Solve by Addition/Elimination 5x+6y=3 , 10x+12y=6

$5 x + 6 y = 3$ , $10 x + 12 y = 6$

Step 1

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

$(- 2) \cdot (5 x + 6 y) = (- 2) (3)$

$10 x + 12 y = 6$

Step 2

Simplify.

Simplify the left side.

Simplify $(- 2) \cdot (5 x + 6 y)$ .

Apply the distributive property.

$- 2 (5 x) - 2 (6 y) = (- 2) (3)$

$10 x + 12 y = 6$

Multiply.

Multiply $5$ by $- 2$ .

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

$10 x + 12 y = 6$

Multiply $6$ by $- 2$ .

$- 10 x - 12 y = (- 2) (3)$

$10 x + 12 y = 6$

$- 10 x - 12 y = (- 2) (3)$

$10 x + 12 y = 6$

$- 10 x - 12 y = (- 2) (3)$

$10 x + 12 y = 6$

$- 10 x - 12 y = (- 2) (3)$

$10 x + 12 y = 6$

Simplify the right side.

Multiply $- 2$ by $3$ .

$- 10 x - 12 y = - 6$

$10 x + 12 y = 6$

$- 10 x - 12 y = - 6$

$10 x + 12 y = 6$

$- 10 x - 12 y = - 6$

$10 x + 12 y = 6$

Step 3

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

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

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 $10 x$ to both sides of the equation.

$- 12 y = - 6 + 10 x$

Divide each term in $- 12 y = - 6 + 10 x$ by $- 12$ and simplify.

Divide each term in $- 12 y = - 6 + 10 x$ by $- 12$ .

$\frac{- 12 y}{- 12} = \frac{- 6}{- 12} + \frac{10 x}{- 12}$

Simplify the left side.

Cancel the common factor of $- 12$ .

Cancel the common factor.

$\frac{- 12 y}{- 12} = \frac{- 6}{- 12} + \frac{10 x}{- 12}$

Divide $y$ by $1$ .

$y = \frac{- 6}{- 12} + \frac{10 x}{- 12}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $- 6$ and $- 12$ .

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

$y = \frac{- 6 (1)}{- 12} + \frac{10 x}{- 12}$

Cancel the common factors.

Factor $- 6$ out of $- 12$ .

$y = \frac{- 6 \cdot 1}{- 6 \cdot 2} + \frac{10 x}{- 12}$

Cancel the common factor.

$y = \frac{- 6 \cdot 1}{- 6 \cdot 2} + \frac{10 x}{- 12}$

Rewrite the expression.

$y = \frac{1}{2} + \frac{10 x}{- 12}$

Cancel the common factor of $10$ and $- 12$ .

Factor $2$ out of $10 x$ .

$y = \frac{1}{2} + \frac{2 (5 x)}{- 12}$

Cancel the common factors.

Factor $2$ out of $- 12$ .

$y = \frac{1}{2} + \frac{2 (5 x)}{2 (- 6)}$

Cancel the common factor.

$y = \frac{1}{2} + \frac{2 (5 x)}{2 \cdot - 6}$

Rewrite the expression.

$y = \frac{1}{2} + \frac{5 x}{- 6}$

Move the negative in front of the fraction.

$y = \frac{1}{2} - \frac{5 x}{6}$

Step 6

The solution is the set of ordered pairs that make $y = \frac{1}{2} - \frac{5 x}{6}$ true.

$(x, \frac{1}{2} - \frac{5 x}{6})$

Step 7

The result can be shown in multiple forms.

Point Form:

$(x, \frac{1}{2} - \frac{5 x}{6})$

Equation Form:

$x = x, y = \frac{1}{2} - \frac{5 x}{6}$

Step 8

Solve by Addition/Elimination 5x+6y=3 , 10x+12y=6

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