Solve by Linear Combination x-y=25 , 2x+3y=180

$x - y = 25$ , $2 x + 3 y = 180$

Step 1

Multiply each equation by the value that makes the coefficients of $x$ opposite.

$(- 2) \cdot (x - y) = (- 2) (25)$

$2 x + 3 y = 180$

Step 2

Simplify.

Simplify the left side.

Simplify $(- 2) \cdot (x - y)$ .

Apply the distributive property.

$- 2 x - 2 (- y) = (- 2) (25)$

$2 x + 3 y = 180$

Multiply $- 1$ by $- 2$ .

$- 2 x + 2 y = (- 2) (25)$

$2 x + 3 y = 180$

$- 2 x + 2 y = (- 2) (25)$

$2 x + 3 y = 180$

$- 2 x + 2 y = (- 2) (25)$

$2 x + 3 y = 180$

Simplify the right side.

Multiply $- 2$ by $25$ .

$- 2 x + 2 y = - 50$

$2 x + 3 y = 180$

$- 2 x + 2 y = - 50$

$2 x + 3 y = 180$

$- 2 x + 2 y = - 50$

$2 x + 3 y = 180$

Step 3

Add the two equations together to eliminate $x$ from the system.

	$-$	$2$	$x$	$+$	$2$	$y$	$=$	$-$	$5$	$0$
$+$		$2$	$x$	$+$	$3$	$y$	$=$	$1$	$8$	$0$
					$5$	$y$	$=$	$1$	$3$	$0$

Step 4

Divide each term in $5 y = 130$ by $5$ and simplify.

Divide each term in $5 y = 130$ by $5$ .

$\frac{5 y}{5} = \frac{130}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 y}{5} = \frac{130}{5}$

Divide $y$ by $1$ .

$y = \frac{130}{5}$

Simplify the right side.

Divide $130$ by $5$ .

$y = 26$

Step 5

Substitute the value found for $y$ into one of the original equations, then solve for $x$ .

Substitute the value found for $y$ into one of the original equations to solve for $x$ .

$- 2 x + 2 (26) = - 50$

Multiply $2$ by $26$ .

$- 2 x + 52 = - 50$

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

Subtract $52$ from both sides of the equation.

$- 2 x = - 50 - 52$

Subtract $52$ from $- 50$ .

$- 2 x = - 102$

Divide each term in $- 2 x = - 102$ by $- 2$ and simplify.

Divide each term in $- 2 x = - 102$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{- 102}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{- 102}{- 2}$

Divide $x$ by $1$ .

$x = \frac{- 102}{- 2}$

Simplify the right side.

Divide $- 102$ by $- 2$ .

$x = 51$

Step 6

The solution to the independent system of equations can be represented as a point.

$(51, 26)$

Step 7

The result can be shown in multiple forms.

Point Form:

$(51, 26)$

Equation Form:

$x = 51, y = 26$

Step 8

Solve by Linear Combination x-y=25 , 2x+3y=180

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