Precalculus Examples

$x - 3 y + 4 z = - 3$ , $3 x + y - z = 4$ , $- 2 x + 2 y + 3 z = - 5$

Step 1

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

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Step 1.1

Add $3 y$ to both sides of the equation.

$x + 4 z = - 3 + 3 y$

$3 x + y - z = 4$

$- 2 x + 2 y + 3 z = - 5$

Step 1.2

Subtract $4 z$ from both sides of the equation.

$x = - 3 + 3 y - 4 z$

$3 x + y - z = 4$

$- 2 x + 2 y + 3 z = - 5$

$x = - 3 + 3 y - 4 z$

$3 x + y - z = 4$

$- 2 x + 2 y + 3 z = - 5$

Step 2

Replace all occurrences of $x$ with $- 3 + 3 y - 4 z$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $3 x + y - z = 4$ with $- 3 + 3 y - 4 z$ .

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

$x = - 3 + 3 y - 4 z$

$- 2 x + 2 y + 3 z = - 5$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $3 (- 3 + 3 y - 4 z) + y - z$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Apply the distributive property.

$3 \cdot - 3 + 3 (3 y) + 3 (- 4 z) + y - z = 4$

$x = - 3 + 3 y - 4 z$

$- 2 x + 2 y + 3 z = - 5$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $3$ by $- 3$ .

$- 9 + 3 (3 y) + 3 (- 4 z) + y - z = 4$

$x = - 3 + 3 y - 4 z$

$- 2 x + 2 y + 3 z = - 5$

Step 2.2.1.1.2.2

Multiply $3$ by $3$ .

$- 9 + 9 y + 3 (- 4 z) + y - z = 4$

$x = - 3 + 3 y - 4 z$

$- 2 x + 2 y + 3 z = - 5$

Step 2.2.1.1.2.3

Multiply $- 4$ by $3$ .

$- 9 + 9 y - 12 z + y - z = 4$

$x = - 3 + 3 y - 4 z$

$- 2 x + 2 y + 3 z = - 5$

$- 9 + 9 y - 12 z + y - z = 4$

$x = - 3 + 3 y - 4 z$

$- 2 x + 2 y + 3 z = - 5$

$- 9 + 9 y - 12 z + y - z = 4$

$x = - 3 + 3 y - 4 z$

$- 2 x + 2 y + 3 z = - 5$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Add $9 y$ and $y$ .

$- 9 + 10 y - 12 z - z = 4$

$x = - 3 + 3 y - 4 z$

$- 2 x + 2 y + 3 z = - 5$

Step 2.2.1.2.2

Subtract $z$ from $- 12 z$ .

$- 9 + 10 y - 13 z = 4$

$x = - 3 + 3 y - 4 z$

$- 2 x + 2 y + 3 z = - 5$

$- 9 + 10 y - 13 z = 4$

$x = - 3 + 3 y - 4 z$

$- 2 x + 2 y + 3 z = - 5$

$- 9 + 10 y - 13 z = 4$

$x = - 3 + 3 y - 4 z$

$- 2 x + 2 y + 3 z = - 5$

$- 9 + 10 y - 13 z = 4$

$x = - 3 + 3 y - 4 z$

$- 2 x + 2 y + 3 z = - 5$

Step 2.3

Replace all occurrences of $x$ in $- 2 x + 2 y + 3 z = - 5$ with $- 3 + 3 y - 4 z$ .

$- 2 (- 3 + 3 y - 4 z) + 2 y + 3 z = - 5$

$- 9 + 10 y - 13 z = 4$

$x = - 3 + 3 y - 4 z$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $- 2 (- 3 + 3 y - 4 z) + 2 y + 3 z$ .

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Step 2.4.1.1

Simplify each term.

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Step 2.4.1.1.1

Apply the distributive property.

$- 2 \cdot - 3 - 2 (3 y) - 2 (- 4 z) + 2 y + 3 z = - 5$

$- 9 + 10 y - 13 z = 4$

$x = - 3 + 3 y - 4 z$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $- 2$ by $- 3$ .

