Finite Math Examples

$14 x - 3 y + 5 z = - 15$ , $3 x + 2 y - 6 z = 10$ , $7 x - y + 4 z = - 5$

Step 1

Solve for $x$ in $14 x - 3 y + 5 z = - 15$ .

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Step 1.1

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

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Step 1.1.1

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

$14 x + 5 z = - 15 + 3 y$

$3 x + 2 y - 6 z = 10$

$7 x - y + 4 z = - 5$

Step 1.1.2

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

$14 x = - 15 + 3 y - 5 z$

$3 x + 2 y - 6 z = 10$

$7 x - y + 4 z = - 5$

$14 x = - 15 + 3 y - 5 z$

$3 x + 2 y - 6 z = 10$

$7 x - y + 4 z = - 5$

Step 1.2

Divide each term in $14 x = - 15 + 3 y - 5 z$ by $14$ and simplify.

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Step 1.2.1

Divide each term in $14 x = - 15 + 3 y - 5 z$ by $14$ .

$\frac{14 x}{14} = \frac{- 15}{14} + \frac{3 y}{14} + \frac{- 5 z}{14}$

$3 x + 2 y - 6 z = 10$

$7 x - y + 4 z = - 5$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $14$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{14 x}{14} = \frac{- 15}{14} + \frac{3 y}{14} + \frac{- 5 z}{14}$

$3 x + 2 y - 6 z = 10$

$7 x - y + 4 z = - 5$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 15}{14} + \frac{3 y}{14} + \frac{- 5 z}{14}$

$3 x + 2 y - 6 z = 10$

$7 x - y + 4 z = - 5$

$x = \frac{- 15}{14} + \frac{3 y}{14} + \frac{- 5 z}{14}$

$3 x + 2 y - 6 z = 10$

$7 x - y + 4 z = - 5$

$x = \frac{- 15}{14} + \frac{3 y}{14} + \frac{- 5 z}{14}$

$3 x + 2 y - 6 z = 10$

$7 x - y + 4 z = - 5$

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

Move the negative in front of the fraction.

$x = - \frac{15}{14} + \frac{3 y}{14} + \frac{- 5 z}{14}$

$3 x + 2 y - 6 z = 10$

$7 x - y + 4 z = - 5$

Step 1.2.3.1.2

Move the negative in front of the fraction.

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$3 x + 2 y - 6 z = 10$

$7 x - y + 4 z = - 5$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$3 x + 2 y - 6 z = 10$

$7 x - y + 4 z = - 5$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$3 x + 2 y - 6 z = 10$

$7 x - y + 4 z = - 5$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$3 x + 2 y - 6 z = 10$

$7 x - y + 4 z = - 5$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$3 x + 2 y - 6 z = 10$

$7 x - y + 4 z = - 5$

Step 2

Replace all occurrences of $x$ with $- \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $3 x + 2 y - 6 z = 10$ with $- \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$ .

$3 (- \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}) + 2 y - 6 z = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$7 x - y + 4 z = - 5$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $3 (- \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}) + 2 y - 6 z$ .

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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.

$3 (- \frac{15}{14}) + 3 (\frac{3 y}{14}) + 3 (- \frac{5 z}{14}) + 2 y - 6 z = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$7 x - y + 4 z = - 5$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $3 (- \frac{15}{14})$ .

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Step 2.2.1.1.2.1.1

Multiply $- 1$ by $3$ .

$- 3 (\frac{15}{14}) + 3 (\frac{3 y}{14}) + 3 (- \frac{5 z}{14}) + 2 y - 6 z = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$7 x - y + 4 z = - 5$

Step 2.2.1.1.2.1.2

Combine $- 3$ and $\frac{15}{14}$ .

$\frac{- 3 \cdot 15}{14} + 3 (\frac{3 y}{14}) + 3 (- \frac{5 z}{14}) + 2 y - 6 z = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$7 x - y + 4 z = - 5$

Step 2.2.1.1.2.1.3

Multiply $- 3$ by $15$ .

$\frac{- 45}{14} + 3 (\frac{3 y}{14}) + 3 (- \frac{5 z}{14}) + 2 y - 6 z = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$7 x - y + 4 z = - 5$

$\frac{- 45}{14} + 3 (\frac{3 y}{14}) + 3 (- \frac{5 z}{14}) + 2 y - 6 z = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$7 x - y + 4 z = - 5$

Step 2.2.1.1.2.2

Multiply $3 \frac{3 y}{14}$ .

