Precalculus Examples

$2 x - 4 x + z = 8$ , $3 x + 3 y + 4 z = 20$ , $5 x + y - 3 z = - 13$

Step 1

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

$2 x - 4 x + z = 8$

$3 x + 3 y + 4 z = 20$

Step 2

Eliminate $z$ from the system.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Subtract $4 x$ from $2 x$ .

$- 2 x + z = 8$

$3 x + 3 y + 4 z = 20$

$- 2 x + z = 8$

$3 x + 3 y + 4 z = 20$

Step 2.2

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

$(- 4) \cdot (- 2 x + z) = (- 4) (8)$

$3 x + 3 y + 4 z = 20$

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

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Step 2.3.1.1.1

Apply the distributive property.

$- 4 (- 2 x) - 4 z = (- 4) (8)$

$3 x + 3 y + 4 z = 20$

Step 2.3.1.1.2

Multiply $- 2$ by $- 4$ .

$8 x - 4 z = (- 4) (8)$

$3 x + 3 y + 4 z = 20$

$8 x - 4 z = (- 4) (8)$

$3 x + 3 y + 4 z = 20$

$8 x - 4 z = (- 4) (8)$

$3 x + 3 y + 4 z = 20$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Multiply $- 4$ by $8$ .

$8 x - 4 z = - 32$

$3 x + 3 y + 4 z = 20$

$8 x - 4 z = - 32$

$3 x + 3 y + 4 z = 20$

$8 x - 4 z = - 32$

$3 x + 3 y + 4 z = 20$

Step 2.4

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

			$8$	$x$	$-$	$4$	$z$	$=$	$-$	$3$	$2$
	$+$		$3$	$x$	$+$	$4$	$z$	$=$		$2$	$0$
$3$	$y$	$1$	$1$	$x$	$+$			$=$	$-$	$1$	$2$

Step 2.5

The resultant equation has $z$ eliminated.

$11 x + 3 y = - 12$

Step 3

Choose another two equations and eliminate $z$ .

$3 x + 3 y + 4 z = 20$

$5 x + y - 3 z = - 13$

Step 4

Eliminate $z$ from the system.

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Step 4.1

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

$(3) \cdot (3 x + 3 y + 4 z) = (3) (20)$

$(4) \cdot (5 x + y - 3 z) = (4) (- 13)$

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

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Step 4.2.1.1.1

Apply the distributive property.

$3 (3 x) + 3 (3 y) + 3 (4 z) = (3) (20)$

$(4) \cdot (5 x + y - 3 z) = (4) (- 13)$

Step 4.2.1.1.2

Simplify.

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Step 4.2.1.1.2.1

Multiply $3$ by $3$ .

$9 x + 3 (3 y) + 3 (4 z) = (3) (20)$

$(4) \cdot (5 x + y - 3 z) = (4) (- 13)$

Step 4.2.1.1.2.2

Multiply $3$ by $3$ .

$9 x + 9 y + 3 (4 z) = (3) (20)$

$(4) \cdot (5 x + y - 3 z) = (4) (- 13)$

Step 4.2.1.1.2.3

Multiply $4$ by $3$ .

$9 x + 9 y + 12 z = (3) (20)$

$(4) \cdot (5 x + y - 3 z) = (4) (- 13)$

$9 x + 9 y + 12 z = (3) (20)$

$(4) \cdot (5 x + y - 3 z) = (4) (- 13)$

$9 x + 9 y + 12 z = (3) (20)$

$(4) \cdot (5 x + y - 3 z) = (4) (- 13)$

$9 x + 9 y + 12 z = (3) (20)$

$(4) \cdot (5 x + y - 3 z) = (4) (- 13)$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Multiply $3$ by $20$ .

$9 x + 9 y + 12 z = 60$

$(4) \cdot (5 x + y - 3 z) = (4) (- 13)$

$9 x + 9 y + 12 z = 60$

$(4) \cdot (5 x + y - 3 z) = (4) (- 13)$

Step 4.2.3

Simplify the left side.

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Step 4.2.3.1

Simplify $(4) \cdot (5 x + y - 3 z)$ .

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Step 4.2.3.1.1

Apply the distributive property.