$6 - 2 (3 y) - 2 (- 4 z) + 2 y + 3 z = - 5$

$- 9 + 10 y - 13 z = 4$

$x = - 3 + 3 y - 4 z$

Step 2.4.1.1.2.2

Multiply $3$ by $- 2$ .

$6 - 6 y - 2 (- 4 z) + 2 y + 3 z = - 5$

$- 9 + 10 y - 13 z = 4$

$x = - 3 + 3 y - 4 z$

Step 2.4.1.1.2.3

Multiply $- 4$ by $- 2$ .

$6 - 6 y + 8 z + 2 y + 3 z = - 5$

$- 9 + 10 y - 13 z = 4$

$x = - 3 + 3 y - 4 z$

$6 - 6 y + 8 z + 2 y + 3 z = - 5$

$- 9 + 10 y - 13 z = 4$

$x = - 3 + 3 y - 4 z$

$6 - 6 y + 8 z + 2 y + 3 z = - 5$

$- 9 + 10 y - 13 z = 4$

$x = - 3 + 3 y - 4 z$

Step 2.4.1.2

Simplify by adding terms.

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Step 2.4.1.2.1

Add $- 6 y$ and $2 y$ .

$6 - 4 y + 8 z + 3 z = - 5$

$- 9 + 10 y - 13 z = 4$

$x = - 3 + 3 y - 4 z$

Step 2.4.1.2.2

Add $8 z$ and $3 z$ .

$6 - 4 y + 11 z = - 5$

$- 9 + 10 y - 13 z = 4$

$x = - 3 + 3 y - 4 z$

$6 - 4 y + 11 z = - 5$

$- 9 + 10 y - 13 z = 4$

$x = - 3 + 3 y - 4 z$

$6 - 4 y + 11 z = - 5$

$- 9 + 10 y - 13 z = 4$

$x = - 3 + 3 y - 4 z$

$6 - 4 y + 11 z = - 5$

$- 9 + 10 y - 13 z = 4$

$x = - 3 + 3 y - 4 z$

$6 - 4 y + 11 z = - 5$

$- 9 + 10 y - 13 z = 4$

$x = - 3 + 3 y - 4 z$

Step 3

Solve for $y$ in $6 - 4 y + 11 z = - 5$ .

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Step 3.1

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

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Step 3.1.1

Subtract $6$ from both sides of the equation.

$- 4 y + 11 z = - 5 - 6$

$- 9 + 10 y - 13 z = 4$

$x = - 3 + 3 y - 4 z$

Step 3.1.2

Subtract $11 z$ from both sides of the equation.

$- 4 y = - 5 - 6 - 11 z$

$- 9 + 10 y - 13 z = 4$

$x = - 3 + 3 y - 4 z$

Step 3.1.3

Subtract $6$ from $- 5$ .

$- 4 y = - 11 - 11 z$

$- 9 + 10 y - 13 z = 4$

$x = - 3 + 3 y - 4 z$

$- 4 y = - 11 - 11 z$

$- 9 + 10 y - 13 z = 4$

$x = - 3 + 3 y - 4 z$

Step 3.2

Divide each term in $- 4 y = - 11 - 11 z$ by $- 4$ and simplify.

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Step 3.2.1

Divide each term in $- 4 y = - 11 - 11 z$ by $- 4$ .

$\frac{- 4 y}{- 4} = \frac{- 11}{- 4} + \frac{- 11 z}{- 4}$

$- 9 + 10 y - 13 z = 4$

$x = - 3 + 3 y - 4 z$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $- 4$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{- 4 y}{- 4} = \frac{- 11}{- 4} + \frac{- 11 z}{- 4}$

$- 9 + 10 y - 13 z = 4$

$x = - 3 + 3 y - 4 z$

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 11}{- 4} + \frac{- 11 z}{- 4}$

$- 9 + 10 y - 13 z = 4$

$x = - 3 + 3 y - 4 z$

$y = \frac{- 11}{- 4} + \frac{- 11 z}{- 4}$

$- 9 + 10 y - 13 z = 4$

$x = - 3 + 3 y - 4 z$

$y = \frac{- 11}{- 4} + \frac{- 11 z}{- 4}$

$- 9 + 10 y - 13 z = 4$

$x = - 3 + 3 y - 4 z$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

Dividing two negative values results in a positive value.

