Precalculus Examples Systems of Equations

Solve in Terms of the Arbitrary Variable x

$x - 4 y + 5 z = 0$ , $x + z - 3 y = 0$

Step 1

Solve the equation for $y$ .

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Step 1.1

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

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Step 1.1.1

Subtract $x$ from both sides of the equation.

$- 4 y + 5 z = - x$

$x + z - 3 y = 0$

Step 1.1.2

Subtract $5 z$ from both sides of the equation.

$- 4 y = - x - 5 z$

$x + z - 3 y = 0$

$- 4 y = - x - 5 z$

$x + z - 3 y = 0$

Step 1.2

Divide each term in $- 4 y = - x - 5 z$ by $- 4$ and simplify.

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Step 1.2.1

Divide each term in $- 4 y = - x - 5 z$ by $- 4$ .

$\frac{- 4 y}{- 4} = \frac{- x}{- 4} + \frac{- 5 z}{- 4}$

$x + z - 3 y = 0$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $- 4$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{- 4 y}{- 4} = \frac{- x}{- 4} + \frac{- 5 z}{- 4}$

$x + z - 3 y = 0$

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- x}{- 4} + \frac{- 5 z}{- 4}$

$x + z - 3 y = 0$

$y = \frac{- x}{- 4} + \frac{- 5 z}{- 4}$

$x + z - 3 y = 0$

$y = \frac{- x}{- 4} + \frac{- 5 z}{- 4}$

$x + z - 3 y = 0$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Dividing two negative values results in a positive value.

$y = \frac{x}{4} + \frac{- 5 z}{- 4}$

$x + z - 3 y = 0$

Step 1.2.3.1.2

Dividing two negative values results in a positive value.

$y = \frac{x}{4} + \frac{5 z}{4}$

$x + z - 3 y = 0$

$y = \frac{x}{4} + \frac{5 z}{4}$

$x + z - 3 y = 0$

$y = \frac{x}{4} + \frac{5 z}{4}$

$x + z - 3 y = 0$

$y = \frac{x}{4} + \frac{5 z}{4}$

$x + z - 3 y = 0$

$y = \frac{x}{4} + \frac{5 z}{4}$

$x + z - 3 y = 0$

Step 2

Solve the equation for $z$ .

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Step 2.1

Simplify $x + z - 3 (\frac{x}{4} + \frac{5 z}{4})$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Apply the distributive property.

$x + z - 3 \frac{x}{4} - 3 \frac{5 z}{4} = 0$

$y = \frac{x}{4} + \frac{5 z}{4}$

Step 2.1.1.2

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

$x + z + \frac{- 3 x}{4} - 3 \frac{5 z}{4} = 0$

$y = \frac{x}{4} + \frac{5 z}{4}$

Step 2.1.1.3

Multiply $- 3 \frac{5 z}{4}$ .

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Step 2.1.1.3.1

Combine $- 3$ and $\frac{5 z}{4}$ .

$x + z + \frac{- 3 x}{4} + \frac{- 3 (5 z)}{4} = 0$

$y = \frac{x}{4} + \frac{5 z}{4}$

Step 2.1.1.3.2

Multiply $5$ by $- 3$ .

$x + z + \frac{- 3 x}{4} + \frac{- 15 z}{4} = 0$

$y = \frac{x}{4} + \frac{5 z}{4}$

$x + z + \frac{- 3 x}{4} + \frac{- 15 z}{4} = 0$

$y = \frac{x}{4} + \frac{5 z}{4}$

Step 2.1.1.4

Simplify each term.

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Step 2.1.1.4.1

Move the negative in front of the fraction.

$x + z - \frac{(3) x}{4} + \frac{- 15 z}{4} = 0$

$y = \frac{x}{4} + \frac{5 z}{4}$

Step 2.1.1.4.2

Move the negative in front of the fraction.