$9 x + 9 y + 12 z = 60$

$4 (5 x) + 4 y + 4 (- 3 z) = (4) (- 13)$

Step 4.2.3.1.2

Simplify.

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Step 4.2.3.1.2.1

Multiply $5$ by $4$ .

$9 x + 9 y + 12 z = 60$

$20 x + 4 y + 4 (- 3 z) = (4) (- 13)$

Step 4.2.3.1.2.2

Multiply $- 3$ by $4$ .

$9 x + 9 y + 12 z = 60$

$20 x + 4 y - 12 z = (4) (- 13)$

$9 x + 9 y + 12 z = 60$

$20 x + 4 y - 12 z = (4) (- 13)$

$9 x + 9 y + 12 z = 60$

$20 x + 4 y - 12 z = (4) (- 13)$

$9 x + 9 y + 12 z = 60$

$20 x + 4 y - 12 z = (4) (- 13)$

Step 4.2.4

Simplify the right side.

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Step 4.2.4.1

Multiply $4$ by $- 13$ .

$9 x + 9 y + 12 z = 60$

$20 x + 4 y - 12 z = - 52$

$9 x + 9 y + 12 z = 60$

$20 x + 4 y - 12 z = - 52$

$9 x + 9 y + 12 z = 60$

$20 x + 4 y - 12 z = - 52$

Step 4.3

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

		$9$	$x$	$+$		$9$	$y$	$+$	$1$	$2$	$z$	$=$		$6$	$0$
$+$	$2$	$0$	$x$	$+$		$4$	$y$	$-$	$1$	$2$	$z$	$=$	$-$	$5$	$2$
	$2$	$9$	$x$	$+$	$1$	$3$	$y$					$=$			$8$

Step 4.4

The resultant equation has $z$ eliminated.

$29 x + 13 y = 8$

Step 5

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

$11 x + 3 y = - 12$

$29 x + 13 y = 8$

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.

$(- 13) \cdot (11 x + 3 y) = (- 13) (- 12)$

$(3) \cdot (29 x + 13 y) = (3) (8)$

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 $(- 13) \cdot (11 x + 3 y)$ .

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Step 6.2.1.1.1

Apply the distributive property.

$- 13 (11 x) - 13 (3 y) = (- 13) (- 12)$

$(3) \cdot (29 x + 13 y) = (3) (8)$

Step 6.2.1.1.2

Multiply.

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Step 6.2.1.1.2.1

Multiply $11$ by $- 13$ .

$- 143 x - 13 (3 y) = (- 13) (- 12)$

$(3) \cdot (29 x + 13 y) = (3) (8)$

Step 6.2.1.1.2.2

Multiply $3$ by $- 13$ .

$- 143 x - 39 y = (- 13) (- 12)$

$(3) \cdot (29 x + 13 y) = (3) (8)$

$- 143 x - 39 y = (- 13) (- 12)$

$(3) \cdot (29 x + 13 y) = (3) (8)$

$- 143 x - 39 y = (- 13) (- 12)$

$(3) \cdot (29 x + 13 y) = (3) (8)$

$- 143 x - 39 y = (- 13) (- 12)$

$(3) \cdot (29 x + 13 y) = (3) (8)$

Step 6.2.2

Simplify the right side.

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Step 6.2.2.1

Multiply $- 13$ by $- 12$ .

$- 143 x - 39 y = 156$

$(3) \cdot (29 x + 13 y) = (3) (8)$

$- 143 x - 39 y = 156$

$(3) \cdot (29 x + 13 y) = (3) (8)$

Step 6.2.3

Simplify the left side.

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Step 6.2.3.1

Simplify $(3) \cdot (29 x + 13 y)$ .

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Step 6.2.3.1.1

Apply the distributive property.

$- 143 x - 39 y = 156$

$3 (29 x) + 3 (13 y) = (3) (8)$

Step 6.2.3.1.2

Multiply.

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Step 6.2.3.1.2.1

Multiply $29$ by $3$ .

$- 143 x - 39 y = 156$

$87 x + 3 (13 y) = (3) (8)$

Step 6.2.3.1.2.2

Multiply $13$ by $3$ .