$y = \frac{11}{4} + \frac{- 11 z}{- 4}$

$- 9 + 10 y - 13 z = 4$

$x = - 3 + 3 y - 4 z$

Step 3.2.3.1.2

Dividing two negative values results in a positive value.

$y = \frac{11}{4} + \frac{11 z}{4}$

$- 9 + 10 y - 13 z = 4$

$x = - 3 + 3 y - 4 z$

$y = \frac{11}{4} + \frac{11 z}{4}$

$- 9 + 10 y - 13 z = 4$

$x = - 3 + 3 y - 4 z$

$y = \frac{11}{4} + \frac{11 z}{4}$

$- 9 + 10 y - 13 z = 4$

$x = - 3 + 3 y - 4 z$

$y = \frac{11}{4} + \frac{11 z}{4}$

$- 9 + 10 y - 13 z = 4$

$x = - 3 + 3 y - 4 z$

$y = \frac{11}{4} + \frac{11 z}{4}$

$- 9 + 10 y - 13 z = 4$

$x = - 3 + 3 y - 4 z$

Step 4

Replace all occurrences of $y$ with $\frac{11}{4} + \frac{11 z}{4}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $- 9 + 10 y - 13 z = 4$ with $\frac{11}{4} + \frac{11 z}{4}$ .

$- 9 + 10 (\frac{11}{4} + \frac{11 z}{4}) - 13 z = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

$x = - 3 + 3 y - 4 z$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $- 9 + 10 (\frac{11}{4} + \frac{11 z}{4}) - 13 z$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Apply the distributive property.

$- 9 + 10 (\frac{11}{4}) + 10 (\frac{11 z}{4}) - 13 z = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

$x = - 3 + 3 y - 4 z$

Step 4.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.1.1.2.1

Factor $2$ out of $10$ .

$- 9 + 2 (5) (\frac{11}{4}) + 10 (\frac{11 z}{4}) - 13 z = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

$x = - 3 + 3 y - 4 z$

Step 4.2.1.1.2.2

Factor $2$ out of $4$ .

$- 9 + 2 \cdot (5 (\frac{11}{2 \cdot 2})) + 10 (\frac{11 z}{4}) - 13 z = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

$x = - 3 + 3 y - 4 z$

Step 4.2.1.1.2.3

Cancel the common factor.

$- 9 + 2 \cdot (5 (\frac{11}{2 \cdot 2})) + 10 (\frac{11 z}{4}) - 13 z = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

$x = - 3 + 3 y - 4 z$

Step 4.2.1.1.2.4

Rewrite the expression.

$- 9 + 5 (\frac{11}{2}) + 10 (\frac{11 z}{4}) - 13 z = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

$x = - 3 + 3 y - 4 z$

$- 9 + 5 (\frac{11}{2}) + 10 (\frac{11 z}{4}) - 13 z = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

$x = - 3 + 3 y - 4 z$

Step 4.2.1.1.3

Combine $5$ and $\frac{11}{2}$ .

$- 9 + \frac{5 \cdot 11}{2} + 10 (\frac{11 z}{4}) - 13 z = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

$x = - 3 + 3 y - 4 z$

Step 4.2.1.1.4

Multiply $5$ by $11$ .

$- 9 + \frac{55}{2} + 10 (\frac{11 z}{4}) - 13 z = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

$x = - 3 + 3 y - 4 z$

Step 4.2.1.1.5

Cancel the common factor of $2$ .

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Step 4.2.1.1.5.1

Factor $2$ out of $10$ .

$- 9 + \frac{55}{2} + 2 (5) (\frac{11 z}{4}) - 13 z = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

$x = - 3 + 3 y - 4 z$

Step 4.2.1.1.5.2

Factor $2$ out of $4$ .

$- 9 + \frac{55}{2} + 2 \cdot (5 (\frac{11 z}{2 \cdot 2})) - 13 z = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

$x = - 3 + 3 y - 4 z$

Step 4.2.1.1.5.3

Cancel the common factor.

