Precalculus Examples

$5 x + y + z = 10$ , $x + y - z = 6$ , $2 x + 4 y + z = 0$

Step 1

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

$5 x + y + z = 10$

$x + y - z = 6$

Step 2

Eliminate $y$ from the system.

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Step 2.1

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

$5 x + y + z = 10$

$(- 1) \cdot (x + y - z) = (- 1) (6)$

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

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Step 2.2.1.1.1

Apply the distributive property.

$5 x + y + z = 10$

$- 1 x - 1 y - 1 (- z) = (- 1) (6)$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Rewrite $- 1 x$ as $- x$ .

$5 x + y + z = 10$

$- x - 1 y - 1 (- z) = (- 1) (6)$

Step 2.2.1.1.2.2

Rewrite $- 1 y$ as $- y$ .

$5 x + y + z = 10$

$- x - y - 1 (- z) = (- 1) (6)$

Step 2.2.1.1.2.3

Multiply $- 1 (- z)$ .

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Step 2.2.1.1.2.3.1

Multiply $- 1$ by $- 1$ .

$5 x + y + z = 10$

$- x - y + 1 z = (- 1) (6)$

Step 2.2.1.1.2.3.2

Multiply $z$ by $1$ .

$5 x + y + z = 10$

$- x - y + z = (- 1) (6)$

$5 x + y + z = 10$

$- x - y + z = (- 1) (6)$

$5 x + y + z = 10$

$- x - y + z = (- 1) (6)$

$5 x + y + z = 10$

$- x - y + z = (- 1) (6)$

$5 x + y + z = 10$

$- x - y + z = (- 1) (6)$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Multiply $- 1$ by $6$ .

$5 x + y + z = 10$

$- x - y + z = - 6$

$5 x + y + z = 10$

$- x - y + z = - 6$

$5 x + y + z = 10$

$- x - y + z = - 6$

Step 2.3

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

$5$

$x$

$+$

$y$

$+$

$z$

$=$

$1$

$0$

$+$

$-$

$x$

$-$

$y$

$+$

$z$

$=$

$-$

$6$

$4$

$x$

$+$

$2$

$z$

$=$

$4$

Step 2.4

The resultant equation has $y$ eliminated.

$4 x + 2 z = 4$

Step 3

Choose another two equations and eliminate $y$ .

$x + y - z = 6$

$2 x + 4 y + z = 0$

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.

$(- 4) \cdot (x + y - z) = (- 4) (6)$

$2 x + 4 y + z = 0$

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

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Step 4.2.1.1.1

Apply the distributive property.

$- 4 x - 4 y - 4 (- z) = (- 4) (6)$

$2 x + 4 y + z = 0$

Step 4.2.1.1.2

Multiply $- 1$ by $- 4$ .

$- 4 x - 4 y + 4 z = (- 4) (6)$

$2 x + 4 y + z = 0$

$- 4 x - 4 y + 4 z = (- 4) (6)$

$2 x + 4 y + z = 0$

$- 4 x - 4 y + 4 z = (- 4) (6)$

$2 x + 4 y + z = 0$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Multiply $- 4$ by $6$ .

$- 4 x - 4 y + 4 z = - 24$

$2 x + 4 y + z = 0$

$- 4 x - 4 y + 4 z = - 24$

$2 x + 4 y + z = 0$

$- 4 x - 4 y + 4 z = - 24$

$2 x + 4 y + z = 0$

Step 4.3

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

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

Step 4.4

The resultant equation has $y$ eliminated.

$- 2 x + 5 z = - 24$

Step 5

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

$4 x + 2 z = 4$

$- 2 x + 5 z = - 24$

Step 6

Eliminate $x$ from the system.

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Step 6.1

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

$4 x + 2 z = 4$

$(2) \cdot (- 2 x + 5 z) = (2) (- 24)$

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

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Step 6.2.1.1.1

Apply the distributive property.

$4 x + 2 z = 4$

$2 (- 2 x) + 2 (5 z) = (2) (- 24)$

Step 6.2.1.1.2

Multiply.

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Step 6.2.1.1.2.1

Multiply $- 2$ by $2$ .

$4 x + 2 z = 4$

$- 4 x + 2 (5 z) = (2) (- 24)$

Step 6.2.1.1.2.2

Multiply $5$ by $2$ .