$- 143 x - 39 y = 156$

$87 x + 39 y = (3) (8)$

$- 143 x - 39 y = 156$

$87 x + 39 y = (3) (8)$

$- 143 x - 39 y = 156$

$87 x + 39 y = (3) (8)$

$- 143 x - 39 y = 156$

$87 x + 39 y = (3) (8)$

Step 6.2.4

Simplify the right side.

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Step 6.2.4.1

Multiply $3$ by $8$ .

$- 143 x - 39 y = 156$

$87 x + 39 y = 24$

$- 143 x - 39 y = 156$

$87 x + 39 y = 24$

$- 143 x - 39 y = 156$

$87 x + 39 y = 24$

Step 6.3

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

	$-$	$1$	$4$	$3$	$x$	$-$	$3$	$9$	$y$	$=$	$1$	$5$	$6$
$+$			$8$	$7$	$x$	$+$	$3$	$9$	$y$	$=$		$2$	$4$
		$-$	$5$	$6$	$x$					$=$	$1$	$8$	$0$

Step 6.4

The resultant equation has $y$ eliminated.

$- 56 x = 180$

Step 6.5

Divide each term in $- 56 x = 180$ by $- 56$ and simplify.

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Step 6.5.1

Divide each term in $- 56 x = 180$ by $- 56$ .

$\frac{- 56 x}{- 56} = \frac{180}{- 56}$

Step 6.5.2

Simplify the left side.

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Step 6.5.2.1

Cancel the common factor of $- 56$ .

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Step 6.5.2.1.1

Cancel the common factor.

$\frac{- 56 x}{- 56} = \frac{180}{- 56}$

Step 6.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{180}{- 56}$

Step 6.5.3

Simplify the right side.

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Step 6.5.3.1

Cancel the common factor of $180$ and $- 56$ .

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Step 6.5.3.1.1

Factor $4$ out of $180$ .

$x = \frac{4 (45)}{- 56}$

Step 6.5.3.1.2

Cancel the common factors.

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Step 6.5.3.1.2.1

Factor $4$ out of $- 56$ .

$x = \frac{4 \cdot 45}{4 \cdot - 14}$

Step 6.5.3.1.2.2

Cancel the common factor.

$x = \frac{4 \cdot 45}{4 \cdot - 14}$

Step 6.5.3.1.2.3

Rewrite the expression.

$x = \frac{45}{- 14}$

Step 6.5.3.2

Move the negative in front of the fraction.

$x = - \frac{45}{14}$

Step 7

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

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Step 7.1

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

$11 (- \frac{45}{14}) + 3 y = - 12$

Step 7.2

Solve for $y$ .

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

Multiply $11 (- \frac{45}{14})$ .

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Step 7.2.1.1.1

Multiply $- 1$ by $11$ .

$- 11 (\frac{45}{14}) + 3 y = - 12$

Step 7.2.1.1.2

Combine $- 11$ and $\frac{45}{14}$ .

$\frac{- 11 \cdot 45}{14} + 3 y = - 12$

Step 7.2.1.1.3

Multiply $- 11$ by $45$ .

$\frac{- 495}{14} + 3 y = - 12$

Step 7.2.1.2

Move the negative in front of the fraction.

$- \frac{495}{14} + 3 y = - 12$

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 $\frac{495}{14}$ to both sides of the equation.

$3 y = - 12 + \frac{495}{14}$

Step 7.2.2.2

To write $- 12$ as a fraction with a common denominator, multiply by $\frac{14}{14}$ .

$3 y = - 12 \cdot \frac{14}{14} + \frac{495}{14}$

Step 7.2.2.3

Combine $- 12$ and $\frac{14}{14}$ .

$3 y = \frac{- 12 \cdot 14}{14} + \frac{495}{14}$

Step 7.2.2.4

Combine the numerators over the common denominator.

$3 y = \frac{- 12 \cdot 14 + 495}{14}$

Step 7.2.2.5

Simplify the numerator.

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Step 7.2.2.5.1

Multiply $- 12$ by $14$ .

$3 y = \frac{- 168 + 495}{14}$

Step 7.2.2.5.2

Add $- 168$ and $495$ .