$- 9 + \frac{55}{2} + 2 \cdot (5 (\frac{11 z}{2 \cdot 2})) - 13 z = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

$x = - 3 + 3 y - 4 z$

Step 4.2.1.1.5.4

Rewrite the expression.

$- 9 + \frac{55}{2} + 5 (\frac{11 z}{2}) - 13 z = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

$x = - 3 + 3 y - 4 z$

$- 9 + \frac{55}{2} + 5 (\frac{11 z}{2}) - 13 z = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

$x = - 3 + 3 y - 4 z$

Step 4.2.1.1.6

Combine $5$ and $\frac{11 z}{2}$ .

$- 9 + \frac{55}{2} + \frac{5 (11 z)}{2} - 13 z = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

$x = - 3 + 3 y - 4 z$

Step 4.2.1.1.7

Multiply $11$ by $5$ .

$- 9 + \frac{55}{2} + \frac{55 z}{2} - 13 z = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

$x = - 3 + 3 y - 4 z$

$- 9 + \frac{55}{2} + \frac{55 z}{2} - 13 z = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

$x = - 3 + 3 y - 4 z$

Step 4.2.1.2

To write $- 9$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- 9 \cdot \frac{2}{2} + \frac{55}{2} + \frac{55 z}{2} - 13 z = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

$x = - 3 + 3 y - 4 z$

Step 4.2.1.3

Combine $- 9$ and $\frac{2}{2}$ .

$\frac{- 9 \cdot 2}{2} + \frac{55}{2} + \frac{55 z}{2} - 13 z = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

$x = - 3 + 3 y - 4 z$

Step 4.2.1.4

Combine the numerators over the common denominator.

$\frac{- 9 \cdot 2 + 55}{2} + \frac{55 z}{2} - 13 z = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

$x = - 3 + 3 y - 4 z$

Step 4.2.1.5

Simplify the numerator.

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Step 4.2.1.5.1

Multiply $- 9$ by $2$ .

$\frac{- 18 + 55}{2} + \frac{55 z}{2} - 13 z = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

$x = - 3 + 3 y - 4 z$

Step 4.2.1.5.2

Add $- 18$ and $55$ .

$\frac{37}{2} + \frac{55 z}{2} - 13 z = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

$x = - 3 + 3 y - 4 z$

$\frac{37}{2} + \frac{55 z}{2} - 13 z = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

$x = - 3 + 3 y - 4 z$

Step 4.2.1.6

To write $- 13 z$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{37}{2} + \frac{55 z}{2} - 13 z \cdot \frac{2}{2} = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

$x = - 3 + 3 y - 4 z$

Step 4.2.1.7

Combine $- 13 z$ and $\frac{2}{2}$ .

$\frac{37}{2} + \frac{55 z}{2} + \frac{- 13 z \cdot 2}{2} = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

$x = - 3 + 3 y - 4 z$

Step 4.2.1.8

Combine the numerators over the common denominator.

$\frac{37}{2} + \frac{55 z - 13 z \cdot 2}{2} = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

$x = - 3 + 3 y - 4 z$

Step 4.2.1.9

Combine the numerators over the common denominator.

$\frac{37 + 55 z - 13 z \cdot 2}{2} = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

$x = - 3 + 3 y - 4 z$

Step 4.2.1.10

Multiply $2$ by $- 13$ .

$\frac{37 + 55 z - 26 z}{2} = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

$x = - 3 + 3 y - 4 z$

Step 4.2.1.11

Subtract $26 z$ from $55 z$ .

$\frac{37 + 29 z}{2} = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

$x = - 3 + 3 y - 4 z$

$\frac{37 + 29 z}{2} = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

$x = - 3 + 3 y - 4 z$

$\frac{37 + 29 z}{2} = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

$x = - 3 + 3 y - 4 z$

Step 4.3

Replace all occurrences of $y$ in $x = - 3 + 3 y - 4 z$ with $\frac{11}{4} + \frac{11 z}{4}$ .

$x = - 3 + 3 (\frac{11}{4} + \frac{11 z}{4}) - 4 z$

$\frac{37 + 29 z}{2} = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $- 3 + 3 (\frac{11}{4} + \frac{11 z}{4}) - 4 z$ .

