Precalculus Examples

$2 x + 1 y - 5 z = - 29$ , $6 x + 4 y - z = - 19$ , $x + y + 3 z = 12$

Step 1

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

$2 x + 1 y - 5 z = - 29$

$6 x + 4 y - z = - 19$

Step 2

Eliminate $y$ from the system.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Multiply $y$ by $1$ .

$2 x + y - 5 z = - 29$

$6 x + 4 y - z = - 19$

$2 x + y - 5 z = - 29$

$6 x + 4 y - z = - 19$

Step 2.2

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

$(- 4) \cdot (2 x + y - 5 z) = (- 4) (- 29)$

$6 x + 4 y - z = - 19$

Step 2.3

Simplify.

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Step 2.3.1

Simplify the left side.

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Step 2.3.1.1

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

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Step 2.3.1.1.1

Apply the distributive property.

$- 4 (2 x) - 4 y - 4 (- 5 z) = (- 4) (- 29)$

$6 x + 4 y - z = - 19$

Step 2.3.1.1.2

Simplify.

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Step 2.3.1.1.2.1

Multiply $2$ by $- 4$ .

$- 8 x - 4 y - 4 (- 5 z) = (- 4) (- 29)$

$6 x + 4 y - z = - 19$

Step 2.3.1.1.2.2

Multiply $- 5$ by $- 4$ .

$- 8 x - 4 y + 20 z = (- 4) (- 29)$

$6 x + 4 y - z = - 19$

$- 8 x - 4 y + 20 z = (- 4) (- 29)$

$6 x + 4 y - z = - 19$

$- 8 x - 4 y + 20 z = (- 4) (- 29)$

$6 x + 4 y - z = - 19$

$- 8 x - 4 y + 20 z = (- 4) (- 29)$

$6 x + 4 y - z = - 19$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Multiply $- 4$ by $- 29$ .

$- 8 x - 4 y + 20 z = 116$

$6 x + 4 y - z = - 19$

$- 8 x - 4 y + 20 z = 116$

$6 x + 4 y - z = - 19$

$- 8 x - 4 y + 20 z = 116$

$6 x + 4 y - z = - 19$

Step 2.4

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

	$-$	$8$	$x$	$-$	$4$	$y$	$+$	$2$	$0$	$z$	$=$	$1$	$1$	$6$
$+$		$6$	$x$	$+$	$4$	$y$			$-$	$z$	$=$	$-$	$1$	$9$
	$-$	$2$	$x$	$+$				$1$	$9$	$z$	$=$		$9$	$7$

Step 2.5

The resultant equation has $y$ eliminated.

$- 2 x + 19 z = 97$

Step 3

Choose another two equations and eliminate $y$ .

$6 x + 4 y - z = - 19$

$x + y + 3 z = 12$

Step 4

Eliminate $y$ from the system.

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Step 4.1

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

$6 x + 4 y - z = - 19$

$(- 4) \cdot (x + y + 3 z) = (- 4) (12)$

Step 4.2

Simplify.

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

Simplify $(- 4) \cdot (x + y + 3 z)$ .

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Step 4.2.1.1.1

Apply the distributive property.

$6 x + 4 y - z = - 19$

$- 4 x - 4 y - 4 (3 z) = (- 4) (12)$

Step 4.2.1.1.2

Multiply $3$ by $- 4$ .

$6 x + 4 y - z = - 19$

$- 4 x - 4 y - 12 z = (- 4) (12)$

$6 x + 4 y - z = - 19$

$- 4 x - 4 y - 12 z = (- 4) (12)$

$6 x + 4 y - z = - 19$

$- 4 x - 4 y - 12 z = (- 4) (12)$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Multiply $- 4$ by $12$ .

$6 x + 4 y - z = - 19$

$- 4 x - 4 y - 12 z = - 48$

$6 x + 4 y - z = - 19$

$- 4 x - 4 y - 12 z = - 48$

$6 x + 4 y - z = - 19$

$- 4 x - 4 y - 12 z = - 48$

Step 4.3

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

		$6$	$x$	$+$	$4$	$y$			$-$	$z$	$=$	$-$	$1$	$9$
$+$	$-$	$4$	$x$	$-$	$4$	$y$	$-$	$1$	$2$	$z$	$=$	$-$	$4$	$8$
		$2$	$x$				$-$	$1$	$3$	$z$	$=$	$-$	$6$	$7$

Step 4.4

The resultant equation has $y$ eliminated.

$2 x - 13 z = - 67$

Step 5

Take the resultant equations and eliminate another variable. In this case, eliminate $x$ .

$- 2 x + 19 z = 97$

$2 x - 13 z = - 67$

Step 6

Eliminate $x$ from the system.

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Step 6.1

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

	$-$	$2$	$x$	$+$	$1$	$9$	$z$	$=$		$9$	$7$
$+$		$2$	$x$	$-$	$1$	$3$	$z$	$=$	$-$	$6$	$7$
						$6$	$z$	$=$		$3$	$0$

Step 6.2

The resultant equation has $x$ eliminated.

$6 z = 30$

Step 6.3

Divide each term in $6 z = 30$ by $6$ and simplify.

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Step 6.3.1

Divide each term in $6 z = 30$ by $6$ .

$\frac{6 z}{6} = \frac{30}{6}$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Cancel the common factor of $6$ .

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Step 6.3.2.1.1

Cancel the common factor.

$\frac{6 z}{6} = \frac{30}{6}$

Step 6.3.2.1.2

Divide $z$ by $1$ .

$z = \frac{30}{6}$

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

Divide $30$ by $6$ .

$z = 5$

Step 7

Substitute the value of $z$ into an equation with $y$ eliminated already and solve for the remaining variable.