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Step 2.2.1.1.2.2.1

Combine $3$ and $\frac{3 y}{14}$ .

$\frac{- 45}{14} + \frac{3 (3 y)}{14} + 3 (- \frac{5 z}{14}) + 2 y - 6 z = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$7 x - y + 4 z = - 5$

Step 2.2.1.1.2.2.2

Multiply $3$ by $3$ .

$\frac{- 45}{14} + \frac{9 y}{14} + 3 (- \frac{5 z}{14}) + 2 y - 6 z = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$7 x - y + 4 z = - 5$

$\frac{- 45}{14} + \frac{9 y}{14} + 3 (- \frac{5 z}{14}) + 2 y - 6 z = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$7 x - y + 4 z = - 5$

Step 2.2.1.1.2.3

Multiply $3 (- \frac{5 z}{14})$ .

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Step 2.2.1.1.2.3.1

Multiply $- 1$ by $3$ .

$\frac{- 45}{14} + \frac{9 y}{14} - 3 \frac{5 z}{14} + 2 y - 6 z = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$7 x - y + 4 z = - 5$

Step 2.2.1.1.2.3.2

Combine $- 3$ and $\frac{5 z}{14}$ .

$\frac{- 45}{14} + \frac{9 y}{14} + \frac{- 3 (5 z)}{14} + 2 y - 6 z = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$7 x - y + 4 z = - 5$

Step 2.2.1.1.2.3.3

Multiply $5$ by $- 3$ .

$\frac{- 45}{14} + \frac{9 y}{14} + \frac{- 15 z}{14} + 2 y - 6 z = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$7 x - y + 4 z = - 5$

$\frac{- 45}{14} + \frac{9 y}{14} + \frac{- 15 z}{14} + 2 y - 6 z = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$7 x - y + 4 z = - 5$

$\frac{- 45}{14} + \frac{9 y}{14} + \frac{- 15 z}{14} + 2 y - 6 z = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$7 x - y + 4 z = - 5$

Step 2.2.1.1.3

Simplify each term.

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Step 2.2.1.1.3.1

Move the negative in front of the fraction.

$- \frac{45}{14} + \frac{9 y}{14} + \frac{- 15 z}{14} + 2 y - 6 z = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$7 x - y + 4 z = - 5$

Step 2.2.1.1.3.2

Move the negative in front of the fraction.

$- \frac{45}{14} + \frac{9 y}{14} - \frac{15 z}{14} + 2 y - 6 z = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$7 x - y + 4 z = - 5$

$- \frac{45}{14} + \frac{9 y}{14} - \frac{15 z}{14} + 2 y - 6 z = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$7 x - y + 4 z = - 5$

$- \frac{45}{14} + \frac{9 y}{14} - \frac{15 z}{14} + 2 y - 6 z = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$7 x - y + 4 z = - 5$

Step 2.2.1.2

To write $2 y$ as a fraction with a common denominator, multiply by $\frac{14}{14}$ .

$- \frac{45}{14} - \frac{15 z}{14} + \frac{9 y}{14} + 2 y \cdot \frac{14}{14} - 6 z = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$7 x - y + 4 z = - 5$

Step 2.2.1.3

Combine $2 y$ and $\frac{14}{14}$ .

$- \frac{45}{14} - \frac{15 z}{14} + \frac{9 y}{14} + \frac{2 y \cdot 14}{14} - 6 z = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$7 x - y + 4 z = - 5$

Step 2.2.1.4

Combine the numerators over the common denominator.

$- \frac{45}{14} - \frac{15 z}{14} + \frac{9 y + 2 y \cdot 14}{14} - 6 z = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$7 x - y + 4 z = - 5$

Step 2.2.1.5

Combine the numerators over the common denominator.

$- 6 z + \frac{- 45 - 15 z + 9 y + 2 y \cdot 14}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$7 x - y + 4 z = - 5$

Step 2.2.1.6

Multiply $14$ by $2$ .

$- 6 z + \frac{- 45 - 15 z + 9 y + 28 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$7 x - y + 4 z = - 5$

Step 2.2.1.7

Add $9 y$ and $28 y$ .

