Precalculus Examples

$2 x + 6 z = - 9$ , $3 x - 2 y + 11 z = - 16$ , $3 x - y + 7 z = - 11$

Step 1

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

$2 x + 6 z = - 9$

$3 x - 2 y + 11 z = - 16$

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.

$(- 3) \cdot (2 x + 6 z) = (- 3) (- 9)$

$(2) \cdot (3 x - 2 y + 11 z) = (2) (- 16)$

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 $(- 3) \cdot (2 x + 6 z)$ .

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Step 2.2.1.1.1

Apply the distributive property.

$- 3 (2 x) - 3 (6 z) = (- 3) (- 9)$

$(2) \cdot (3 x - 2 y + 11 z) = (2) (- 16)$

Step 2.2.1.1.2

Multiply.

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Step 2.2.1.1.2.1

Multiply $2$ by $- 3$ .

$- 6 x - 3 (6 z) = (- 3) (- 9)$

$(2) \cdot (3 x - 2 y + 11 z) = (2) (- 16)$

Step 2.2.1.1.2.2

Multiply $6$ by $- 3$ .

$- 6 x - 18 z = (- 3) (- 9)$

$(2) \cdot (3 x - 2 y + 11 z) = (2) (- 16)$

$- 6 x - 18 z = (- 3) (- 9)$

$(2) \cdot (3 x - 2 y + 11 z) = (2) (- 16)$

$- 6 x - 18 z = (- 3) (- 9)$

$(2) \cdot (3 x - 2 y + 11 z) = (2) (- 16)$

$- 6 x - 18 z = (- 3) (- 9)$

$(2) \cdot (3 x - 2 y + 11 z) = (2) (- 16)$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Multiply $- 3$ by $- 9$ .

$- 6 x - 18 z = 27$

$(2) \cdot (3 x - 2 y + 11 z) = (2) (- 16)$

$- 6 x - 18 z = 27$

$(2) \cdot (3 x - 2 y + 11 z) = (2) (- 16)$

Step 2.2.3

Simplify the left side.

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Step 2.2.3.1

Simplify $(2) \cdot (3 x - 2 y + 11 z)$ .

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Step 2.2.3.1.1

Apply the distributive property.

$- 6 x - 18 z = 27$

$2 (3 x) + 2 (- 2 y) + 2 (11 z) = (2) (- 16)$

Step 2.2.3.1.2

Simplify.

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Step 2.2.3.1.2.1

Multiply $3$ by $2$ .

$- 6 x - 18 z = 27$

$6 x + 2 (- 2 y) + 2 (11 z) = (2) (- 16)$

Step 2.2.3.1.2.2

Multiply $- 2$ by $2$ .

$- 6 x - 18 z = 27$

$6 x - 4 y + 2 (11 z) = (2) (- 16)$

Step 2.2.3.1.2.3

Multiply $11$ by $2$ .

$- 6 x - 18 z = 27$

$6 x - 4 y + 22 z = (2) (- 16)$

$- 6 x - 18 z = 27$

$6 x - 4 y + 22 z = (2) (- 16)$

$- 6 x - 18 z = 27$

$6 x - 4 y + 22 z = (2) (- 16)$

$- 6 x - 18 z = 27$

$6 x - 4 y + 22 z = (2) (- 16)$

Step 2.2.4

Simplify the right side.

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Step 2.2.4.1

Multiply $2$ by $- 16$ .

$- 6 x - 18 z = 27$

$6 x - 4 y + 22 z = - 32$

$- 6 x - 18 z = 27$

$6 x - 4 y + 22 z = - 32$

$- 6 x - 18 z = 27$

$6 x - 4 y + 22 z = - 32$

Step 2.3

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

				$-$	$6$	$x$	$-$	$1$	$8$	$z$	$=$		$2$	$7$
		$+$			$6$	$x$	$+$	$2$	$2$	$z$	$=$	$-$	$3$	$2$
$-$	$4$	$y$	$+$						$4$	$z$	$=$		$-$	$5$

Step 2.4

The resultant equation has $x$ eliminated.

$- 4 y + 4 z = - 5$

Step 3

Choose another two equations and eliminate $x$ .

$3 x - 2 y + 11 z = - 16$

$3 x - y + 7 z = - 11$

Step 4

Eliminate $x$ from the system.