$4 x + 2 z = 4$

$- 4 x + 10 z = (2) (- 24)$

$4 x + 2 z = 4$

$- 4 x + 10 z = (2) (- 24)$

$4 x + 2 z = 4$

$- 4 x + 10 z = (2) (- 24)$

$4 x + 2 z = 4$

$- 4 x + 10 z = (2) (- 24)$

Step 6.2.2

Simplify the right side.

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Step 6.2.2.1

Multiply $2$ by $- 24$ .

$4 x + 2 z = 4$

$- 4 x + 10 z = - 48$

$4 x + 2 z = 4$

$- 4 x + 10 z = - 48$

$4 x + 2 z = 4$

$- 4 x + 10 z = - 48$

Step 6.3

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

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

Step 6.4

The resultant equation has $x$ eliminated.

$12 z = - 44$

Step 6.5

Divide each term in $12 z = - 44$ by $12$ and simplify.

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Step 6.5.1

Divide each term in $12 z = - 44$ by $12$ .

$\frac{12 z}{12} = \frac{- 44}{12}$

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.

$\frac{12 z}{12} = \frac{- 44}{12}$

Step 6.5.2.1.2

Divide $z$ by $1$ .

$z = \frac{- 44}{12}$

Step 6.5.3

Simplify the right side.

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Step 6.5.3.1

Cancel the common factor of $- 44$ and $12$ .

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Step 6.5.3.1.1

Factor $4$ out of $- 44$ .

$z = \frac{4 (- 11)}{12}$

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 $12$ .

$z = \frac{4 \cdot - 11}{4 \cdot 3}$

Step 6.5.3.1.2.2

Cancel the common factor.

$z = \frac{4 \cdot - 11}{4 \cdot 3}$

Step 6.5.3.1.2.3

Rewrite the expression.

$z = \frac{- 11}{3}$

Step 6.5.3.2

Move the negative in front of the fraction.

$z = - \frac{11}{3}$

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.

$4 x + 2 (- \frac{11}{3}) = 4$

Step 7.2

Solve for $x$ .

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

Multiply $2 (- \frac{11}{3})$ .

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Step 7.2.1.1.1

Multiply $- 1$ by $2$ .

$4 x - 2 (\frac{11}{3}) = 4$

Step 7.2.1.1.2

Combine $- 2$ and $\frac{11}{3}$ .

$4 x + \frac{- 2 \cdot 11}{3} = 4$

Step 7.2.1.1.3

Multiply $- 2$ by $11$ .

$4 x + \frac{- 22}{3} = 4$

Step 7.2.1.2

Move the negative in front of the fraction.

$4 x - \frac{22}{3} = 4$

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

Add $\frac{22}{3}$ to both sides of the equation.

$4 x = 4 + \frac{22}{3}$

Step 7.2.2.2

To write $4$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$4 x = 4 \cdot \frac{3}{3} + \frac{22}{3}$

Step 7.2.2.3

Combine $4$ and $\frac{3}{3}$ .

$4 x = \frac{4 \cdot 3}{3} + \frac{22}{3}$

Step 7.2.2.4

Combine the numerators over the common denominator.

$4 x = \frac{4 \cdot 3 + 22}{3}$

Step 7.2.2.5

Simplify the numerator.

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Step 7.2.2.5.1

Multiply $4$ by $3$ .

$4 x = \frac{12 + 22}{3}$

Step 7.2.2.5.2

Add $12$ and $22$ .

$4 x = \frac{34}{3}$

Step 7.2.3

Divide each term in $4 x = \frac{34}{3}$ by $4$ and simplify.

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Step 7.2.3.1

Divide each term in $4 x = \frac{34}{3}$ by $4$ .

$\frac{4 x}{4} = \frac{\frac{34}{3}}{4}$

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.

$\frac{4 x}{4} = \frac{\frac{34}{3}}{4}$

Step 7.2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{34}{3}}{4}$

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.

$x = \frac{34}{3} \cdot \frac{1}{4}$

Step 7.2.3.3.2

Cancel the common factor of $2$ .

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Step 7.2.3.3.2.1

Factor $2$ out of $34$ .

$x = \frac{2 (17)}{3} \cdot \frac{1}{4}$

Step 7.2.3.3.2.2

Factor $2$ out of $4$ .

