Algebra Examples

$- x - 5 y + z = 17$ $- 5 x - 5 y + 5 z = 5$ $2 x + 5 y - 3 z = - 10$

Step 1

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

$- x - 5 y + z = 17$

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

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.

$(- 5) \cdot (- x - 5 y + z) = (- 5) (17)$

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

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

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Step 2.2.1.1.1

Apply the distributive property.

$- 5 (- x) - 5 (- 5 y) - 5 z = (- 5) (17)$

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

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $- 1$ by $- 5$ .

$5 x - 5 (- 5 y) - 5 z = (- 5) (17)$

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

Step 2.2.1.1.2.2

Multiply $- 5$ by $- 5$ .

$5 x + 25 y - 5 z = (- 5) (17)$

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

$5 x + 25 y - 5 z = (- 5) (17)$

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

$5 x + 25 y - 5 z = (- 5) (17)$

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

$5 x + 25 y - 5 z = (- 5) (17)$

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

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Multiply $- 5$ by $17$ .

$5 x + 25 y - 5 z = - 85$

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

$5 x + 25 y - 5 z = - 85$

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

$5 x + 25 y - 5 z = - 85$

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

Step 2.3

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

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

Step 2.4

The resultant equation has $x$ eliminated.

$20 y = - 80$

Step 3

Choose another two equations and eliminate $x$ .

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

$2 x + 5 y - 3 z = - 10$

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.

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

$(5) \cdot (2 x + 5 y - 3 z) = (5) (- 10)$

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

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Step 4.2.1.1.1

Apply the distributive property.

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

$(5) \cdot (2 x + 5 y - 3 z) = (5) (- 10)$

Step 4.2.1.1.2

Simplify.

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Step 4.2.1.1.2.1

Multiply $- 5$ by $2$ .

$- 10 x + 2 (- 5 y) + 2 (5 z) = (2) (5)$

$(5) \cdot (2 x + 5 y - 3 z) = (5) (- 10)$

Step 4.2.1.1.2.2

Multiply $- 5$ by $2$ .

$- 10 x - 10 y + 2 (5 z) = (2) (5)$

$(5) \cdot (2 x + 5 y - 3 z) = (5) (- 10)$

Step 4.2.1.1.2.3

Multiply $5$ by $2$ .

$- 10 x - 10 y + 10 z = (2) (5)$

$(5) \cdot (2 x + 5 y - 3 z) = (5) (- 10)$

$- 10 x - 10 y + 10 z = (2) (5)$

$(5) \cdot (2 x + 5 y - 3 z) = (5) (- 10)$

$- 10 x - 10 y + 10 z = (2) (5)$

$(5) \cdot (2 x + 5 y - 3 z) = (5) (- 10)$

$- 10 x - 10 y + 10 z = (2) (5)$

$(5) \cdot (2 x + 5 y - 3 z) = (5) (- 10)$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Multiply $2$ by $5$ .

$- 10 x - 10 y + 10 z = 10$

$(5) \cdot (2 x + 5 y - 3 z) = (5) (- 10)$

$- 10 x - 10 y + 10 z = 10$

$(5) \cdot (2 x + 5 y - 3 z) = (5) (- 10)$

Step 4.2.3

Simplify the left side.

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Step 4.2.3.1

Simplify $(5) \cdot (2 x + 5 y - 3 z)$ .

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Step 4.2.3.1.1

Apply the distributive property.

$- 10 x - 10 y + 10 z = 10$

$5 (2 x) + 5 (5 y) + 5 (- 3 z) = (5) (- 10)$

Step 4.2.3.1.2

Simplify.

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Step 4.2.3.1.2.1

Multiply $2$ by $5$ .

$- 10 x - 10 y + 10 z = 10$

$10 x + 5 (5 y) + 5 (- 3 z) = (5) (- 10)$

Step 4.2.3.1.2.2

Multiply $5$ by $5$ .

$- 10 x - 10 y + 10 z = 10$

$10 x + 25 y + 5 (- 3 z) = (5) (- 10)$

Step 4.2.3.1.2.3

Multiply $- 3$ by $5$ .

$- 10 x - 10 y + 10 z = 10$

$10 x + 25 y - 15 z = (5) (- 10)$

$- 10 x - 10 y + 10 z = 10$

$10 x + 25 y - 15 z = (5) (- 10)$

$- 10 x - 10 y + 10 z = 10$

$10 x + 25 y - 15 z = (5) (- 10)$

$- 10 x - 10 y + 10 z = 10$

$10 x + 25 y - 15 z = (5) (- 10)$

Step 4.2.4

Simplify the right side.

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Step 4.2.4.1

Multiply $5$ by $- 10$ .

$- 10 x - 10 y + 10 z = 10$

$10 x + 25 y - 15 z = - 50$

$- 10 x - 10 y + 10 z = 10$

$10 x + 25 y - 15 z = - 50$

$- 10 x - 10 y + 10 z = 10$

$10 x + 25 y - 15 z = - 50$

Step 4.3

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

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

Step 4.4

The resultant equation has $x$ eliminated.

