Solve by Addition/Elimination 4x-5y=6 , 20x-25y=30

$4 x - 5 y = 6$ , $20 x - 25 y = 30$

Step 1

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

$(- 5) \cdot (4 x - 5 y) = (- 5) (6)$

$20 x - 25 y = 30$

Step 2

Simplify.

Simplify the left side.

Simplify $(- 5) \cdot (4 x - 5 y)$ .

Apply the distributive property.

$- 5 (4 x) - 5 (- 5 y) = (- 5) (6)$

$20 x - 25 y = 30$

Multiply.

Multiply $4$ by $- 5$ .

$- 20 x - 5 (- 5 y) = (- 5) (6)$

$20 x - 25 y = 30$

Multiply $- 5$ by $- 5$ .

$- 20 x + 25 y = (- 5) (6)$

$20 x - 25 y = 30$

$- 20 x + 25 y = (- 5) (6)$

$20 x - 25 y = 30$

$- 20 x + 25 y = (- 5) (6)$

$20 x - 25 y = 30$

$- 20 x + 25 y = (- 5) (6)$

$20 x - 25 y = 30$

Simplify the right side.

Multiply $- 5$ by $6$ .

$- 20 x + 25 y = - 30$

$20 x - 25 y = 30$

$- 20 x + 25 y = - 30$

$20 x - 25 y = 30$

$- 20 x + 25 y = - 30$

$20 x - 25 y = 30$

Step 3

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

	$-$	$2$	$0$	$x$	$+$	$2$	$5$	$y$	$=$	$-$	$3$	$0$
$+$		$2$	$0$	$x$	$-$	$2$	$5$	$y$	$=$		$3$	$0$
								$0$	$=$			$0$

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 $20 x$ to both sides of the equation.

$25 y = - 30 + 20 x$

Divide each term in $25 y = - 30 + 20 x$ by $25$ and simplify.

Divide each term in $25 y = - 30 + 20 x$ by $25$ .

$\frac{25 y}{25} = \frac{- 30}{25} + \frac{20 x}{25}$

Simplify the left side.

Cancel the common factor of $25$ .

Cancel the common factor.

$\frac{25 y}{25} = \frac{- 30}{25} + \frac{20 x}{25}$

Divide $y$ by $1$ .

$y = \frac{- 30}{25} + \frac{20 x}{25}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $- 30$ and $25$ .

Factor $5$ out of $- 30$ .

$y = \frac{5 (- 6)}{25} + \frac{20 x}{25}$

Cancel the common factors.

Factor $5$ out of $25$ .

$y = \frac{5 \cdot - 6}{5 \cdot 5} + \frac{20 x}{25}$

Cancel the common factor.

$y = \frac{5 \cdot - 6}{5 \cdot 5} + \frac{20 x}{25}$

Rewrite the expression.

$y = \frac{- 6}{5} + \frac{20 x}{25}$

Move the negative in front of the fraction.

$y = - \frac{6}{5} + \frac{20 x}{25}$

Cancel the common factor of $20$ and $25$ .

Factor $5$ out of $20 x$ .

$y = - \frac{6}{5} + \frac{5 (4 x)}{25}$

Cancel the common factors.

Factor $5$ out of $25$ .

$y = - \frac{6}{5} + \frac{5 (4 x)}{5 (5)}$

Cancel the common factor.

$y = - \frac{6}{5} + \frac{5 (4 x)}{5 \cdot 5}$

Rewrite the expression.

$y = - \frac{6}{5} + \frac{4 x}{5}$

Step 6

The solution is the set of ordered pairs that make $y = - \frac{6}{5} + \frac{4 x}{5}$ true.

$(x, - \frac{6}{5} + \frac{4 x}{5})$

Step 7

The result can be shown in multiple forms.

Point Form:

$(x, - \frac{6}{5} + \frac{4 x}{5})$

Equation Form:

$x = x, y = - \frac{6}{5} + \frac{4 x}{5}$

Step 8

Solve by Addition/Elimination 4x-5y=6 , 20x-25y=30

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