$3 y = \frac{327}{14}$

Step 7.2.3

Divide each term in $3 y = \frac{327}{14}$ by $3$ and simplify.

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Step 7.2.3.1

Divide each term in $3 y = \frac{327}{14}$ by $3$ .

$\frac{3 y}{3} = \frac{\frac{327}{14}}{3}$

Step 7.2.3.2

Simplify the left side.

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Step 7.2.3.2.1

Cancel the common factor of $3$ .

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Step 7.2.3.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{\frac{327}{14}}{3}$

Step 7.2.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{\frac{327}{14}}{3}$

Step 7.2.3.3

Simplify the right side.

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Step 7.2.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{327}{14} \cdot \frac{1}{3}$

Step 7.2.3.3.2

Cancel the common factor of $3$ .

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Step 7.2.3.3.2.1

Factor $3$ out of $327$ .

$y = \frac{3 (109)}{14} \cdot \frac{1}{3}$

Step 7.2.3.3.2.2

Cancel the common factor.

$y = \frac{3 \cdot 109}{14} \cdot \frac{1}{3}$

Step 7.2.3.3.2.3

Rewrite the expression.

$y = \frac{109}{14}$

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.

$2 (- \frac{45}{14}) - 4 (- \frac{45}{14}) + z = 8$

Step 8.2

Solve for $z$ .

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Step 8.2.1

Simplify $2 (- \frac{45}{14}) - 4 (- \frac{45}{14}) + z$ .

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Step 8.2.1.1

Simplify each term.

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Step 8.2.1.1.1

Cancel the common factor of $2$ .

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Step 8.2.1.1.1.1

Move the leading negative in $- \frac{45}{14}$ into the numerator.

$2 (\frac{- 45}{14}) - 4 (- \frac{45}{14}) + z = 8$

Step 8.2.1.1.1.2

Factor $2$ out of $14$ .

$2 \frac{- 45}{2 (7)} - 4 (- \frac{45}{14}) + z = 8$

Step 8.2.1.1.1.3

Cancel the common factor.

$2 \frac{- 45}{2 \cdot 7} - 4 (- \frac{45}{14}) + z = 8$

Step 8.2.1.1.1.4

Rewrite the expression.

$\frac{- 45}{7} - 4 (- \frac{45}{14}) + z = 8$

Step 8.2.1.1.2

Move the negative in front of the fraction.

$- \frac{45}{7} - 4 (- \frac{45}{14}) + z = 8$

Step 8.2.1.1.3

Cancel the common factor of $2$ .

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Step 8.2.1.1.3.1

Move the leading negative in $- \frac{45}{14}$ into the numerator.

$- \frac{45}{7} - 4 (\frac{- 45}{14}) + z = 8$

Step 8.2.1.1.3.2

Factor $2$ out of $- 4$ .

$- \frac{45}{7} + 2 (- 2) \frac{- 45}{14} + z = 8$

Step 8.2.1.1.3.3

Factor $2$ out of $14$ .

$- \frac{45}{7} + 2 \cdot - 2 \frac{- 45}{2 \cdot 7} + z = 8$

Step 8.2.1.1.3.4

Cancel the common factor.

$- \frac{45}{7} + 2 \cdot - 2 \frac{- 45}{2 \cdot 7} + z = 8$

Step 8.2.1.1.3.5

Rewrite the expression.

$- \frac{45}{7} - 2 (\frac{- 45}{7}) + z = 8$

Step 8.2.1.1.4

Combine $- 2$ and $\frac{- 45}{7}$ .

$- \frac{45}{7} + \frac{- 2 \cdot - 45}{7} + z = 8$

Step 8.2.1.1.5

Multiply $- 2$ by $- 45$ .

$- \frac{45}{7} + \frac{90}{7} + z = 8$

Step 8.2.1.2

Combine fractions.

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Step 8.2.1.2.1

Combine the numerators over the common denominator.

$z + \frac{- 45 + 90}{7} = 8$

Step 8.2.1.2.2

Add $- 45$ and $90$ .

