Solve by Addition/Elimination 2x+6y=10 , 3x+9y=15

$2 x + 6 y = 10$ , $3 x + 9 y = 15$

Step 1

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

$(- 3) \cdot (2 x + 6 y) = (- 3) (10)$

$(2) \cdot (3 x + 9 y) = (2) (15)$

Step 2

Simplify.

Simplify the left side.

Simplify $(- 3) \cdot (2 x + 6 y)$ .

Apply the distributive property.

$- 3 (2 x) - 3 (6 y) = (- 3) (10)$

$(2) \cdot (3 x + 9 y) = (2) (15)$

Multiply.

Multiply $2$ by $- 3$ .

$- 6 x - 3 (6 y) = (- 3) (10)$

$(2) \cdot (3 x + 9 y) = (2) (15)$

Multiply $6$ by $- 3$ .

$- 6 x - 18 y = (- 3) (10)$

$(2) \cdot (3 x + 9 y) = (2) (15)$

$- 6 x - 18 y = (- 3) (10)$

$(2) \cdot (3 x + 9 y) = (2) (15)$

$- 6 x - 18 y = (- 3) (10)$

$(2) \cdot (3 x + 9 y) = (2) (15)$

$- 6 x - 18 y = (- 3) (10)$

$(2) \cdot (3 x + 9 y) = (2) (15)$

Simplify the right side.

Multiply $- 3$ by $10$ .

$- 6 x - 18 y = - 30$

$(2) \cdot (3 x + 9 y) = (2) (15)$

$- 6 x - 18 y = - 30$

$(2) \cdot (3 x + 9 y) = (2) (15)$

Simplify the left side.

Simplify $(2) \cdot (3 x + 9 y)$ .

Apply the distributive property.

$- 6 x - 18 y = - 30$

$2 (3 x) + 2 (9 y) = (2) (15)$

Multiply.

Multiply $3$ by $2$ .

$- 6 x - 18 y = - 30$

$6 x + 2 (9 y) = (2) (15)$

Multiply $9$ by $2$ .

$- 6 x - 18 y = - 30$

$6 x + 18 y = (2) (15)$

$- 6 x - 18 y = - 30$

$6 x + 18 y = (2) (15)$

$- 6 x - 18 y = - 30$

$6 x + 18 y = (2) (15)$

$- 6 x - 18 y = - 30$

$6 x + 18 y = (2) (15)$

Simplify the right side.

Multiply $2$ by $15$ .

$- 6 x - 18 y = - 30$

$6 x + 18 y = 30$

$- 6 x - 18 y = - 30$

$6 x + 18 y = 30$

$- 6 x - 18 y = - 30$

$6 x + 18 y = 30$

Step 3

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

	$-$	$6$	$x$	$-$	$1$	$8$	$y$	$=$	$-$	$3$	$0$
$+$		$6$	$x$	$+$	$1$	$8$	$y$	$=$		$3$	$0$
							$0$	$=$			$0$

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 $6 x$ to both sides of the equation.

$- 18 y = - 30 + 6 x$

Divide each term in $- 18 y = - 30 + 6 x$ by $- 18$ and simplify.

Divide each term in $- 18 y = - 30 + 6 x$ by $- 18$ .

$\frac{- 18 y}{- 18} = \frac{- 30}{- 18} + \frac{6 x}{- 18}$

Simplify the left side.

Cancel the common factor of $- 18$ .

Cancel the common factor.

$\frac{- 18 y}{- 18} = \frac{- 30}{- 18} + \frac{6 x}{- 18}$

Divide $y$ by $1$ .

$y = \frac{- 30}{- 18} + \frac{6 x}{- 18}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $- 30$ and $- 18$ .

Factor $- 6$ out of $- 30$ .

$y = \frac{- 6 (5)}{- 18} + \frac{6 x}{- 18}$

Cancel the common factors.

Factor $- 6$ out of $- 18$ .

$y = \frac{- 6 \cdot 5}{- 6 \cdot 3} + \frac{6 x}{- 18}$

Cancel the common factor.

$y = \frac{- 6 \cdot 5}{- 6 \cdot 3} + \frac{6 x}{- 18}$

Rewrite the expression.

$y = \frac{5}{3} + \frac{6 x}{- 18}$

Cancel the common factor of $6$ and $- 18$ .

Factor $6$ out of $6 x$ .

$y = \frac{5}{3} + \frac{6 (x)}{- 18}$

Cancel the common factors.

Factor $6$ out of $- 18$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

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 2x+6y=10 , 3x+9y=15

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