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Step 4.1

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

$(- 1) \cdot (3 x - 2 y + 11 z) = (- 1) (- 16)$

$3 x - y + 7 z = - 11$

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 $(- 1) \cdot (3 x - 2 y + 11 z)$ .

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Step 4.2.1.1.1

Apply the distributive property.

$- 1 (3 x) - 1 (- 2 y) - 1 (11 z) = (- 1) (- 16)$

$3 x - y + 7 z = - 11$

Step 4.2.1.1.2

Simplify.

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Step 4.2.1.1.2.1

Multiply $3$ by $- 1$ .

$- 3 x - 1 (- 2 y) - 1 (11 z) = (- 1) (- 16)$

$3 x - y + 7 z = - 11$

Step 4.2.1.1.2.2

Multiply $- 2$ by $- 1$ .

$- 3 x + 2 y - 1 (11 z) = (- 1) (- 16)$

$3 x - y + 7 z = - 11$

Step 4.2.1.1.2.3

Multiply $11$ by $- 1$ .

$- 3 x + 2 y - 11 z = (- 1) (- 16)$

$3 x - y + 7 z = - 11$

$- 3 x + 2 y - 11 z = (- 1) (- 16)$

$3 x - y + 7 z = - 11$

$- 3 x + 2 y - 11 z = (- 1) (- 16)$

$3 x - y + 7 z = - 11$

$- 3 x + 2 y - 11 z = (- 1) (- 16)$

$3 x - y + 7 z = - 11$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Multiply $- 1$ by $- 16$ .

$- 3 x + 2 y - 11 z = 16$

$3 x - y + 7 z = - 11$

$- 3 x + 2 y - 11 z = 16$

$3 x - y + 7 z = - 11$

$- 3 x + 2 y - 11 z = 16$

$3 x - y + 7 z = - 11$

Step 4.3

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

	$-$	$3$	$x$	$+$	$2$	$y$	$-$	$1$	$1$	$z$	$=$		$1$	$6$
$+$		$3$	$x$		$-$	$y$	$+$		$7$	$z$	$=$	$-$	$1$	$1$
						$y$		$-$	$4$	$z$	$=$			$5$

Step 4.4

The resultant equation has $x$ eliminated.

$y - 4 z = 5$

Step 5

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

$- 4 y + 4 z = - 5$

$y - 4 z = 5$

Step 6

Eliminate $y$ from the system.

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Step 6.1

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

$- 4 y + 4 z = - 5$

$(4) \cdot (y - 4 z) = (4) (5)$

Step 6.2

Simplify.

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Step 6.2.1

Simplify the left side.

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Step 6.2.1.1

Simplify $(4) \cdot (y - 4 z)$ .

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Step 6.2.1.1.1

Apply the distributive property.

$- 4 y + 4 z = - 5$

$4 y + 4 (- 4 z) = (4) (5)$

Step 6.2.1.1.2

Multiply $- 4$ by $4$ .

$- 4 y + 4 z = - 5$

$4 y - 16 z = (4) (5)$

$- 4 y + 4 z = - 5$

$4 y - 16 z = (4) (5)$

$- 4 y + 4 z = - 5$

$4 y - 16 z = (4) (5)$

Step 6.2.2

Simplify the right side.

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Step 6.2.2.1

Multiply $4$ by $5$ .

$- 4 y + 4 z = - 5$

$4 y - 16 z = 20$

$- 4 y + 4 z = - 5$

$4 y - 16 z = 20$

$- 4 y + 4 z = - 5$

$4 y - 16 z = 20$

Step 6.3

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

	$-$	$4$	$y$	$+$		$4$	$z$	$=$	$-$	$5$
$+$		$4$	$y$	$-$	$1$	$6$	$z$	$=$	$2$	$0$
				$-$	$1$	$2$	$z$	$=$	$1$	$5$

Step 6.4

The resultant equation has $y$ eliminated.

$- 12 z = 15$

Step 6.5

Divide each term in $- 12 z = 15$ by $- 12$ and simplify.

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Step 6.5.1

Divide each term in $- 12 z = 15$ by $- 12$ .

$\frac{- 12 z}{- 12} = \frac{15}{- 12}$

Step 6.5.2

Simplify the left side.

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Step 6.5.2.1

Cancel the common factor of $- 12$ .

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Step 6.5.2.1.1

Cancel the common factor.

$\frac{- 12 z}{- 12} = \frac{15}{- 12}$

Step 6.5.2.1.2

Divide $z$ by $1$ .