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Step 7.1

Substitute the value of $z$ into an equation with $y$ eliminated already.

$- 2 x + 19 (5) = 97$

Step 7.2

Solve for $x$ .

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Step 7.2.1

Multiply $19$ by $5$ .

$- 2 x + 95 = 97$

Step 7.2.2

Move all terms not containing $x$ to the right side of the equation.

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Step 7.2.2.1

Subtract $95$ from both sides of the equation.

$- 2 x = 97 - 95$

Step 7.2.2.2

Subtract $95$ from $97$ .

$- 2 x = 2$

Step 7.2.3

Divide each term in $- 2 x = 2$ by $- 2$ and simplify.

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Step 7.2.3.1

Divide each term in $- 2 x = 2$ by $- 2$ .

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

Step 7.2.3.2

Simplify the left side.

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Step 7.2.3.2.1

Cancel the common factor of $- 2$ .

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Step 7.2.3.2.1.1

Cancel the common factor.

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

Step 7.2.3.2.1.2

Divide $x$ by $1$ .

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

Step 7.2.3.3

Simplify the right side.

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Step 7.2.3.3.1

Divide $2$ by $- 2$ .

$x = - 1$

Step 8

Substitute the value of each known variable into one of the initial equations and solve for the last variable.

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Step 8.1

Substitute the value of each known variable into one of the initial equations.

$2 (- 1) + 1 y - 5 \cdot 5 = - 29$

Step 8.2

Solve for $y$ .

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Step 8.2.1

Simplify $2 (- 1) + 1 y - 5 \cdot 5$ .

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Step 8.2.1.1

Simplify each term.

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Step 8.2.1.1.1

Multiply $2$ by $- 1$ .

$- 2 + 1 y - 5 \cdot 5 = - 29$

Step 8.2.1.1.2

Multiply $y$ by $1$ .

$- 2 + y - 5 \cdot 5 = - 29$

Step 8.2.1.1.3

Multiply $- 5$ by $5$ .

$- 2 + y - 25 = - 29$

Step 8.2.1.2

Subtract $25$ from $- 2$ .

$y - 27 = - 29$

Step 8.2.2

Move all terms not containing $y$ to the right side of the equation.

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Step 8.2.2.1

Add $27$ to both sides of the equation.

$y = - 29 + 27$

Step 8.2.2.2

Add $- 29$ and $27$ .

$y = - 2$

Step 9

The solution to the system of equations can be represented as a point.

$(- 1, - 2, 5)$

Step 10

The result can be shown in multiple forms.

Point Form:

$(- 1, - 2, 5)$

Equation Form:

$x = - 1, y = - 2, z = 5$

	$-$	$8$	$x$	$-$	$4$	$y$	$+$	$2$	$0$	$z$	$=$	$1$	$1$	$6$
$+$		$6$	$x$	$+$	$4$	$y$			$-$	$z$	$=$	$-$	$1$	$9$
	$-$	$2$	$x$	$+$				$1$	$9$	$z$	$=$		$9$	$7$

		$6$	$x$	$+$	$4$	$y$			$-$	$z$	$=$	$-$	$1$	$9$
$+$	$-$	$4$	$x$	$-$	$4$	$y$	$-$	$1$	$2$	$z$	$=$	$-$	$4$	$8$
		$2$	$x$				$-$	$1$	$3$	$z$	$=$	$-$	$6$	$7$

	$-$	$2$	$x$	$+$	$1$	$9$	$z$	$=$		$9$	$7$
$+$		$2$	$x$	$-$	$1$	$3$	$z$	$=$	$-$	$6$	$7$
						$6$	$z$	$=$		$3$	$0$

	$-$	$8$	$x$	$-$	$4$	$y$	$+$	$2$	$0$	$z$	$=$	$1$	$1$	$6$
$+$		$6$	$x$	$+$	$4$	$y$			$-$	$z$	$=$	$-$	$1$	$9$
	$-$	$2$	$x$	$+$				$1$	$9$	$z$	$=$		$9$	$7$

		$6$	$x$	$+$	$4$	$y$			$-$	$z$	$=$	$-$	$1$	$9$
$+$	$-$	$4$	$x$	$-$	$4$	$y$	$-$	$1$	$2$	$z$	$=$	$-$	$4$	$8$
		$2$	$x$				$-$	$1$	$3$	$z$	$=$	$-$	$6$	$7$

	$-$	$2$	$x$	$+$	$1$	$9$	$z$	$=$		$9$	$7$
$+$		$2$	$x$	$-$	$1$	$3$	$z$	$=$	$-$	$6$	$7$
						$6$	$z$	$=$		$3$	$0$

	$-$	$8$	$x$	$-$	$4$	$y$	$+$	$2$	$0$	$z$	$=$	$1$	$1$	$6$
$+$		$6$	$x$	$+$	$4$	$y$			$-$	$z$	$=$	$-$	$1$	$9$
	$-$	$2$	$x$	$+$				$1$	$9$	$z$	$=$		$9$	$7$

		$6$	$x$	$+$	$4$	$y$			$-$	$z$	$=$	$-$	$1$	$9$
$+$	$-$	$4$	$x$	$-$	$4$	$y$	$-$	$1$	$2$	$z$	$=$	$-$	$4$	$8$
		$2$	$x$				$-$	$1$	$3$	$z$	$=$	$-$	$6$	$7$

	$-$	$2$	$x$	$+$	$1$	$9$	$z$	$=$		$9$	$7$
$+$		$2$	$x$	$-$	$1$	$3$	$z$	$=$	$-$	$6$	$7$
						$6$	$z$	$=$		$3$	$0$