Solve by Addition/Elimination x+3y=9 , 6x+18y=54

$x + 3 y = 9$ , $6 x + 18 y = 54$

Step 1

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

$(- 6) \cdot (x + 3 y) = (- 6) (9)$

$6 x + 18 y = 54$

Step 2

Simplify.

Simplify the left side.

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

Apply the distributive property.

$- 6 x - 6 (3 y) = (- 6) (9)$

$6 x + 18 y = 54$

Multiply $3$ by $- 6$ .

$- 6 x - 18 y = (- 6) (9)$

$6 x + 18 y = 54$

$- 6 x - 18 y = (- 6) (9)$

$6 x + 18 y = 54$

$- 6 x - 18 y = (- 6) (9)$

$6 x + 18 y = 54$

Simplify the right side.

Multiply $- 6$ by $9$ .

$- 6 x - 18 y = - 54$

$6 x + 18 y = 54$

$- 6 x - 18 y = - 54$

$6 x + 18 y = 54$

$- 6 x - 18 y = - 54$

$6 x + 18 y = 54$

Step 3

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

	$-$	$6$	$x$	$-$	$1$	$8$	$y$	$=$	$-$	$5$	$4$
$+$		$6$	$x$	$+$	$1$	$8$	$y$	$=$		$5$	$4$
							$0$	$=$			$0$

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.

$- 18 y = - 54 + 6 x$

Divide each term in $- 18 y = - 54 + 6 x$ by $- 18$ and simplify.

Divide each term in $- 18 y = - 54 + 6 x$ by $- 18$ .

$\frac{- 18 y}{- 18} = \frac{- 54}{- 18} + \frac{6 x}{- 18}$

Simplify the left side.

Cancel the common factor of $- 18$ .

Cancel the common factor.

$\frac{- 18 y}{- 18} = \frac{- 54}{- 18} + \frac{6 x}{- 18}$

Divide $y$ by $1$ .

$y = \frac{- 54}{- 18} + \frac{6 x}{- 18}$

Simplify the right side.

Simplify each term.

Divide $- 54$ by $- 18$ .

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

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

Factor $6$ out of $6 x$ .

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

Cancel the common factors.

Factor $6$ out of $- 18$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

Step 6

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

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

Step 7

The result can be shown in multiple forms.

Point Form:

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

Equation Form:

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

Step 8

Solve by Addition/Elimination x+3y=9 , 6x+18y=54

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