$x + z - \frac{3 x}{4} - \frac{15 z}{4} = 0$

$y = \frac{x}{4} + \frac{5 z}{4}$

$x + z - \frac{3 x}{4} - \frac{15 z}{4} = 0$

$y = \frac{x}{4} + \frac{5 z}{4}$

$x + z - \frac{3 x}{4} - \frac{15 z}{4} = 0$

$y = \frac{x}{4} + \frac{5 z}{4}$

Step 2.1.2

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

$z + x \cdot \frac{4}{4} - \frac{3 x}{4} - \frac{15 z}{4} = 0$

$y = \frac{x}{4} + \frac{5 z}{4}$

Step 2.1.3

Simplify terms.

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Step 2.1.3.1

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

$z + \frac{x \cdot 4}{4} - \frac{3 x}{4} - \frac{15 z}{4} = 0$

$y = \frac{x}{4} + \frac{5 z}{4}$

Step 2.1.3.2

Combine the numerators over the common denominator.

$z + \frac{x \cdot 4 - 3 x}{4} - \frac{15 z}{4} = 0$

$y = \frac{x}{4} + \frac{5 z}{4}$

Step 2.1.3.3

Combine the numerators over the common denominator.

$z + \frac{x \cdot 4 - 3 x - 15 z}{4} = 0$

$y = \frac{x}{4} + \frac{5 z}{4}$

$z + \frac{x \cdot 4 - 3 x - 15 z}{4} = 0$

$y = \frac{x}{4} + \frac{5 z}{4}$

Step 2.1.4

Move $4$ to the left of $x$ .

$z + \frac{4 x - 3 x - 15 z}{4} = 0$

$y = \frac{x}{4} + \frac{5 z}{4}$

Step 2.1.5

Subtract $3 x$ from $4 x$ .

$z + \frac{x - 15 z}{4} = 0$

$y = \frac{x}{4} + \frac{5 z}{4}$

Step 2.1.6

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

$z \cdot \frac{4}{4} + \frac{x - 15 z}{4} = 0$

$y = \frac{x}{4} + \frac{5 z}{4}$

Step 2.1.7

Simplify terms.

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Step 2.1.7.1

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

$\frac{z \cdot 4}{4} + \frac{x - 15 z}{4} = 0$

$y = \frac{x}{4} + \frac{5 z}{4}$

Step 2.1.7.2

Combine the numerators over the common denominator.

$\frac{z \cdot 4 + x - 15 z}{4} = 0$

$y = \frac{x}{4} + \frac{5 z}{4}$

$\frac{z \cdot 4 + x - 15 z}{4} = 0$

$y = \frac{x}{4} + \frac{5 z}{4}$

Step 2.1.8

Simplify the numerator.

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Step 2.1.8.1

Move $4$ to the left of $z$ .

$\frac{4 \cdot z + x - 15 z}{4} = 0$

$y = \frac{x}{4} + \frac{5 z}{4}$

Step 2.1.8.2

Subtract $15 z$ from $4 z$ .

$\frac{x - 11 z}{4} = 0$

$y = \frac{x}{4} + \frac{5 z}{4}$

$\frac{x - 11 z}{4} = 0$

$y = \frac{x}{4} + \frac{5 z}{4}$

$\frac{x - 11 z}{4} = 0$

$y = \frac{x}{4} + \frac{5 z}{4}$

Step 2.2

Set the numerator equal to zero.

$x - 11 z = 0$

$y = \frac{x}{4} + \frac{5 z}{4}$

Step 2.3

Solve the equation for $z$ .

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Step 2.3.1

Subtract $x$ from both sides of the equation.

$- 11 z = - x$

$y = \frac{x}{4} + \frac{5 z}{4}$

Step 2.3.2

Divide each term in $- 11 z = - x$ by $- 11$ and simplify.

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Step 2.3.2.1

Divide each term in $- 11 z = - x$ by $- 11$ .

