Precalculus Examples

$3 x - 4 y + z = 14$ , $x + y = 7$

Step 1

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

$3 x - 4 y + z = 14$

$(- 3) \cdot (x + y) = (- 3) (7)$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Apply the distributive property.

$3 x - 4 y + z = 14$

$- 3 x - 3 y = (- 3) (7)$

$3 x - 4 y + z = 14$

$- 3 x - 3 y = (- 3) (7)$

Step 2.2

Simplify the right side.

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Step 2.2.1

Multiply $- 3$ by $7$ .

$3 x - 4 y + z = 14$

$- 3 x - 3 y = - 21$

$3 x - 4 y + z = 14$

$- 3 x - 3 y = - 21$

$3 x - 4 y + z = 14$

$- 3 x - 3 y = - 21$

Step 3

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

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

Step 4

Simplify the equation and solve for $y$ .

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Step 4.1

Subtract $z$ from both sides of the equation.

$- 7 y = - 7 - z$

Step 4.2

Divide each term in $- 7 y = - 7 - z$ by $- 7$ and simplify.

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Step 4.2.1

Divide each term in $- 7 y = - 7 - z$ by $- 7$ .

$\frac{- 7 y}{- 7} = \frac{- 7}{- 7} + \frac{- z}{- 7}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $- 7$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{- 7 y}{- 7} = \frac{- 7}{- 7} + \frac{- z}{- 7}$

Step 4.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 7}{- 7} + \frac{- z}{- 7}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Simplify each term.

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Step 4.2.3.1.1

Divide $- 7$ by $- 7$ .

$y = 1 + \frac{- z}{- 7}$

Step 4.2.3.1.2

Dividing two negative values results in a positive value.

$y = 1 + \frac{z}{7}$

Step 5

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

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Step 5.1

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

$3 x - 4 (1 + \frac{z}{7}) + z = 14$

Step 5.2

Simplify $3 x - 4 (1 + \frac{z}{7}) + z$ .

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Apply the distributive property.

$3 x - 4 \cdot 1 - 4 \frac{z}{7} + z = 14$

Step 5.2.1.2

Multiply $- 4$ by $1$ .

$3 x - 4 - 4 \frac{z}{7} + z = 14$

Step 5.2.1.3

Combine $- 4$ and $\frac{z}{7}$ .

$3 x - 4 + \frac{- 4 z}{7} + z = 14$

Step 5.2.1.4

Move the negative in front of the fraction.

$3 x - 4 - \frac{4 z}{7} + z = 14$

Step 5.2.2

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

$3 x - 4 - \frac{4 z}{7} + z \cdot \frac{7}{7} = 14$

Step 5.2.3

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

$3 x - 4 - \frac{4 z}{7} + \frac{z \cdot 7}{7} = 14$

Step 5.2.4

Combine the numerators over the common denominator.

$3 x - 4 + \frac{- 4 z + z \cdot 7}{7} = 14$

Step 5.2.5

Add $- 4 z$ and $z \cdot 7$ .

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Step 5.2.5.1

Reorder $z$ and $7$ .

$3 x - 4 + \frac{- 4 z + 7 \cdot z}{7} = 14$

Step 5.2.5.2

Add $- 4 z$ and $7 \cdot z$ .

$3 x - 4 + \frac{3 z}{7} = 14$

Step 5.3

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

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Step 5.3.1

Add $4$ to both sides of the equation.

$3 x + \frac{3 z}{7} = 14 + 4$

Step 5.3.2

Subtract $\frac{3 z}{7}$ from both sides of the equation.

$3 x = 14 + 4 - \frac{3 z}{7}$

Step 5.3.3

Add $14$ and $4$ .

$3 x = 18 - \frac{3 z}{7}$

Step 5.4

Divide each term in $3 x = 18 - \frac{3 z}{7}$ by $3$ and simplify.

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Step 5.4.1

Divide each term in $3 x = 18 - \frac{3 z}{7}$ by $3$ .

$\frac{3 x}{3} = \frac{18}{3} + \frac{- \frac{3 z}{7}}{3}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $3$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{18}{3} + \frac{- \frac{3 z}{7}}{3}$

Step 5.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{18}{3} + \frac{- \frac{3 z}{7}}{3}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Simplify each term.

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Step 5.4.3.1.1

Divide $18$ by $3$ .

$x = 6 + \frac{- \frac{3 z}{7}}{3}$

Step 5.4.3.1.2

Multiply the numerator by the reciprocal of the denominator.

$x = 6 - \frac{3 z}{7} \cdot \frac{1}{3}$

Step 5.4.3.1.3

Cancel the common factor of $3$ .

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Step 5.4.3.1.3.1

Move the leading negative in $- \frac{3 z}{7}$ into the numerator.

$x = 6 + \frac{- 3 z}{7} \cdot \frac{1}{3}$

Step 5.4.3.1.3.2

Factor $3$ out of $- 3 z$ .

$x = 6 + \frac{3 (- z)}{7} \cdot \frac{1}{3}$

Step 5.4.3.1.3.3

Cancel the common factor.

$x = 6 + \frac{3 (- z)}{7} \cdot \frac{1}{3}$

Step 5.4.3.1.3.4

Rewrite the expression.

$x = 6 + \frac{- z}{7}$

Step 5.4.3.1.4

Move the negative in front of the fraction.

$x = 6 - \frac{z}{7}$

Step 6

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

$y = 1 + \frac{z}{7}$

$x = 6 - \frac{z}{7}$

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

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

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