Precalculus Examples

$3 x - 3 y + 6 z = 27$ , $3 x + y - 3 z = - 11$ , $47 x - y - 6 z = 0$

Step 1

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

$3 x - 3 y + 6 z = 27$

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

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.

$(- 1) \cdot (3 x - 3 y + 6 z) = (- 1) (27)$

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

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 $(- 1) \cdot (3 x - 3 y + 6 z)$ .

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Step 2.2.1.1.1

Apply the distributive property.

$- 1 (3 x) - 1 (- 3 y) - 1 (6 z) = (- 1) (27)$

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

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $3$ by $- 1$ .

$- 3 x - 1 (- 3 y) - 1 (6 z) = (- 1) (27)$

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

Step 2.2.1.1.2.2

Multiply $- 3$ by $- 1$ .

$- 3 x + 3 y - 1 (6 z) = (- 1) (27)$

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

Step 2.2.1.1.2.3

Multiply $6$ by $- 1$ .

$- 3 x + 3 y - 6 z = (- 1) (27)$

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

$- 3 x + 3 y - 6 z = (- 1) (27)$

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

$- 3 x + 3 y - 6 z = (- 1) (27)$

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

$- 3 x + 3 y - 6 z = (- 1) (27)$

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

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Multiply $- 1$ by $27$ .

$- 3 x + 3 y - 6 z = - 27$

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

$- 3 x + 3 y - 6 z = - 27$

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

$- 3 x + 3 y - 6 z = - 27$

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

Step 2.3

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

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

Step 2.4

The resultant equation has $x$ eliminated.

$4 y - 9 z = - 38$

Step 3

Choose another two equations and eliminate $x$ .

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

$47 x - y - 6 z = 0$

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.

$(- 47) \cdot (3 x + y - 3 z) = (- 47) (- 11)$

$(3) \cdot (47 x - y - 6 z) = (3) (0)$

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 $(- 47) \cdot (3 x + y - 3 z)$ .

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Step 4.2.1.1.1

Apply the distributive property.

$- 47 (3 x) - 47 y - 47 (- 3 z) = (- 47) (- 11)$

$(3) \cdot (47 x - y - 6 z) = (3) (0)$

Step 4.2.1.1.2

Simplify.

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Step 4.2.1.1.2.1

Multiply $3$ by $- 47$ .

$- 141 x - 47 y - 47 (- 3 z) = (- 47) (- 11)$

$(3) \cdot (47 x - y - 6 z) = (3) (0)$

Step 4.2.1.1.2.2

Multiply $- 3$ by $- 47$ .

$- 141 x - 47 y + 141 z = (- 47) (- 11)$

$(3) \cdot (47 x - y - 6 z) = (3) (0)$

$- 141 x - 47 y + 141 z = (- 47) (- 11)$

$(3) \cdot (47 x - y - 6 z) = (3) (0)$

$- 141 x - 47 y + 141 z = (- 47) (- 11)$

$(3) \cdot (47 x - y - 6 z) = (3) (0)$

$- 141 x - 47 y + 141 z = (- 47) (- 11)$

$(3) \cdot (47 x - y - 6 z) = (3) (0)$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Multiply $- 47$ by $- 11$ .

$- 141 x - 47 y + 141 z = 517$

$(3) \cdot (47 x - y - 6 z) = (3) (0)$

$- 141 x - 47 y + 141 z = 517$

$(3) \cdot (47 x - y - 6 z) = (3) (0)$

Step 4.2.3

Simplify the left side.

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Step 4.2.3.1

Simplify $(3) \cdot (47 x - y - 6 z)$ .

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Step 4.2.3.1.1

Apply the distributive property.

$- 141 x - 47 y + 141 z = 517$

$3 (47 x) + 3 (- y) + 3 (- 6 z) = (3) (0)$

Step 4.2.3.1.2

Simplify.

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Step 4.2.3.1.2.1

Multiply $47$ by $3$ .

$- 141 x - 47 y + 141 z = 517$

$141 x + 3 (- y) + 3 (- 6 z) = (3) (0)$

Step 4.2.3.1.2.2

Multiply $- 1$ by $3$ .

$- 141 x - 47 y + 141 z = 517$

$141 x - 3 y + 3 (- 6 z) = (3) (0)$

Step 4.2.3.1.2.3

Multiply $- 6$ by $3$ .

$- 141 x - 47 y + 141 z = 517$

$141 x - 3 y - 18 z = (3) (0)$

$- 141 x - 47 y + 141 z = 517$

$141 x - 3 y - 18 z = (3) (0)$

$- 141 x - 47 y + 141 z = 517$

$141 x - 3 y - 18 z = (3) (0)$

$- 141 x - 47 y + 141 z = 517$

$141 x - 3 y - 18 z = (3) (0)$

Step 4.2.4

Simplify the right side.