$z = \frac{15}{- 12}$

Step 6.5.3

Simplify the right side.

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Step 6.5.3.1

Cancel the common factor of $15$ and $- 12$ .

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Step 6.5.3.1.1

Factor $3$ out of $15$ .

$z = \frac{3 (5)}{- 12}$

Step 6.5.3.1.2

Cancel the common factors.

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Step 6.5.3.1.2.1

Factor $3$ out of $- 12$ .

$z = \frac{3 \cdot 5}{3 \cdot - 4}$

Step 6.5.3.1.2.2

Cancel the common factor.

$z = \frac{3 \cdot 5}{3 \cdot - 4}$

Step 6.5.3.1.2.3

Rewrite the expression.

$z = \frac{5}{- 4}$

Step 6.5.3.2

Move the negative in front of the fraction.

$z = - \frac{5}{4}$

Step 7

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

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Step 7.1

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

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

Step 7.2

Solve for $y$ .

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Step 7.2.1

Cancel the common factor of $4$ .

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Step 7.2.1.1

Move the leading negative in $- \frac{5}{4}$ into the numerator.

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

Step 7.2.1.2

Cancel the common factor.

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

Step 7.2.1.3

Rewrite the expression.

$- 4 y - 5 = - 5$

Step 7.2.2

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

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Step 7.2.2.1

Add $5$ to both sides of the equation.

$- 4 y = - 5 + 5$

Step 7.2.2.2

Add $- 5$ and $5$ .

$- 4 y = 0$

Step 7.2.3

Divide each term in $- 4 y = 0$ by $- 4$ and simplify.

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Step 7.2.3.1

Divide each term in $- 4 y = 0$ by $- 4$ .

$\frac{- 4 y}{- 4} = \frac{0}{- 4}$

Step 7.2.3.2

Simplify the left side.

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Step 7.2.3.2.1

Cancel the common factor of $- 4$ .

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Step 7.2.3.2.1.1

Cancel the common factor.

$\frac{- 4 y}{- 4} = \frac{0}{- 4}$

Step 7.2.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{0}{- 4}$

Step 7.2.3.3

Simplify the right side.

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Step 7.2.3.3.1

Divide $0$ by $- 4$ .

$y = 0$

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.

$2 x + 6 (- \frac{5}{4}) = - 9$

Step 8.2

Solve for $x$ .

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Step 8.2.1

Simplify each term.

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Step 8.2.1.1

Cancel the common factor of $2$ .

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Step 8.2.1.1.1

Move the leading negative in $- \frac{5}{4}$ into the numerator.

$2 x + 6 (\frac{- 5}{4}) = - 9$

Step 8.2.1.1.2

Factor $2$ out of $6$ .

$2 x + 2 (3) \frac{- 5}{4} = - 9$

Step 8.2.1.1.3

Factor $2$ out of $4$ .

$2 x + 2 \cdot 3 \frac{- 5}{2 \cdot 2} = - 9$

Step 8.2.1.1.4

Cancel the common factor.

$2 x + 2 \cdot 3 \frac{- 5}{2 \cdot 2} = - 9$

Step 8.2.1.1.5

Rewrite the expression.

$2 x + 3 (\frac{- 5}{2}) = - 9$

Step 8.2.1.2

Combine $3$ and $\frac{- 5}{2}$ .

$2 x + \frac{3 \cdot - 5}{2} = - 9$

Step 8.2.1.3

Multiply $3$ by $- 5$ .

$2 x + \frac{- 15}{2} = - 9$

Step 8.2.1.4

Move the negative in front of the fraction.

$2 x - \frac{15}{2} = - 9$

Step 8.2.2

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

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Step 8.2.2.1

Add $\frac{15}{2}$ to both sides of the equation.

$2 x = - 9 + \frac{15}{2}$

Step 8.2.2.2

To write $- 9$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 x = - 9 \cdot \frac{2}{2} + \frac{15}{2}$

Step 8.2.2.3

Combine $- 9$ and $\frac{2}{2}$ .

$2 x = \frac{- 9 \cdot 2}{2} + \frac{15}{2}$

Step 8.2.2.4

Combine the numerators over the common denominator.

$2 x = \frac{- 9 \cdot 2 + 15}{2}$

Step 8.2.2.5

Simplify the numerator.

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Step 8.2.2.5.1

Multiply $- 9$ by $2$ .