$z + \frac{45}{7} = 8$

Step 8.2.2

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

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Step 8.2.2.1

Subtract $\frac{45}{7}$ from both sides of the equation.

$z = 8 - \frac{45}{7}$

Step 8.2.2.2

To write $8$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$z = 8 \cdot \frac{7}{7} - \frac{45}{7}$

Step 8.2.2.3

Combine $8$ and $\frac{7}{7}$ .

$z = \frac{8 \cdot 7}{7} - \frac{45}{7}$

Step 8.2.2.4

Combine the numerators over the common denominator.

$z = \frac{8 \cdot 7 - 45}{7}$

Step 8.2.2.5

Simplify the numerator.

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Step 8.2.2.5.1

Multiply $8$ by $7$ .

$z = \frac{56 - 45}{7}$

Step 8.2.2.5.2

Subtract $45$ from $56$ .

$z = \frac{11}{7}$

Step 9

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

$(- \frac{45}{14}, \frac{109}{14}, \frac{11}{7})$

Step 10

The result can be shown in multiple forms.

Point Form:

$(- \frac{45}{14}, \frac{109}{14}, \frac{11}{7})$

Equation Form:

$x = - \frac{45}{14}, y = \frac{109}{14}, z = \frac{11}{7}$

			$8$	$x$	$-$	$4$	$z$	$=$	$-$	$3$	$2$
	$+$		$3$	$x$	$+$	$4$	$z$	$=$		$2$	$0$
$3$	$y$	$1$	$1$	$x$	$+$			$=$	$-$	$1$	$2$

		$9$	$x$	$+$		$9$	$y$	$+$	$1$	$2$	$z$	$=$		$6$	$0$
$+$	$2$	$0$	$x$	$+$		$4$	$y$	$-$	$1$	$2$	$z$	$=$	$-$	$5$	$2$
	$2$	$9$	$x$	$+$	$1$	$3$	$y$					$=$			$8$

	$-$	$1$	$4$	$3$	$x$	$-$	$3$	$9$	$y$	$=$	$1$	$5$	$6$
$+$			$8$	$7$	$x$	$+$	$3$	$9$	$y$	$=$		$2$	$4$
		$-$	$5$	$6$	$x$					$=$	$1$	$8$	$0$

			$8$	$x$	$-$	$4$	$z$	$=$	$-$	$3$	$2$
	$+$		$3$	$x$	$+$	$4$	$z$	$=$		$2$	$0$
$3$	$y$	$1$	$1$	$x$	$+$			$=$	$-$	$1$	$2$

		$9$	$x$	$+$		$9$	$y$	$+$	$1$	$2$	$z$	$=$		$6$	$0$
$+$	$2$	$0$	$x$	$+$		$4$	$y$	$-$	$1$	$2$	$z$	$=$	$-$	$5$	$2$
	$2$	$9$	$x$	$+$	$1$	$3$	$y$					$=$			$8$

	$-$	$1$	$4$	$3$	$x$	$-$	$3$	$9$	$y$	$=$	$1$	$5$	$6$
$+$			$8$	$7$	$x$	$+$	$3$	$9$	$y$	$=$		$2$	$4$
		$-$	$5$	$6$	$x$					$=$	$1$	$8$	$0$

			$8$	$x$	$-$	$4$	$z$	$=$	$-$	$3$	$2$
	$+$		$3$	$x$	$+$	$4$	$z$	$=$		$2$	$0$
$3$	$y$	$1$	$1$	$x$	$+$			$=$	$-$	$1$	$2$

		$9$	$x$	$+$		$9$	$y$	$+$	$1$	$2$	$z$	$=$		$6$	$0$
$+$	$2$	$0$	$x$	$+$		$4$	$y$	$-$	$1$	$2$	$z$	$=$	$-$	$5$	$2$
	$2$	$9$	$x$	$+$	$1$	$3$	$y$					$=$			$8$

	$-$	$1$	$4$	$3$	$x$	$-$	$3$	$9$	$y$	$=$	$1$	$5$	$6$
$+$			$8$	$7$	$x$	$+$	$3$	$9$	$y$	$=$		$2$	$4$
		$-$	$5$	$6$	$x$					$=$	$1$	$8$	$0$