Solve by Addition/Elimination 4x+5y=2 , 8x+10y=4

$4 x + 5 y = 2$ , $8 x + 10 y = 4$

Step 1

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

$(- 2) \cdot (4 x + 5 y) = (- 2) (2)$

$8 x + 10 y = 4$

Step 2

Simplify.

Simplify the left side.

Simplify $(- 2) \cdot (4 x + 5 y)$ .

Apply the distributive property.

$- 2 (4 x) - 2 (5 y) = (- 2) (2)$

$8 x + 10 y = 4$

Multiply.

Multiply $4$ by $- 2$ .

$- 8 x - 2 (5 y) = (- 2) (2)$

$8 x + 10 y = 4$

Multiply $5$ by $- 2$ .

$- 8 x - 10 y = (- 2) (2)$

$8 x + 10 y = 4$

$- 8 x - 10 y = (- 2) (2)$

$8 x + 10 y = 4$

$- 8 x - 10 y = (- 2) (2)$

$8 x + 10 y = 4$

$- 8 x - 10 y = (- 2) (2)$

$8 x + 10 y = 4$

Simplify the right side.

Multiply $- 2$ by $2$ .

$- 8 x - 10 y = - 4$

$8 x + 10 y = 4$

$- 8 x - 10 y = - 4$

$8 x + 10 y = 4$

$- 8 x - 10 y = - 4$

$8 x + 10 y = 4$

Step 3

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

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

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 $8 x$ to both sides of the equation.

$- 10 y = - 4 + 8 x$

Divide each term in $- 10 y = - 4 + 8 x$ by $- 10$ and simplify.

Divide each term in $- 10 y = - 4 + 8 x$ by $- 10$ .

$\frac{- 10 y}{- 10} = \frac{- 4}{- 10} + \frac{8 x}{- 10}$

Simplify the left side.

Cancel the common factor of $- 10$ .

Cancel the common factor.

$\frac{- 10 y}{- 10} = \frac{- 4}{- 10} + \frac{8 x}{- 10}$

Divide $y$ by $1$ .

$y = \frac{- 4}{- 10} + \frac{8 x}{- 10}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $- 4$ and $- 10$ .

Factor $- 2$ out of $- 4$ .

$y = \frac{- 2 (2)}{- 10} + \frac{8 x}{- 10}$

Cancel the common factors.

Factor $- 2$ out of $- 10$ .

$y = \frac{- 2 \cdot 2}{- 2 \cdot 5} + \frac{8 x}{- 10}$

Cancel the common factor.

$y = \frac{- 2 \cdot 2}{- 2 \cdot 5} + \frac{8 x}{- 10}$

Rewrite the expression.

$y = \frac{2}{5} + \frac{8 x}{- 10}$

Cancel the common factor of $8$ and $- 10$ .

Factor $2$ out of $8 x$ .

$y = \frac{2}{5} + \frac{2 (4 x)}{- 10}$

Cancel the common factors.

Factor $2$ out of $- 10$ .

$y = \frac{2}{5} + \frac{2 (4 x)}{2 (- 5)}$

Cancel the common factor.

$y = \frac{2}{5} + \frac{2 (4 x)}{2 \cdot - 5}$

Rewrite the expression.

$y = \frac{2}{5} + \frac{4 x}{- 5}$

Move the negative in front of the fraction.

$y = \frac{2}{5} - \frac{4 x}{5}$

Step 6

The solution is the set of ordered pairs that make $y = \frac{2}{5} - \frac{4 x}{5}$ true.

$(x, \frac{2}{5} - \frac{4 x}{5})$

Step 7

The result can be shown in multiple forms.

Point Form:

$(x, \frac{2}{5} - \frac{4 x}{5})$

Equation Form:

$x = x, y = \frac{2}{5} - \frac{4 x}{5}$

Step 8

Solve by Addition/Elimination 4x+5y=2 , 8x+10y=4

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