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Step 4.2.4.1

Multiply $3$ by $0$ .

$- 141 x - 47 y + 141 z = 517$

$141 x - 3 y - 18 z = 0$

$- 141 x - 47 y + 141 z = 517$

$141 x - 3 y - 18 z = 0$

$- 141 x - 47 y + 141 z = 517$

$141 x - 3 y - 18 z = 0$

Step 4.3

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

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

Step 4.4

The resultant equation has $x$ eliminated.

$- 50 y + 123 z = 517$

Step 5

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

$4 y - 9 z = - 38$

$- 50 y + 123 z = 517$

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.

$(25) \cdot (4 y - 9 z) = (25) (- 38)$

$(2) \cdot (- 50 y + 123 z) = (2) (517)$

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 $(25) \cdot (4 y - 9 z)$ .

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Step 6.2.1.1.1

Apply the distributive property.

$25 (4 y) + 25 (- 9 z) = (25) (- 38)$

$(2) \cdot (- 50 y + 123 z) = (2) (517)$

Step 6.2.1.1.2

Multiply.

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Step 6.2.1.1.2.1

Multiply $4$ by $25$ .

$100 y + 25 (- 9 z) = (25) (- 38)$

$(2) \cdot (- 50 y + 123 z) = (2) (517)$

Step 6.2.1.1.2.2

Multiply $- 9$ by $25$ .

$100 y - 225 z = (25) (- 38)$

$(2) \cdot (- 50 y + 123 z) = (2) (517)$

$100 y - 225 z = (25) (- 38)$

$(2) \cdot (- 50 y + 123 z) = (2) (517)$

$100 y - 225 z = (25) (- 38)$

$(2) \cdot (- 50 y + 123 z) = (2) (517)$

$100 y - 225 z = (25) (- 38)$

$(2) \cdot (- 50 y + 123 z) = (2) (517)$

Step 6.2.2

Simplify the right side.

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Step 6.2.2.1

Multiply $25$ by $- 38$ .

$100 y - 225 z = - 950$

$(2) \cdot (- 50 y + 123 z) = (2) (517)$

$100 y - 225 z = - 950$

$(2) \cdot (- 50 y + 123 z) = (2) (517)$

Step 6.2.3

Simplify the left side.

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Step 6.2.3.1

Simplify $(2) \cdot (- 50 y + 123 z)$ .

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Step 6.2.3.1.1

Apply the distributive property.

$100 y - 225 z = - 950$

$2 (- 50 y) + 2 (123 z) = (2) (517)$

Step 6.2.3.1.2

Multiply.

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Step 6.2.3.1.2.1

Multiply $- 50$ by $2$ .

$100 y - 225 z = - 950$

$- 100 y + 2 (123 z) = (2) (517)$

Step 6.2.3.1.2.2

Multiply $123$ by $2$ .

$100 y - 225 z = - 950$

$- 100 y + 246 z = (2) (517)$

$100 y - 225 z = - 950$

$- 100 y + 246 z = (2) (517)$

$100 y - 225 z = - 950$

$- 100 y + 246 z = (2) (517)$

$100 y - 225 z = - 950$

$- 100 y + 246 z = (2) (517)$

Step 6.2.4

Simplify the right side.

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Step 6.2.4.1

Multiply $2$ by $517$ .

$100 y - 225 z = - 950$

$- 100 y + 246 z = 1034$

$100 y - 225 z = - 950$

$- 100 y + 246 z = 1034$

$100 y - 225 z = - 950$

$- 100 y + 246 z = 1034$

Step 6.3

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

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

Step 6.4

The resultant equation has $y$ eliminated.

$21 z = 84$

Step 6.5

Divide each term in $21 z = 84$ by $21$ and simplify.

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Step 6.5.1

Divide each term in $21 z = 84$ by $21$ .

$\frac{21 z}{21} = \frac{84}{21}$

Step 6.5.2

Simplify the left side.

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Step 6.5.2.1

Cancel the common factor of $21$ .

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Step 6.5.2.1.1

Cancel the common factor.

$\frac{21 z}{21} = \frac{84}{21}$

Step 6.5.2.1.2

Divide $z$ by $1$ .

$z = \frac{84}{21}$

Step 6.5.3

Simplify the right side.

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Step 6.5.3.1

Divide $84$ by $21$ .

$z = 4$

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.

$4 y - 9 \cdot 4 = - 38$

Step 7.2

Solve for $y$ .

