Solve by Addition/Elimination 2x-3y=5 -4x+6y=-10

$2 x - 3 y = 5$ $- 4 x + 6 y = - 10$

Step 1

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

$(2) \cdot (2 x - 3 y) = (2) (5)$

$- 4 x + 6 y = - 10$

Step 2

Simplify.

Simplify the left side.

Simplify $(2) \cdot (2 x - 3 y)$ .

Apply the distributive property.

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

$- 4 x + 6 y = - 10$

Multiply.

Multiply $2$ by $2$ .

$4 x + 2 (- 3 y) = (2) (5)$

$- 4 x + 6 y = - 10$

Multiply $- 3$ by $2$ .

$4 x - 6 y = (2) (5)$

$- 4 x + 6 y = - 10$

$4 x - 6 y = (2) (5)$

$- 4 x + 6 y = - 10$

$4 x - 6 y = (2) (5)$

$- 4 x + 6 y = - 10$

$4 x - 6 y = (2) (5)$

$- 4 x + 6 y = - 10$

Simplify the right side.

Multiply $2$ by $5$ .

$4 x - 6 y = 10$

$- 4 x + 6 y = - 10$

$4 x - 6 y = 10$

$- 4 x + 6 y = - 10$

$4 x - 6 y = 10$

$- 4 x + 6 y = - 10$

Step 3

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

		$4$	$x$	$-$	$6$	$y$	$=$		$1$	$0$
$+$	$-$	$4$	$x$	$+$	$6$	$y$	$=$	$-$	$1$	$0$
						$0$	$=$			$0$

Step 4

Since $0 = 0$ , the equations intersect at an infinite number of points.

Infinite number of solutions

Step 5

Solve one of the equations for $y$ .

Subtract $4 x$ from both sides of the equation.

$- 6 y = 10 - 4 x$

Divide each term in $- 6 y = 10 - 4 x$ by $- 6$ and simplify.

Divide each term in $- 6 y = 10 - 4 x$ by $- 6$ .

$\frac{- 6 y}{- 6} = \frac{10}{- 6} + \frac{- 4 x}{- 6}$

Simplify the left side.

Cancel the common factor of $- 6$ .

Cancel the common factor.

$\frac{- 6 y}{- 6} = \frac{10}{- 6} + \frac{- 4 x}{- 6}$

Divide $y$ by $1$ .

$y = \frac{10}{- 6} + \frac{- 4 x}{- 6}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $10$ and $- 6$ .

Factor $2$ out of $10$ .

$y = \frac{2 (5)}{- 6} + \frac{- 4 x}{- 6}$

Cancel the common factors.

Factor $2$ out of $- 6$ .

$y = \frac{2 \cdot 5}{2 \cdot - 3} + \frac{- 4 x}{- 6}$

Cancel the common factor.

$y = \frac{2 \cdot 5}{2 \cdot - 3} + \frac{- 4 x}{- 6}$

Rewrite the expression.

$y = \frac{5}{- 3} + \frac{- 4 x}{- 6}$

Move the negative in front of the fraction.

$y = - \frac{5}{3} + \frac{- 4 x}{- 6}$

Cancel the common factor of $- 4$ and $- 6$ .

Factor $- 2$ out of $- 4 x$ .

$y = - \frac{5}{3} + \frac{- 2 (2 x)}{- 6}$

Cancel the common factors.

Factor $- 2$ out of $- 6$ .

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

Cancel the common factor.

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

Rewrite the expression.

$y = - \frac{5}{3} + \frac{2 x}{3}$

Step 6

The solution is the set of ordered pairs that make $y = - \frac{5}{3} + \frac{2 x}{3}$ true.

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

Step 7

The result can be shown in multiple forms.

Point Form:

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

Equation Form:

$x = x, y = - \frac{5}{3} + \frac{2 x}{3}$

Step 8

Solve by Addition/Elimination 2x-3y=5 -4x+6y=-10

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