Precalculus Examples

$- 5 x + 5 y - 25 z = - 40$ , $x - y + 5 z = 8$ , $2 x - 2 y + 10 z = 16$

Step 1

Choose two equations and eliminate one variable. In this case, eliminate $x$ .

$- 5 x + 5 y - 25 z = - 40$

$x - y + 5 z = 8$

Step 2

Eliminate $x$ from the system.

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Step 2.1

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

$- 5 x + 5 y - 25 z = - 40$

$(5) \cdot (x - y + 5 z) = (5) (8)$

Step 2.2

Simplify.

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Step 2.2.1

Simplify the left side.

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Step 2.2.1.1

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

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Step 2.2.1.1.1

Apply the distributive property.

$- 5 x + 5 y - 25 z = - 40$

$5 x + 5 (- y) + 5 (5 z) = (5) (8)$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $- 1$ by $5$ .

$- 5 x + 5 y - 25 z = - 40$

$5 x - 5 y + 5 (5 z) = (5) (8)$

Step 2.2.1.1.2.2

Multiply $5$ by $5$ .

$- 5 x + 5 y - 25 z = - 40$

$5 x - 5 y + 25 z = (5) (8)$

$- 5 x + 5 y - 25 z = - 40$

$5 x - 5 y + 25 z = (5) (8)$

$- 5 x + 5 y - 25 z = - 40$

$5 x - 5 y + 25 z = (5) (8)$

$- 5 x + 5 y - 25 z = - 40$

$5 x - 5 y + 25 z = (5) (8)$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Multiply $5$ by $8$ .

$- 5 x + 5 y - 25 z = - 40$

$5 x - 5 y + 25 z = 40$

$- 5 x + 5 y - 25 z = - 40$

$5 x - 5 y + 25 z = 40$

$- 5 x + 5 y - 25 z = - 40$

$5 x - 5 y + 25 z = 40$

Step 2.3

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

	$-$	$5$	$x$	$+$	$5$	$y$	$-$	$2$	$5$	$z$	$=$	$-$	$4$	$0$
$+$		$5$	$x$	$-$	$5$	$y$	$+$	$2$	$5$	$z$	$=$		$4$	$0$
										$0$	$=$			$0$

Step 2.4

The resultant equation has $x$ eliminated.

$0 = 0$

Step 3

Because the resultant equation includes no variables and is true, the system of equations has an infinite number of solutions.

Infinite number of solutions

	$-$	$5$	$x$	$+$	$5$	$y$	$-$	$2$	$5$	$z$	$=$	$-$	$4$	$0$
$+$		$5$	$x$	$-$	$5$	$y$	$+$	$2$	$5$	$z$	$=$		$4$	$0$
										$0$	$=$			$0$

	$-$	$5$	$x$	$+$	$5$	$y$	$-$	$2$	$5$	$z$	$=$	$-$	$4$	$0$
$+$		$5$	$x$	$-$	$5$	$y$	$+$	$2$	$5$	$z$	$=$		$4$	$0$
										$0$	$=$			$0$

	$-$	$5$	$x$	$+$	$5$	$y$	$-$	$2$	$5$	$z$	$=$	$-$	$4$	$0$
$+$		$5$	$x$	$-$	$5$	$y$	$+$	$2$	$5$	$z$	$=$		$4$	$0$
										$0$	$=$			$0$