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Step 7.2.1

Multiply $- 9$ by $4$ .

$4 y - 36 = - 38$

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 $36$ to both sides of the equation.

$4 y = - 38 + 36$

Step 7.2.2.2

Add $- 38$ and $36$ .

$4 y = - 2$

Step 7.2.3

Divide each term in $4 y = - 2$ by $4$ and simplify.

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Step 7.2.3.1

Divide each term in $4 y = - 2$ by $4$ .

$\frac{4 y}{4} = \frac{- 2}{4}$

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.

$\frac{4 y}{4} = \frac{- 2}{4}$

Step 7.2.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 2}{4}$

Step 7.2.3.3

Simplify the right side.

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Step 7.2.3.3.1

Cancel the common factor of $- 2$ and $4$ .

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Step 7.2.3.3.1.1

Factor $2$ out of $- 2$ .

$y = \frac{2 (- 1)}{4}$

Step 7.2.3.3.1.2

Cancel the common factors.

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Step 7.2.3.3.1.2.1

Factor $2$ out of $4$ .

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

Step 7.2.3.3.1.2.2

Cancel the common factor.

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

Step 7.2.3.3.1.2.3

Rewrite the expression.

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

Step 7.2.3.3.2

Move the negative in front of the fraction.

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

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.

$3 x - 3 (- \frac{1}{2}) + 6 (4) = 27$

Step 8.2

Solve for $x$ .

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Step 8.2.1

Simplify $3 x - 3 (- \frac{1}{2}) + 6 (4)$ .

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Step 8.2.1.1

Simplify each term.

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Step 8.2.1.1.1

Multiply $- 3 (- \frac{1}{2})$ .

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Step 8.2.1.1.1.1

Multiply $- 1$ by $- 3$ .

$3 x + 3 (\frac{1}{2}) + 6 (4) = 27$

Step 8.2.1.1.1.2

Combine $3$ and $\frac{1}{2}$ .

$3 x + \frac{3}{2} + 6 (4) = 27$

Step 8.2.1.1.2

Multiply $6$ by $4$ .

$3 x + \frac{3}{2} + 24 = 27$

Step 8.2.1.2

To write $24$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$3 x + \frac{3}{2} + 24 \cdot \frac{2}{2} = 27$

Step 8.2.1.3

Combine $24$ and $\frac{2}{2}$ .

$3 x + \frac{3}{2} + \frac{24 \cdot 2}{2} = 27$

Step 8.2.1.4

Combine the numerators over the common denominator.

$3 x + \frac{3 + 24 \cdot 2}{2} = 27$

Step 8.2.1.5

Simplify the numerator.

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Step 8.2.1.5.1

Multiply $24$ by $2$ .

$3 x + \frac{3 + 48}{2} = 27$

Step 8.2.1.5.2

Add $3$ and $48$ .

$3 x + \frac{51}{2} = 27$

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 $\frac{51}{2}$ from both sides of the equation.

$3 x = 27 - \frac{51}{2}$

Step 8.2.2.2

To write $27$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$3 x = 27 \cdot \frac{2}{2} - \frac{51}{2}$

Step 8.2.2.3

Combine $27$ and $\frac{2}{2}$ .

$3 x = \frac{27 \cdot 2}{2} - \frac{51}{2}$

Step 8.2.2.4

Combine the numerators over the common denominator.

$3 x = \frac{27 \cdot 2 - 51}{2}$

Step 8.2.2.5

Simplify the numerator.

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Step 8.2.2.5.1

Multiply $27$ by $2$ .

$3 x = \frac{54 - 51}{2}$

Step 8.2.2.5.2

Subtract $51$ from $54$ .

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

Step 8.2.3

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

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Step 8.2.3.1

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

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

Step 8.2.3.2

Simplify the left side.

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Step 8.2.3.2.1

Cancel the common factor of $3$ .

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Step 8.2.3.2.1.1

Cancel the common factor.

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

Step 8.2.3.2.1.2

Divide $x$ by $1$ .

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

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.

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

Step 8.2.3.3.2

Cancel the common factor of $3$ .

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Step 8.2.3.3.2.1

Cancel the common factor.

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

Step 8.2.3.3.2.2

Rewrite the expression.

$x = \frac{1}{2}$

Step 9

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

$(\frac{1}{2}, - \frac{1}{2}, 4)$

Step 10

The result can be shown in multiple forms.

Point Form:

$(\frac{1}{2}, - \frac{1}{2}, 4)$

Equation Form:

$x = \frac{1}{2}, y = - \frac{1}{2}, z = 4$

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

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

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

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

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

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

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

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

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