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Step 4.4.1.1

Simplify each term.

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Step 4.4.1.1.1

Apply the distributive property.

$x = - 3 + 3 (\frac{11}{4}) + 3 (\frac{11 z}{4}) - 4 z$

$\frac{37 + 29 z}{2} = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

Step 4.4.1.1.2

Multiply $3 (\frac{11}{4})$ .

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Step 4.4.1.1.2.1

Combine $3$ and $\frac{11}{4}$ .

$x = - 3 + \frac{3 \cdot 11}{4} + 3 (\frac{11 z}{4}) - 4 z$

$\frac{37 + 29 z}{2} = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

Step 4.4.1.1.2.2

Multiply $3$ by $11$ .

$x = - 3 + \frac{33}{4} + 3 (\frac{11 z}{4}) - 4 z$

$\frac{37 + 29 z}{2} = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

$x = - 3 + \frac{33}{4} + 3 (\frac{11 z}{4}) - 4 z$

$\frac{37 + 29 z}{2} = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

Step 4.4.1.1.3

Multiply $3 \frac{11 z}{4}$ .

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Step 4.4.1.1.3.1

Combine $3$ and $\frac{11 z}{4}$ .

$x = - 3 + \frac{33}{4} + \frac{3 (11 z)}{4} - 4 z$

$\frac{37 + 29 z}{2} = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

Step 4.4.1.1.3.2

Multiply $11$ by $3$ .

$x = - 3 + \frac{33}{4} + \frac{33 z}{4} - 4 z$

$\frac{37 + 29 z}{2} = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

$x = - 3 + \frac{33}{4} + \frac{33 z}{4} - 4 z$

$\frac{37 + 29 z}{2} = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

$x = - 3 + \frac{33}{4} + \frac{33 z}{4} - 4 z$

$\frac{37 + 29 z}{2} = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

Step 4.4.1.2

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = - 3 \cdot \frac{4}{4} + \frac{33}{4} + \frac{33 z}{4} - 4 z$

$\frac{37 + 29 z}{2} = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

Step 4.4.1.3

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

$x = \frac{- 3 \cdot 4}{4} + \frac{33}{4} + \frac{33 z}{4} - 4 z$

$\frac{37 + 29 z}{2} = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

Step 4.4.1.4

Combine the numerators over the common denominator.

$x = \frac{- 3 \cdot 4 + 33}{4} + \frac{33 z}{4} - 4 z$

$\frac{37 + 29 z}{2} = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

Step 4.4.1.5

Simplify the numerator.

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Step 4.4.1.5.1

Multiply $- 3$ by $4$ .

$x = \frac{- 12 + 33}{4} + \frac{33 z}{4} - 4 z$

$\frac{37 + 29 z}{2} = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

Step 4.4.1.5.2

Add $- 12$ and $33$ .

$x = \frac{21}{4} + \frac{33 z}{4} - 4 z$

$\frac{37 + 29 z}{2} = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

$x = \frac{21}{4} + \frac{33 z}{4} - 4 z$

$\frac{37 + 29 z}{2} = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

Step 4.4.1.6

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

$x = \frac{21}{4} + \frac{33 z}{4} - 4 z \cdot \frac{4}{4}$

$\frac{37 + 29 z}{2} = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

Step 4.4.1.7

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

$x = \frac{21}{4} + \frac{33 z}{4} + \frac{- 4 z \cdot 4}{4}$

$\frac{37 + 29 z}{2} = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

Step 4.4.1.8

Combine the numerators over the common denominator.

$x = \frac{21}{4} + \frac{33 z - 4 z \cdot 4}{4}$

$\frac{37 + 29 z}{2} = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

Step 4.4.1.9

Combine the numerators over the common denominator.

$x = \frac{21 + 33 z - 4 z \cdot 4}{4}$

$\frac{37 + 29 z}{2} = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

Step 4.4.1.10

Multiply $4$ by $- 4$ .

