Solve by Addition/Elimination 6x+3y=0 , -42x-21y=0

$6 x + 3 y = 0$ , $- 42 x - 21 y = 0$

Step 1

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

$(7) \cdot (6 x + 3 y) = (7) (0)$

$- 42 x - 21 y = 0$

Step 2

Simplify.

Simplify the left side.

Simplify $(7) \cdot (6 x + 3 y)$ .

Apply the distributive property.

$7 (6 x) + 7 (3 y) = (7) (0)$

$- 42 x - 21 y = 0$

Multiply.

Multiply $6$ by $7$ .

$42 x + 7 (3 y) = (7) (0)$

$- 42 x - 21 y = 0$

Multiply $3$ by $7$ .

$42 x + 21 y = (7) (0)$

$- 42 x - 21 y = 0$

$42 x + 21 y = (7) (0)$

$- 42 x - 21 y = 0$

$42 x + 21 y = (7) (0)$

$- 42 x - 21 y = 0$

$42 x + 21 y = (7) (0)$

$- 42 x - 21 y = 0$

Simplify the right side.

Multiply $7$ by $0$ .

$42 x + 21 y = 0$

$- 42 x - 21 y = 0$

$42 x + 21 y = 0$

$- 42 x - 21 y = 0$

$42 x + 21 y = 0$

$- 42 x - 21 y = 0$

Step 3

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

		$4$	$2$	$x$	$+$	$2$	$1$	$y$	$=$	$0$
$+$	$-$	$4$	$2$	$x$	$-$	$2$	$1$	$y$	$=$	$0$
								$0$	$=$	$0$

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 $42 x$ from both sides of the equation.

$21 y = - 42 x$

Divide each term in $21 y = - 42 x$ by $21$ and simplify.

Divide each term in $21 y = - 42 x$ by $21$ .

$\frac{21 y}{21} = \frac{- 42 x}{21}$

Simplify the left side.

Cancel the common factor of $21$ .

Cancel the common factor.

$\frac{21 y}{21} = \frac{- 42 x}{21}$

Divide $y$ by $1$ .

$y = \frac{- 42 x}{21}$

Simplify the right side.

Cancel the common factor of $- 42$ and $21$ .

Factor $21$ out of $- 42 x$ .

$y = \frac{21 (- 2 x)}{21}$

Cancel the common factors.

Factor $21$ out of $21$ .

$y = \frac{21 (- 2 x)}{21 (1)}$

Cancel the common factor.

$y = \frac{21 (- 2 x)}{21 \cdot 1}$

Rewrite the expression.

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

Divide $- 2 x$ by $1$ .

$y = - 2 x$

Step 6

The solution is the set of ordered pairs that make $y = - 2 x$ true.

$(x, - 2 x)$

Step 7

The result can be shown in multiple forms.

Point Form:

$(x, - 2 x)$

Equation Form:

$x = x, y = - 2 x$

Step 8

Solve by Addition/Elimination 6x+3y=0 , -42x-21y=0

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