$- 6 z + \frac{- 45 - 15 z + 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$7 x - y + 4 z = - 5$

Step 2.2.1.8

To write $- 6 z$ as a fraction with a common denominator, multiply by $\frac{14}{14}$ .

$- 6 z \cdot \frac{14}{14} + \frac{- 45 - 15 z + 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$7 x - y + 4 z = - 5$

Step 2.2.1.9

Simplify terms.

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Step 2.2.1.9.1

Combine $- 6 z$ and $\frac{14}{14}$ .

$\frac{- 6 z \cdot 14}{14} + \frac{- 45 - 15 z + 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$7 x - y + 4 z = - 5$

Step 2.2.1.9.2

Combine the numerators over the common denominator.

$\frac{- 6 z \cdot 14 - 45 - 15 z + 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$7 x - y + 4 z = - 5$

$\frac{- 6 z \cdot 14 - 45 - 15 z + 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$7 x - y + 4 z = - 5$

Step 2.2.1.10

Simplify the numerator.

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Step 2.2.1.10.1

Multiply $14$ by $- 6$ .

$\frac{- 84 z - 45 - 15 z + 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$7 x - y + 4 z = - 5$

Step 2.2.1.10.2

Subtract $15 z$ from $- 84 z$ .

$\frac{- 99 z - 45 + 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$7 x - y + 4 z = - 5$

$\frac{- 99 z - 45 + 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$7 x - y + 4 z = - 5$

Step 2.2.1.11

Simplify with factoring out.

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Step 2.2.1.11.1

Factor $- 1$ out of $- 99 z$ .

$\frac{- (99 z) - 45 + 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$7 x - y + 4 z = - 5$

Step 2.2.1.11.2

Rewrite $- 45$ as $- 1 (45)$ .

$\frac{- (99 z) - 1 \cdot 45 + 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$7 x - y + 4 z = - 5$

Step 2.2.1.11.3

Factor $- 1$ out of $- (99 z) - 1 (45)$ .

$\frac{- (99 z + 45) + 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$7 x - y + 4 z = - 5$

Step 2.2.1.11.4

Factor $- 1$ out of $37 y$ .

$\frac{- (99 z + 45) - (- 37 y)}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$7 x - y + 4 z = - 5$

Step 2.2.1.11.5

Factor $- 1$ out of $- (99 z + 45) - (- 37 y)$ .

$\frac{- (99 z + 45 - 37 y)}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$7 x - y + 4 z = - 5$

Step 2.2.1.11.6

Simplify the expression.

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Step 2.2.1.11.6.1

Rewrite $- (99 z + 45 - 37 y)$ as $- 1 (99 z + 45 - 37 y)$ .

$\frac{- 1 (99 z + 45 - 37 y)}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$7 x - y + 4 z = - 5$

Step 2.2.1.11.6.2

Move the negative in front of the fraction.

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$7 x - y + 4 z = - 5$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$7 x - y + 4 z = - 5$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$7 x - y + 4 z = - 5$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$7 x - y + 4 z = - 5$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$7 x - y + 4 z = - 5$

Step 2.3

Replace all occurrences of $x$ in $7 x - y + 4 z = - 5$ with $- \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$ .

$7 (- \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}) - y + 4 z = - 5$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $7 (- \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}) - y + 4 z$ .

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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.

$7 (- \frac{15}{14}) + 7 (\frac{3 y}{14}) + 7 (- \frac{5 z}{14}) - y + 4 z = - 5$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Cancel the common factor of $7$ .

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Step 2.4.1.1.2.1.1

Move the leading negative in $- \frac{15}{14}$ into the numerator.

$7 (\frac{- 15}{14}) + 7 (\frac{3 y}{14}) + 7 (- \frac{5 z}{14}) - y + 4 z = - 5$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 2.4.1.1.2.1.2

Factor $7$ out of $14$ .

$7 (\frac{- 15}{7 (2)}) + 7 (\frac{3 y}{14}) + 7 (- \frac{5 z}{14}) - y + 4 z = - 5$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 2.4.1.1.2.1.3

Cancel the common factor.

$7 (\frac{- 15}{7 \cdot 2}) + 7 (\frac{3 y}{14}) + 7 (- \frac{5 z}{14}) - y + 4 z = - 5$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 2.4.1.1.2.1.4

Rewrite the expression.