$15 y - 5 z = - 40$

Step 5

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

$20 y = - 80$

$15 y - 5 z = - 40$

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.

$(- 3) \cdot (20 y) = (- 3) (- 80)$

$(4) \cdot (15 y - 5 z) = (4) (- 40)$

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

Multiply $20$ by $- 3$ .

$- 60 y = (- 3) (- 80)$

$(4) \cdot (15 y - 5 z) = (4) (- 40)$

$- 60 y = (- 3) (- 80)$

$(4) \cdot (15 y - 5 z) = (4) (- 40)$

Step 6.2.2

Simplify the right side.

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Step 6.2.2.1

Multiply $- 3$ by $- 80$ .

$- 60 y = 240$

$(4) \cdot (15 y - 5 z) = (4) (- 40)$

$- 60 y = 240$

$(4) \cdot (15 y - 5 z) = (4) (- 40)$

Step 6.2.3

Simplify the left side.

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Step 6.2.3.1

Simplify $(4) \cdot (15 y - 5 z)$ .

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Step 6.2.3.1.1

Apply the distributive property.

$- 60 y = 240$

$4 (15 y) + 4 (- 5 z) = (4) (- 40)$

Step 6.2.3.1.2

Multiply.

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Step 6.2.3.1.2.1

Multiply $15$ by $4$ .

$- 60 y = 240$

$60 y + 4 (- 5 z) = (4) (- 40)$

Step 6.2.3.1.2.2

Multiply $- 5$ by $4$ .

$- 60 y = 240$

$60 y - 20 z = (4) (- 40)$

$- 60 y = 240$

$60 y - 20 z = (4) (- 40)$

$- 60 y = 240$

$60 y - 20 z = (4) (- 40)$

$- 60 y = 240$

$60 y - 20 z = (4) (- 40)$

Step 6.2.4

Simplify the right side.

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Step 6.2.4.1

Multiply $4$ by $- 40$ .

$- 60 y = 240$

$60 y - 20 z = - 160$

$- 60 y = 240$

$60 y - 20 z = - 160$

$- 60 y = 240$

$60 y - 20 z = - 160$

Step 6.3

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

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

Step 6.4

The resultant equation has $y$ eliminated.

$- 20 z = 80$

Step 6.5

Divide each term in $- 20 z = 80$ by $- 20$ and simplify.

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Step 6.5.1

Divide each term in $- 20 z = 80$ by $- 20$ .

$\frac{- 20 z}{- 20} = \frac{80}{- 20}$

Step 6.5.2

Simplify the left side.

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Step 6.5.2.1

Cancel the common factor of $- 20$ .

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Step 6.5.2.1.1

Cancel the common factor.

$\frac{- 20 z}{- 20} = \frac{80}{- 20}$

Step 6.5.2.1.2

Divide $z$ by $1$ .

$z = \frac{80}{- 20}$

Step 6.5.3

Simplify the right side.

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Step 6.5.3.1

Divide $80$ by $- 20$ .

$z = - 4$

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.

$20 y = - 80$

Step 7.2

Divide each term in $20 y = - 80$ by $20$ and simplify.

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Step 7.2.1

Divide each term in $20 y = - 80$ by $20$ .

$\frac{20 y}{20} = \frac{- 80}{20}$

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

Cancel the common factor of $20$ .

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Step 7.2.2.1.1

Cancel the common factor.

$\frac{20 y}{20} = \frac{- 80}{20}$

Step 7.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 80}{20}$

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

Divide $- 80$ by $20$ .

$y = - 4$

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.

$- x - 5 \cdot - 4 - 4 = 17$

Step 8.2

Solve for $x$ .

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Step 8.2.1

Simplify $- x - 5 \cdot - 4 - 4$ .

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Step 8.2.1.1

Multiply $- 5$ by $- 4$ .

$- x + 20 - 4 = 17$

Step 8.2.1.2

Subtract $4$ from $20$ .

$- x + 16 = 17$

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

Subtract $16$ from both sides of the equation.

$- x = 17 - 16$

Step 8.2.2.2

Subtract $16$ from $17$ .

$- x = 1$

Step 8.2.3

Divide each term in $- x = 1$ by $- 1$ and simplify.

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Step 8.2.3.1

Divide each term in $- x = 1$ by $- 1$ .

$\frac{- x}{- 1} = \frac{1}{- 1}$

Step 8.2.3.2

Simplify the left side.

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Step 8.2.3.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{1}{- 1}$

Step 8.2.3.2.2

Divide $x$ by $1$ .

$x = \frac{1}{- 1}$

Step 8.2.3.3

Simplify the right side.

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Step 8.2.3.3.1

Divide $1$ by $- 1$ .

$x = - 1$

Step 9

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

$(- 1, - 4, - 4)$

Step 10

The result can be shown in multiple forms.

Point Form:

$(- 1, - 4, - 4)$

Equation Form:

$x = - 1, y = - 4, z = - 4$

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

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

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

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

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

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

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

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

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