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

$y = \frac{x}{4} + \frac{5 z}{4}$

Step 2.3.2.2

Simplify the left side.

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Step 2.3.2.2.1

Cancel the common factor of $- 11$ .

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Step 2.3.2.2.1.1

Cancel the common factor.

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

$y = \frac{x}{4} + \frac{5 z}{4}$

Step 2.3.2.2.1.2

Divide $z$ by $1$ .

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

$y = \frac{x}{4} + \frac{5 z}{4}$

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

$y = \frac{x}{4} + \frac{5 z}{4}$

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

$y = \frac{x}{4} + \frac{5 z}{4}$

Step 2.3.2.3

Simplify the right side.

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Step 2.3.2.3.1

Dividing two negative values results in a positive value.

$z = \frac{x}{11}$

$y = \frac{x}{4} + \frac{5 z}{4}$

$z = \frac{x}{11}$

$y = \frac{x}{4} + \frac{5 z}{4}$

$z = \frac{x}{11}$

$y = \frac{x}{4} + \frac{5 z}{4}$

$z = \frac{x}{11}$

$y = \frac{x}{4} + \frac{5 z}{4}$

$z = \frac{x}{11}$

$y = \frac{x}{4} + \frac{5 z}{4}$

Step 3

Simplify the right side.

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Step 3.1

Simplify $\frac{x}{4} + \frac{5 (\frac{x}{11})}{4}$ .

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Step 3.1.1

Combine the numerators over the common denominator.

$y = \frac{x + 5 (\frac{x}{11})}{4}$

$z = \frac{x}{11}$

Step 3.1.2

Combine $5$ and $\frac{x}{11}$ .

$y = \frac{x + \frac{5 x}{11}}{4}$

$z = \frac{x}{11}$

Step 3.1.3

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

$y = \frac{x \cdot \frac{11}{11} + \frac{5 x}{11}}{4}$

$z = \frac{x}{11}$

Step 3.1.4

Simplify terms.

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Step 3.1.4.1

Combine $x$ and $\frac{11}{11}$ .

$y = \frac{\frac{x \cdot 11}{11} + \frac{5 x}{11}}{4}$

$z = \frac{x}{11}$

Step 3.1.4.2

Combine the numerators over the common denominator.

$y = \frac{\frac{x \cdot 11 + 5 x}{11}}{4}$

$z = \frac{x}{11}$

$y = \frac{\frac{x \cdot 11 + 5 x}{11}}{4}$

$z = \frac{x}{11}$

Step 3.1.5

Simplify the numerator.

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Step 3.1.5.1

Move $11$ to the left of $x$ .

$y = \frac{\frac{11 \cdot x + 5 x}{11}}{4}$

$z = \frac{x}{11}$

Step 3.1.5.2

Add $11 x$ and $5 x$ .

$y = \frac{\frac{16 x}{11}}{4}$

$z = \frac{x}{11}$

$y = \frac{\frac{16 x}{11}}{4}$

$z = \frac{x}{11}$

Step 3.1.6

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{16 x}{11} \cdot \frac{1}{4}$

$z = \frac{x}{11}$

Step 3.1.7

Cancel the common factor of $4$ .

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Step 3.1.7.1

Factor $4$ out of $16 x$ .

$y = \frac{4 (4 x)}{11} \cdot \frac{1}{4}$

$z = \frac{x}{11}$

Step 3.1.7.2

Cancel the common factor.

$y = \frac{4 (4 x)}{11} \cdot \frac{1}{4}$

$z = \frac{x}{11}$

Step 3.1.7.3

Rewrite the expression.

$y = \frac{4 x}{11}$

$z = \frac{x}{11}$

$y = \frac{4 x}{11}$

$z = \frac{x}{11}$

$y = \frac{4 x}{11}$

$z = \frac{x}{11}$

$y = \frac{4 x}{11}$

$z = \frac{x}{11}$

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