$\frac{- 15}{2} + 7 (\frac{3 y}{14}) + 7 (- \frac{5 z}{14}) - y + 4 z = - 5$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$\frac{- 15}{2} + 7 (\frac{3 y}{14}) + 7 (- \frac{5 z}{14}) - y + 4 z = - 5$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 2.4.1.1.2.2

Cancel the common factor of $7$ .

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Step 2.4.1.1.2.2.1

Factor $7$ out of $14$ .

$\frac{- 15}{2} + 7 (\frac{3 y}{7 (2)}) + 7 (- \frac{5 z}{14}) - y + 4 z = - 5$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 2.4.1.1.2.2.2

Cancel the common factor.

$\frac{- 15}{2} + 7 (\frac{3 y}{7 \cdot 2}) + 7 (- \frac{5 z}{14}) - y + 4 z = - 5$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 2.4.1.1.2.2.3

Rewrite the expression.

$\frac{- 15}{2} + \frac{3 y}{2} + 7 (- \frac{5 z}{14}) - y + 4 z = - 5$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$\frac{- 15}{2} + \frac{3 y}{2} + 7 (- \frac{5 z}{14}) - y + 4 z = - 5$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 2.4.1.1.2.3

Cancel the common factor of $7$ .

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Step 2.4.1.1.2.3.1

Move the leading negative in $- \frac{5 z}{14}$ into the numerator.

$\frac{- 15}{2} + \frac{3 y}{2} + 7 (\frac{- 5 z}{14}) - y + 4 z = - 5$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 2.4.1.1.2.3.2

Factor $7$ out of $14$ .

$\frac{- 15}{2} + \frac{3 y}{2} + 7 (\frac{- 5 z}{7 (2)}) - y + 4 z = - 5$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 2.4.1.1.2.3.3

Cancel the common factor.

$\frac{- 15}{2} + \frac{3 y}{2} + 7 (\frac{- 5 z}{7 \cdot 2}) - y + 4 z = - 5$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 2.4.1.1.2.3.4

Rewrite the expression.

$\frac{- 15}{2} + \frac{3 y}{2} + \frac{- 5 z}{2} - y + 4 z = - 5$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$\frac{- 15}{2} + \frac{3 y}{2} + \frac{- 5 z}{2} - y + 4 z = - 5$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$\frac{- 15}{2} + \frac{3 y}{2} + \frac{- 5 z}{2} - y + 4 z = - 5$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 2.4.1.1.3

Simplify each term.

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Step 2.4.1.1.3.1

Move the negative in front of the fraction.

$- \frac{15}{2} + \frac{3 y}{2} + \frac{- 5 z}{2} - y + 4 z = - 5$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 2.4.1.1.3.2

Move the negative in front of the fraction.

$- \frac{15}{2} + \frac{3 y}{2} - \frac{5 z}{2} - y + 4 z = - 5$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$- \frac{15}{2} + \frac{3 y}{2} - \frac{5 z}{2} - y + 4 z = - 5$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$- \frac{15}{2} + \frac{3 y}{2} - \frac{5 z}{2} - y + 4 z = - 5$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 2.4.1.2

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

$- \frac{15}{2} - \frac{5 z}{2} + \frac{3 y}{2} - y \cdot \frac{2}{2} + 4 z = - 5$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 2.4.1.3

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

$- \frac{15}{2} - \frac{5 z}{2} + \frac{3 y}{2} + \frac{- y \cdot 2}{2} + 4 z = - 5$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 2.4.1.4

Combine the numerators over the common denominator.

$- \frac{15}{2} - \frac{5 z}{2} + \frac{3 y - y \cdot 2}{2} + 4 z = - 5$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 2.4.1.5

Combine the numerators over the common denominator.

$4 z + \frac{- 15 - 5 z + 3 y - y \cdot 2}{2} = - 5$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 2.4.1.6

Multiply $2$ by $- 1$ .

$4 z + \frac{- 15 - 5 z + 3 y - 2 y}{2} = - 5$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 2.4.1.7

Subtract $2 y$ from $3 y$ .

$4 z + \frac{- 15 - 5 z + y}{2} = - 5$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 2.4.1.8

To write $4 z$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$4 z \cdot \frac{2}{2} + \frac{- 15 - 5 z + y}{2} = - 5$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 2.4.1.9

Simplify terms.

