{5 x}{2}$

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

$y = \frac{13}{2} - \frac{5 x}{2}$

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

$y = \frac{13}{2} - \frac{5 x}{2}$

Simplify the right side.

Divide $- 8$ by $- 8$ .

$x = 1$

$y = \frac{13}{2} - \frac{5 x}{2}$

$x = 1$

$y = \frac{13}{2} - \frac{5 x}{2}$

$x = 1$

$y = \frac{13}{2} - \frac{5 x}{2}$

$x = 1$

$y = \frac{13}{2} - \frac{5 x}{2}$

Step 4

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

Replace all occurrences of $x$ in $y = \frac{13}{2} - \frac{5 x}{2}$ with $1$ .

$y = \frac{13}{2} - \frac{5 (1)}{2}$

$x = 1$

Simplify the right side.

Simplify $\frac{13}{2} - \frac{5 (1)}{2}$ .

Combine the numerators over the common denominator.

$y = \frac{13 - 5 \cdot 1}{2}$

$x = 1$

Simplify the expression.

Multiply $- 5$ by $1$ .

$y = \frac{13 - 5}{2}$

$x = 1$

Subtract $5$ from $13$ .

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

$x = 1$

Divide $8$ by $2$ .

$y = 4$

$x = 1$

$y = 4$

$x = 1$

$y = 4$

$x = 1$

$y = 4$

$x = 1$

$y = 4$

$x = 1$

Step 5

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

$(1, 4)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(1, 4)$

Equation Form:

$x = 1, y = 4$

Step 7

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