Precalculus Examples

$2 x - 3 y + z = - 2$ , $- 4 x + 9 y = 7$

Step 1

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

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

$- 4 x + 9 y = 7$

Step 2

Simplify.

Tap for more steps...

Step 2.1

Simplify the left side.

Tap for more steps...

Step 2.1.1

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

Tap for more steps...

Step 2.1.1.1

Apply the distributive property.

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

$- 4 x + 9 y = 7$

Step 2.1.1.2

Simplify.

Tap for more steps...

Step 2.1.1.2.1

Multiply $2$ by $2$ .

$4 x + 2 (- 3 y) + 2 z = (2) (- 2)$

$- 4 x + 9 y = 7$

Step 2.1.1.2.2

Multiply $- 3$ by $2$ .

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

$- 4 x + 9 y = 7$

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

$- 4 x + 9 y = 7$

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

$- 4 x + 9 y = 7$

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

$- 4 x + 9 y = 7$

Step 2.2

Simplify the right side.

Tap for more steps...

Step 2.2.1

Multiply $2$ by $- 2$ .

$4 x - 6 y + 2 z = - 4$

$- 4 x + 9 y = 7$

$4 x - 6 y + 2 z = - 4$

$- 4 x + 9 y = 7$

$4 x - 6 y + 2 z = - 4$

$- 4 x + 9 y = 7$

Step 3

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

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

Step 4

Simplify the equation and solve for $y$ .

Tap for more steps...

Step 4.1

Subtract $2 z$ from both sides of the equation.

$3 y = 3 - 2 z$

Step 4.2

Divide each term in $3 y = 3 - 2 z$ by $3$ and simplify.

Tap for more steps...

Step 4.2.1

Divide each term in $3 y = 3 - 2 z$ by $3$ .

$\frac{3 y}{3} = \frac{3}{3} + \frac{- 2 z}{3}$

Step 4.2.2

Simplify the left side.

Tap for more steps...

Step 4.2.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.2.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{3}{3} + \frac{- 2 z}{3}$

Step 4.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{3}{3} + \frac{- 2 z}{3}$

Step 4.2.3

Simplify the right side.

Tap for more steps...

Step 4.2.3.1

Simplify each term.

Tap for more steps...

Step 4.2.3.1.1

Divide $3$ by $3$ .

$y = 1 + \frac{- 2 z}{3}$

Step 4.2.3.1.2

Move the negative in front of the fraction.

$y = 1 - \frac{2 z}{3}$

Step 5

Substitute the value found for $y$ into one of the original equations, then solve for $x$ .

Tap for more steps...

Step 5.1

Substitute the value found for $y$ into one of the original equations to solve for $x$ .

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

Step 5.2

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

Tap for more steps...

Step 5.2.1

Simplify each term.

Tap for more steps...

Step 5.2.1.1

Apply the distributive property.

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

Step 5.2.1.2

Multiply $- 6$ by $1$ .

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

Step 5.2.1.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 5.2.1.3.1

Move the leading negative in $- \frac{2 z}{3}$ into the numerator.

$4 x - 6 - 6 \frac{- 2 z}{3} + 2 z = - 4$

Step 5.2.1.3.2

Factor $3$ out of $- 6$ .

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

Step 5.2.1.3.3

Cancel the common factor.

$4 x - 6 + 3 \cdot - 2 \frac{- 2 z}{3} + 2 z = - 4$

Step 5.2.1.3.4

Rewrite the expression.

$4 x - 6 - 2 (- 2 z) + 2 z = - 4$

Step 5.2.1.4

Multiply $- 2$ by $- 2$ .

$4 x - 6 + 4 z + 2 z = - 4$

Step 5.2.2

Add $4 z$ and $2 z$ .

$4 x - 6 + 6 z = - 4$

Step 5.3

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

Tap for more steps...

Step 5.3.1

Add $6$ to both sides of the equation.

$4 x + 6 z = - 4 + 6$

Step 5.3.2

Subtract $6 z$ from both sides of the equation.

$4 x = - 4 + 6 - 6 z$

Step 5.3.3

Add $- 4$ and $6$ .

$4 x = 2 - 6 z$

Step 5.4

Divide each term in $4 x = 2 - 6 z$ by $4$ and simplify.

Tap for more steps...

Step 5.4.1

Divide each term in $4 x = 2 - 6 z$ by $4$ .

$\frac{4 x}{4} = \frac{2}{4} + \frac{- 6 z}{4}$

Step 5.4.2

Simplify the left side.

Tap for more steps...

Step 5.4.2.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 5.4.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{2}{4} + \frac{- 6 z}{4}$

Step 5.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{2}{4} + \frac{- 6 z}{4}$

Step 5.4.3

Simplify the right side.

Tap for more steps...

Step 5.4.3.1

Simplify each term.

Tap for more steps...

Step 5.4.3.1.1

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

Tap for more steps...

Step 5.4.3.1.1.1

Factor $2$ out of $2$ .

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

Step 5.4.3.1.1.2

Cancel the common factors.

Tap for more steps...

Step 5.4.3.1.1.2.1

Factor $2$ out of $4$ .

$x = \frac{2 \cdot 1}{2 \cdot 2} + \frac{- 6 z}{4}$

Step 5.4.3.1.1.2.2

Cancel the common factor.

$x = \frac{2 \cdot 1}{2 \cdot 2} + \frac{- 6 z}{4}$

Step 5.4.3.1.1.2.3

Rewrite the expression.

$x = \frac{1}{2} + \frac{- 6 z}{4}$

Step 5.4.3.1.2

Cancel the common factor of $- 6$ and $4$ .

Tap for more steps...

Step 5.4.3.1.2.1

Factor $2$ out of $- 6 z$ .

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

Step 5.4.3.1.2.2

Cancel the common factors.

Tap for more steps...

Step 5.4.3.1.2.2.1

Factor $2$ out of $4$ .

$x = \frac{1}{2} + \frac{2 (- 3 z)}{2 (2)}$

Step 5.4.3.1.2.2.2

Cancel the common factor.

$x = \frac{1}{2} + \frac{2 (- 3 z)}{2 \cdot 2}$

Step 5.4.3.1.2.2.3

Rewrite the expression.

$x = \frac{1}{2} + \frac{- 3 z}{2}$

Step 5.4.3.1.3

Move the negative in front of the fraction.

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

Step 6

This is the final solution to the independent system of equations.

$y = 1 - \frac{2 z}{3}$

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

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

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

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