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Step 2.4.1.9.1

Combine $4 z$ and $\frac{2}{2}$ .

$\frac{4 z \cdot 2}{2} + \frac{- 15 - 5 z + y}{2} = - 5$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 2.4.1.9.2

Combine the numerators over the common denominator.

$\frac{4 z \cdot 2 - 15 - 5 z + y}{2} = - 5$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$\frac{4 z \cdot 2 - 15 - 5 z + y}{2} = - 5$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 2.4.1.10

Simplify the numerator.

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Step 2.4.1.10.1

Multiply $2$ by $4$ .

$\frac{8 z - 15 - 5 z + y}{2} = - 5$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 2.4.1.10.2

Subtract $5 z$ from $8 z$ .

$\frac{3 z - 15 + y}{2} = - 5$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$\frac{3 z - 15 + y}{2} = - 5$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$\frac{3 z - 15 + y}{2} = - 5$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$\frac{3 z - 15 + y}{2} = - 5$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$\frac{3 z - 15 + y}{2} = - 5$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 3

Solve for $z$ in $\frac{3 z - 15 + y}{2} = - 5$ .

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Step 3.1

Multiply both sides by $2$ .

$\frac{3 z - 15 + y}{2} \cdot 2 = - 5 \cdot 2$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 3.2

Simplify.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $\frac{3 z - 15 + y}{2} \cdot 2$ .

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Step 3.2.1.1.1

Cancel the common factor of $2$ .

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Step 3.2.1.1.1.1

Cancel the common factor.

$\frac{3 z - 15 + y}{2} \cdot 2 = - 5 \cdot 2$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 3.2.1.1.1.2

Rewrite the expression.

$3 z - 15 + y = - 5 \cdot 2$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$3 z - 15 + y = - 5 \cdot 2$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 3.2.1.1.2

Simplify the expression.

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Step 3.2.1.1.2.1

Move $- 15$ .

$3 z + y - 15 = - 5 \cdot 2$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 3.2.1.1.2.2

Reorder $3 z$ and $y$ .

$y + 3 z - 15 = - 5 \cdot 2$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$y + 3 z - 15 = - 5 \cdot 2$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$y + 3 z - 15 = - 5 \cdot 2$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$y + 3 z - 15 = - 5 \cdot 2$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Multiply $- 5$ by $2$ .

$y + 3 z - 15 = - 10$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$y + 3 z - 15 = - 10$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$y + 3 z - 15 = - 10$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 3.3

Solve for $z$ .

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Step 3.3.1

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

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Step 3.3.1.1

Subtract $y$ from both sides of the equation.

$3 z - 15 = - 10 - y$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 3.3.1.2

Add $15$ to both sides of the equation.

$3 z = - 10 - y + 15$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 3.3.1.3

Add $- 10$ and $15$ .

$3 z = - y + 5$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$3 z = - y + 5$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 3.3.2

Divide each term in $3 z = - y + 5$ by $3$ and simplify.

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Step 3.3.2.1

Divide each term in $3 z = - y + 5$ by $3$ .

$\frac{3 z}{3} = \frac{- y}{3} + \frac{5}{3}$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Cancel the common factor of $3$ .

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Step 3.3.2.2.1.1

Cancel the common factor.

$\frac{3 z}{3} = \frac{- y}{3} + \frac{5}{3}$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 3.3.2.2.1.2

Divide $z$ by $1$ .

$z = \frac{- y}{3} + \frac{5}{3}$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$z = \frac{- y}{3} + \frac{5}{3}$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$z = \frac{- y}{3} + \frac{5}{3}$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 3.3.2.3

Simplify the right side.

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Step 3.3.2.3.1

Move the negative in front of the fraction.

$z = - \frac{y}{3} + \frac{5}{3}$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$z = - \frac{y}{3} + \frac{5}{3}$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$z = - \frac{y}{3} + \frac{5}{3}$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$z = - \frac{y}{3} + \frac{5}{3}$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$z = - \frac{y}{3} + \frac{5}{3}$

$- \frac{99 z + 45 - 37 y}{14} = 10$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 4

Replace all occurrences of $z$ with $- \frac{y}{3} + \frac{5}{3}$ in each equation.