$2 x = \frac{- 18 + 15}{2}$

Step 8.2.2.5.2

Add $- 18$ and $15$ .

$2 x = \frac{- 3}{2}$

Step 8.2.2.6

Move the negative in front of the fraction.

$2 x = - \frac{3}{2}$

Step 8.2.3

Divide each term in $2 x = - \frac{3}{2}$ by $2$ and simplify.

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Step 8.2.3.1

Divide each term in $2 x = - \frac{3}{2}$ by $2$ .

$\frac{2 x}{2} = \frac{- \frac{3}{2}}{2}$

Step 8.2.3.2

Simplify the left side.

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Step 8.2.3.2.1

Cancel the common factor of $2$ .

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Step 8.2.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- \frac{3}{2}}{2}$

Step 8.2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \frac{3}{2}}{2}$

Step 8.2.3.3

Simplify the right side.

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Step 8.2.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{3}{2} \cdot \frac{1}{2}$

Step 8.2.3.3.2

Multiply $- \frac{3}{2} \cdot \frac{1}{2}$ .

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Step 8.2.3.3.2.1

Multiply $\frac{1}{2}$ by $\frac{3}{2}$ .

$x = - \frac{3}{2 \cdot 2}$

Step 8.2.3.3.2.2

Multiply $2$ by $2$ .

$x = - \frac{3}{4}$

Step 9

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

$(- \frac{3}{4}, 0, - \frac{5}{4})$

Step 10

The result can be shown in multiple forms.

Point Form:

$(- \frac{3}{4}, 0, - \frac{5}{4})$

Equation Form:

$x = - \frac{3}{4}, y = 0, z = - \frac{5}{4}$

				$-$	$6$	$x$	$-$	$1$	$8$	$z$	$=$		$2$	$7$
		$+$			$6$	$x$	$+$	$2$	$2$	$z$	$=$	$-$	$3$	$2$
$-$	$4$	$y$	$+$						$4$	$z$	$=$		$-$	$5$

	$-$	$3$	$x$	$+$	$2$	$y$	$-$	$1$	$1$	$z$	$=$		$1$	$6$
$+$		$3$	$x$		$-$	$y$	$+$		$7$	$z$	$=$	$-$	$1$	$1$
						$y$		$-$	$4$	$z$	$=$			$5$

	$-$	$4$	$y$	$+$		$4$	$z$	$=$	$-$	$5$
$+$		$4$	$y$	$-$	$1$	$6$	$z$	$=$	$2$	$0$
				$-$	$1$	$2$	$z$	$=$	$1$	$5$

				$-$	$6$	$x$	$-$	$1$	$8$	$z$	$=$		$2$	$7$
		$+$			$6$	$x$	$+$	$2$	$2$	$z$	$=$	$-$	$3$	$2$
$-$	$4$	$y$	$+$						$4$	$z$	$=$		$-$	$5$

	$-$	$3$	$x$	$+$	$2$	$y$	$-$	$1$	$1$	$z$	$=$		$1$	$6$
$+$		$3$	$x$		$-$	$y$	$+$		$7$	$z$	$=$	$-$	$1$	$1$
						$y$		$-$	$4$	$z$	$=$			$5$

	$-$	$4$	$y$	$+$		$4$	$z$	$=$	$-$	$5$
$+$		$4$	$y$	$-$	$1$	$6$	$z$	$=$	$2$	$0$
				$-$	$1$	$2$	$z$	$=$	$1$	$5$

				$-$	$6$	$x$	$-$	$1$	$8$	$z$	$=$		$2$	$7$
		$+$			$6$	$x$	$+$	$2$	$2$	$z$	$=$	$-$	$3$	$2$
$-$	$4$	$y$	$+$						$4$	$z$	$=$		$-$	$5$

	$-$	$3$	$x$	$+$	$2$	$y$	$-$	$1$	$1$	$z$	$=$		$1$	$6$
$+$		$3$	$x$		$-$	$y$	$+$		$7$	$z$	$=$	$-$	$1$	$1$
						$y$		$-$	$4$	$z$	$=$			$5$

	$-$	$4$	$y$	$+$		$4$	$z$	$=$	$-$	$5$
$+$		$4$	$y$	$-$	$1$	$6$	$z$	$=$	$2$	$0$
				$-$	$1$	$2$	$z$	$=$	$1$	$5$