$x = \frac{21 + 33 z - 16 z}{4}$

$\frac{37 + 29 z}{2} = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

Step 4.4.1.11

Subtract $16 z$ from $33 z$ .

$x = \frac{21 + 17 z}{4}$

$\frac{37 + 29 z}{2} = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

$x = \frac{21 + 17 z}{4}$

$\frac{37 + 29 z}{2} = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

$x = \frac{21 + 17 z}{4}$

$\frac{37 + 29 z}{2} = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

$x = \frac{21 + 17 z}{4}$

$\frac{37 + 29 z}{2} = 4$

$y = \frac{11}{4} + \frac{11 z}{4}$

Step 5

Solve for $z$ in $\frac{37 + 29 z}{2} = 4$ .

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Step 5.1

Multiply both sides by $2$ .

$\frac{37 + 29 z}{2} \cdot 2 = 4 \cdot 2$

$x = \frac{21 + 17 z}{4}$

$y = \frac{11}{4} + \frac{11 z}{4}$

Step 5.2

Simplify.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

Simplify $\frac{37 + 29 z}{2} \cdot 2$ .

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Step 5.2.1.1.1

Cancel the common factor of $2$ .

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Step 5.2.1.1.1.1

Cancel the common factor.

$\frac{37 + 29 z}{2} \cdot 2 = 4 \cdot 2$

$x = \frac{21 + 17 z}{4}$

$y = \frac{11}{4} + \frac{11 z}{4}$

Step 5.2.1.1.1.2

Rewrite the expression.

$37 + 29 z = 4 \cdot 2$

$x = \frac{21 + 17 z}{4}$

$y = \frac{11}{4} + \frac{11 z}{4}$

$37 + 29 z = 4 \cdot 2$

$x = \frac{21 + 17 z}{4}$

$y = \frac{11}{4} + \frac{11 z}{4}$

Step 5.2.1.1.2

Reorder $37$ and $29 z$ .

$29 z + 37 = 4 \cdot 2$

$x = \frac{21 + 17 z}{4}$

$y = \frac{11}{4} + \frac{11 z}{4}$

$29 z + 37 = 4 \cdot 2$

$x = \frac{21 + 17 z}{4}$

$y = \frac{11}{4} + \frac{11 z}{4}$

$29 z + 37 = 4 \cdot 2$

$x = \frac{21 + 17 z}{4}$

$y = \frac{11}{4} + \frac{11 z}{4}$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Multiply $4$ by $2$ .

$29 z + 37 = 8$

$x = \frac{21 + 17 z}{4}$

$y = \frac{11}{4} + \frac{11 z}{4}$

$29 z + 37 = 8$

$x = \frac{21 + 17 z}{4}$

$y = \frac{11}{4} + \frac{11 z}{4}$

$29 z + 37 = 8$

$x = \frac{21 + 17 z}{4}$

$y = \frac{11}{4} + \frac{11 z}{4}$

Step 5.3

Solve for $z$ .

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Step 5.3.1

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

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Step 5.3.1.1

Subtract $37$ from both sides of the equation.

$29 z = 8 - 37$

$x = \frac{21 + 17 z}{4}$

$y = \frac{11}{4} + \frac{11 z}{4}$

Step 5.3.1.2

Subtract $37$ from $8$ .

$29 z = - 29$

$x = \frac{21 + 17 z}{4}$

$y = \frac{11}{4} + \frac{11 z}{4}$

$29 z = - 29$

$x = \frac{21 + 17 z}{4}$

$y = \frac{11}{4} + \frac{11 z}{4}$

Step 5.3.2

Divide each term in $29 z = - 29$ by $29$ and simplify.

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Step 5.3.2.1

Divide each term in $29 z = - 29$ by $29$ .

$\frac{29 z}{29} = \frac{- 29}{29}$

$x = \frac{21 + 17 z}{4}$

$y = \frac{11}{4} + \frac{11 z}{4}$

Step 5.3.2.2

Simplify the left side.

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Step 5.3.2.2.1

Cancel the common factor of $29$ .

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Step 5.3.2.2.1.1

Cancel the common factor.

