Precalculus Examples

x - 4 y + 5 z = 0

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

x + z - 3 y = 0

$x + z - 3 y = 0$

Step 1

Solve the equation for

y

$y$ .

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Step 1.1

Move all terms not containing

y

$y$ to the right side of the equation.

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Step 1.1.1

Subtract

x

$x$ from both sides of the equation.

- 4 y + 5 z = - x

$- 4 y + 5 z = - x$

x + z - 3 y = 0

$x + z - 3 y = 0$

Step 1.1.2

Subtract

5 z

$5 z$ from both sides of the equation.

- 4 y = - x - 5 z

$- 4 y = - x - 5 z$

x + z - 3 y = 0

$x + z - 3 y = 0$

- 4 y = - x - 5 z

$- 4 y = - x - 5 z$

x + z - 3 y = 0

$x + z - 3 y = 0$

Step 1.2

Divide each term in

- 4 y = - x - 5 z

$- 4 y = - x - 5 z$ by

- 4

$- 4$ and simplify.

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Step 1.2.1

Divide each term in

- 4 y = - x - 5 z

$- 4 y = - x - 5 z$ by

- 4

$- 4$ .

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

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

x + z - 3 y = 0

$x + z - 3 y = 0$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of

- 4

$- 4$ .

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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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