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

$x - 3 y = 5$ , $2 x - 6 y = 10$

Step 1

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

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

$2 x - 6 y = 10$

Step 2

Simplify.

Simplify the left side.

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

Apply the distributive property.

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

$2 x - 6 y = 10$

Multiply $- 3$ by $- 2$ .

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

$2 x - 6 y = 10$

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

$2 x - 6 y = 10$

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

$2 x - 6 y = 10$

Simplify the right side.

Multiply $- 2$ by $5$ .

$- 2 x + 6 y = - 10$

$2 x - 6 y = 10$

$- 2 x + 6 y = - 10$

$2 x - 6 y = 10$

$- 2 x + 6 y = - 10$

$2 x - 6 y = 10$

Step 3

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

	$-$	$2$	$x$	$+$	$6$	$y$	$=$	$-$	$1$	$0$
$+$		$2$	$x$	$-$	$6$	$y$	$=$		$1$	$0$
						$0$	$=$			$0$

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 $y$ .

Add $2 x$ to both sides of the equation.

$6 y = - 10 + 2 x$

Divide each term in $6 y = - 10 + 2 x$ by $6$ and simplify.

Divide each term in $6 y = - 10 + 2 x$ by $6$ .

$\frac{6 y}{6} = \frac{- 10}{6} + \frac{2 x}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 y}{6} = \frac{- 10}{6} + \frac{2 x}{6}$

Divide $y$ by $1$ .

$y = \frac{- 10}{6} + \frac{2 x}{6}$

Simplify the right side.

Simplify each term.

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

Factor $2$ out of $- 10$ .

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

Cancel the common factors.

Factor $2$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

Cancel the common factor of $2$ and $6$ .

Factor $2$ out of $2 x$ .

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

Cancel the common factors.

Factor $2$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 6

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

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

Step 7

The result can be shown in multiple forms.

Point Form:

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

Equation Form:

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

Step 8

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

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