$x = \frac{2 \cdot 17}{3} \cdot \frac{1}{2 \cdot 2}$

Step 7.2.3.3.2.3

Cancel the common factor.

$x = \frac{2 \cdot 17}{3} \cdot \frac{1}{2 \cdot 2}$

Step 7.2.3.3.2.4

Rewrite the expression.

$x = \frac{17}{3} \cdot \frac{1}{2}$

Step 7.2.3.3.3

Multiply $\frac{17}{3}$ by $\frac{1}{2}$ .

$x = \frac{17}{3 \cdot 2}$

Step 7.2.3.3.4

Multiply $3$ by $2$ .

$x = \frac{17}{6}$

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.

$5 (\frac{17}{6}) + y - \frac{11}{3} = 10$

Step 8.2

Solve for $y$ .

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Step 8.2.1

Simplify $5 (\frac{17}{6}) + y - \frac{11}{3}$ .

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Step 8.2.1.1

Multiply $5 (\frac{17}{6})$ .

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Step 8.2.1.1.1

Combine $5$ and $\frac{17}{6}$ .

$\frac{5 \cdot 17}{6} + y - \frac{11}{3} = 10$

Step 8.2.1.1.2

Multiply $5$ by $17$ .

$\frac{85}{6} + y - \frac{11}{3} = 10$

Step 8.2.1.2

To write $- \frac{11}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y + \frac{85}{6} - \frac{11}{3} \cdot \frac{2}{2} = 10$

Step 8.2.1.3

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.2.1.3.1

Multiply $\frac{11}{3}$ by $\frac{2}{2}$ .

$y + \frac{85}{6} - \frac{11 \cdot 2}{3 \cdot 2} = 10$

Step 8.2.1.3.2

Multiply $3$ by $2$ .

$y + \frac{85}{6} - \frac{11 \cdot 2}{6} = 10$

Step 8.2.1.4

Combine the numerators over the common denominator.

$y + \frac{85 - 11 \cdot 2}{6} = 10$

Step 8.2.1.5

Simplify the numerator.

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Step 8.2.1.5.1

Multiply $- 11$ by $2$ .

$y + \frac{85 - 22}{6} = 10$

Step 8.2.1.5.2

Subtract $22$ from $85$ .

$y + \frac{63}{6} = 10$

Step 8.2.1.6

Cancel the common factor of $63$ and $6$ .

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Step 8.2.1.6.1

Factor $3$ out of $63$ .

$y + \frac{3 (21)}{6} = 10$

Step 8.2.1.6.2

Cancel the common factors.

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Step 8.2.1.6.2.1

Factor $3$ out of $6$ .

$y + \frac{3 \cdot 21}{3 \cdot 2} = 10$

Step 8.2.1.6.2.2

Cancel the common factor.

$y + \frac{3 \cdot 21}{3 \cdot 2} = 10$

Step 8.2.1.6.2.3

Rewrite the expression.

$y + \frac{21}{2} = 10$

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

Subtract $\frac{21}{2}$ from both sides of the equation.

$y = 10 - \frac{21}{2}$

Step 8.2.2.2

To write $10$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = 10 \cdot \frac{2}{2} - \frac{21}{2}$

Step 8.2.2.3

Combine $10$ and $\frac{2}{2}$ .

$y = \frac{10 \cdot 2}{2} - \frac{21}{2}$

Step 8.2.2.4

Combine the numerators over the common denominator.

$y = \frac{10 \cdot 2 - 21}{2}$

Step 8.2.2.5

Simplify the numerator.

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Step 8.2.2.5.1

Multiply $10$ by $2$ .

$y = \frac{20 - 21}{2}$

Step 8.2.2.5.2

Subtract $21$ from $20$ .

$y = \frac{- 1}{2}$

Step 8.2.2.6

Move the negative in front of the fraction.

$y = - \frac{1}{2}$

Step 9

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

$(\frac{17}{6}, - \frac{1}{2}, - \frac{11}{3})$

Step 10

The result can be shown in multiple forms.

Point Form:

$(\frac{17}{6}, - \frac{1}{2}, - \frac{11}{3})$

Equation Form:

$x = \frac{17}{6}, y = - \frac{1}{2}, z = - \frac{11}{3}$

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

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

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

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

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

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