Precalculus Examples

$2 x + 8 y + 10 z = 112$ , $x + 4 y + 5 z = - 28$ , $x + y + z = - 4$

Step 1

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

$2 x + 8 y + 10 z = 112$

$x + 4 y + 5 z = - 28$

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.

$2 x + 8 y + 10 z = 112$

$(- 2) \cdot (x + 4 y + 5 z) = (- 2) (- 28)$

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

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Step 2.2.1.1.1

Apply the distributive property.

$2 x + 8 y + 10 z = 112$

$- 2 x - 2 (4 y) - 2 (5 z) = (- 2) (- 28)$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $4$ by $- 2$ .

$2 x + 8 y + 10 z = 112$

$- 2 x - 8 y - 2 (5 z) = (- 2) (- 28)$

Step 2.2.1.1.2.2

Multiply $5$ by $- 2$ .

$2 x + 8 y + 10 z = 112$

$- 2 x - 8 y - 10 z = (- 2) (- 28)$

$2 x + 8 y + 10 z = 112$

$- 2 x - 8 y - 10 z = (- 2) (- 28)$

$2 x + 8 y + 10 z = 112$

$- 2 x - 8 y - 10 z = (- 2) (- 28)$

$2 x + 8 y + 10 z = 112$

$- 2 x - 8 y - 10 z = (- 2) (- 28)$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Multiply $- 2$ by $- 28$ .

$2 x + 8 y + 10 z = 112$

$- 2 x - 8 y - 10 z = 56$

$2 x + 8 y + 10 z = 112$

$- 2 x - 8 y - 10 z = 56$

$2 x + 8 y + 10 z = 112$

$- 2 x - 8 y - 10 z = 56$

Step 2.3

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

		$2$	$x$	$+$	$8$	$y$	$+$	$1$	$0$	$z$	$=$	$1$	$1$	$2$
$+$	$-$	$2$	$x$	$-$	$8$	$y$	$-$	$1$	$0$	$z$	$=$		$5$	$6$
										$0$	$=$	$1$	$6$	$8$

Step 2.4

The resultant equation has $x$ eliminated.

$0 = 168$

Step 3

Because the resultant equation is false, the system of equations has no solution.

No solution

		$2$	$x$	$+$	$8$	$y$	$+$	$1$	$0$	$z$	$=$	$1$	$1$	$2$
$+$	$-$	$2$	$x$	$-$	$8$	$y$	$-$	$1$	$0$	$z$	$=$		$5$	$6$
										$0$	$=$	$1$	$6$	$8$

		$2$	$x$	$+$	$8$	$y$	$+$	$1$	$0$	$z$	$=$	$1$	$1$	$2$
$+$	$-$	$2$	$x$	$-$	$8$	$y$	$-$	$1$	$0$	$z$	$=$		$5$	$6$
										$0$	$=$	$1$	$6$	$8$

		$2$	$x$	$+$	$8$	$y$	$+$	$1$	$0$	$z$	$=$	$1$	$1$	$2$
$+$	$-$	$2$	$x$	$-$	$8$	$y$	$-$	$1$	$0$	$z$	$=$		$5$	$6$
										$0$	$=$	$1$	$6$	$8$