$\frac{29 z}{29} = \frac{- 29}{29}$

$x = \frac{21 + 17 z}{4}$

$y = \frac{11}{4} + \frac{11 z}{4}$

Step 5.3.2.2.1.2

Divide $z$ by $1$ .

$z = \frac{- 29}{29}$

$x = \frac{21 + 17 z}{4}$

$y = \frac{11}{4} + \frac{11 z}{4}$

$z = \frac{- 29}{29}$

$x = \frac{21 + 17 z}{4}$

$y = \frac{11}{4} + \frac{11 z}{4}$

$z = \frac{- 29}{29}$

$x = \frac{21 + 17 z}{4}$

$y = \frac{11}{4} + \frac{11 z}{4}$

Step 5.3.2.3

Simplify the right side.

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Step 5.3.2.3.1

Divide $- 29$ by $29$ .

$z = - 1$

$x = \frac{21 + 17 z}{4}$

$y = \frac{11}{4} + \frac{11 z}{4}$

$z = - 1$

$x = \frac{21 + 17 z}{4}$

$y = \frac{11}{4} + \frac{11 z}{4}$

$z = - 1$

$x = \frac{21 + 17 z}{4}$

$y = \frac{11}{4} + \frac{11 z}{4}$

$z = - 1$

$x = \frac{21 + 17 z}{4}$

$y = \frac{11}{4} + \frac{11 z}{4}$

$z = - 1$

$x = \frac{21 + 17 z}{4}$

$y = \frac{11}{4} + \frac{11 z}{4}$

Step 6

Replace all occurrences of $z$ with $- 1$ in each equation.

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Step 6.1

Replace all occurrences of $z$ in $x = \frac{21 + 17 z}{4}$ with $- 1$ .

$x = \frac{21 + 17 (- 1)}{4}$

$z = - 1$

$y = \frac{11}{4} + \frac{11 z}{4}$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $\frac{21 + 17 (- 1)}{4}$ .

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Step 6.2.1.1

Simplify the numerator.

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Step 6.2.1.1.1

Multiply $17$ by $- 1$ .

$x = \frac{21 - 17}{4}$

$z = - 1$

$y = \frac{11}{4} + \frac{11 z}{4}$

Step 6.2.1.1.2

Subtract $17$ from $21$ .

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

$z = - 1$

$y = \frac{11}{4} + \frac{11 z}{4}$

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

$z = - 1$

$y = \frac{11}{4} + \frac{11 z}{4}$

Step 6.2.1.2

Divide $4$ by $4$ .

$x = 1$

$z = - 1$

$y = \frac{11}{4} + \frac{11 z}{4}$

$x = 1$

$z = - 1$

$y = \frac{11}{4} + \frac{11 z}{4}$

$x = 1$

$z = - 1$

$y = \frac{11}{4} + \frac{11 z}{4}$

Step 6.3

Replace all occurrences of $z$ in $y = \frac{11}{4} + \frac{11 z}{4}$ with $- 1$ .

$y = \frac{11}{4} + \frac{11 (- 1)}{4}$

$x = 1$

$z = - 1$

Step 6.4

Simplify the right side.

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Step 6.4.1

Simplify $\frac{11}{4} + \frac{11 (- 1)}{4}$ .

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Step 6.4.1.1

Combine the numerators over the common denominator.

$y = \frac{11 + 11 (- 1)}{4}$

$x = 1$

$z = - 1$

Step 6.4.1.2

Simplify the expression.

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Step 6.4.1.2.1

Multiply $11$ by $- 1$ .

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

$x = 1$

$z = - 1$

Step 6.4.1.2.2

Subtract $11$ from $11$ .

$y = \frac{0}{4}$

$x = 1$

$z = - 1$

Step 6.4.1.2.3

Divide $0$ by $4$ .

$y = 0$

$x = 1$

$z = - 1$

$y = 0$

$x = 1$

$z = - 1$

$y = 0$

$x = 1$

$z = - 1$

$y = 0$

$x = 1$

$z = - 1$

$y = 0$

$x = 1$

$z = - 1$

Step 7

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(1, 0, - 1)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(1, 0, - 1)$

Equation Form:

$x = 1, y = 0, z = - 1$