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Step 4.1

Replace all occurrences of $z$ in $- \frac{99 z + 45 - 37 y}{14} = 10$ with $- \frac{y}{3} + \frac{5}{3}$ .

$- \frac{99 (- \frac{y}{3} + \frac{5}{3}) + 45 - 37 y}{14} = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $- \frac{99 (- \frac{y}{3} + \frac{5}{3}) + 45 - 37 y}{14}$ .

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Step 4.2.1.1

Simplify the numerator.

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Step 4.2.1.1.1

Apply the distributive property.

$- \frac{99 (- \frac{y}{3}) + 99 (\frac{5}{3}) + 45 - 37 y}{14} = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 4.2.1.1.2

Cancel the common factor of $3$ .

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Step 4.2.1.1.2.1

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

$- \frac{99 (\frac{- y}{3}) + 99 (\frac{5}{3}) + 45 - 37 y}{14} = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 4.2.1.1.2.2

Factor $3$ out of $99$ .

$- \frac{3 (33) (\frac{- y}{3}) + 99 (\frac{5}{3}) + 45 - 37 y}{14} = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 4.2.1.1.2.3

Cancel the common factor.

$- \frac{3 \cdot (33 (\frac{- y}{3})) + 99 (\frac{5}{3}) + 45 - 37 y}{14} = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 4.2.1.1.2.4

Rewrite the expression.

$- \frac{33 (- y) + 99 (\frac{5}{3}) + 45 - 37 y}{14} = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$- \frac{33 (- y) + 99 (\frac{5}{3}) + 45 - 37 y}{14} = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 4.2.1.1.3

Multiply $- 1$ by $33$ .

$- \frac{- 33 y + 99 (\frac{5}{3}) + 45 - 37 y}{14} = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 4.2.1.1.4

Cancel the common factor of $3$ .

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Step 4.2.1.1.4.1

Factor $3$ out of $99$ .

$- \frac{- 33 y + 3 (33) (\frac{5}{3}) + 45 - 37 y}{14} = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 4.2.1.1.4.2

Cancel the common factor.

$- \frac{- 33 y + 3 \cdot (33 (\frac{5}{3})) + 45 - 37 y}{14} = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 4.2.1.1.4.3

Rewrite the expression.

$- \frac{- 33 y + 33 \cdot 5 + 45 - 37 y}{14} = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$- \frac{- 33 y + 33 \cdot 5 + 45 - 37 y}{14} = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 4.2.1.1.5

Multiply $33$ by $5$ .

$- \frac{- 33 y + 165 + 45 - 37 y}{14} = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 4.2.1.1.6

Subtract $37 y$ from $- 33 y$ .

$- \frac{- 70 y + 165 + 45}{14} = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 4.2.1.1.7

Add $165$ and $45$ .

$- \frac{- 70 y + 210}{14} = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 4.2.1.1.8

Factor $70$ out of $- 70 y + 210$ .

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Step 4.2.1.1.8.1

Factor $70$ out of $- 70 y$ .

$- \frac{70 (- y) + 210}{14} = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 4.2.1.1.8.2

Factor $70$ out of $210$ .

$- \frac{70 (- y) + 70 (3)}{14} = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 4.2.1.1.8.3

Factor $70$ out of $70 (- y) + 70 (3)$ .

$- \frac{70 (- y + 3)}{14} = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$- \frac{70 (- y + 3)}{14} = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$- \frac{70 (- y + 3)}{14} = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 4.2.1.2

Simplify terms.

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Step 4.2.1.2.1

Cancel the common factor of $70$ and $14$ .

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Step 4.2.1.2.1.1

Factor $14$ out of $70 (- y + 3)$ .

$- \frac{14 (5 (- y + 3))}{14} = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 4.2.1.2.1.2

Cancel the common factors.

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Step 4.2.1.2.1.2.1

Factor $14$ out of $14$ .

$- \frac{14 (5 (- y + 3))}{14 (1)} = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 4.2.1.2.1.2.2

Cancel the common factor.

$- \frac{14 (5 (- y + 3))}{14 \cdot 1} = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 4.2.1.2.1.2.3

Rewrite the expression.

$- \frac{5 (- y + 3)}{1} = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 4.2.1.2.1.2.4

Divide $5 (- y + 3)$ by $1$ .

$- (5 (- y + 3)) = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$- (5 (- y + 3)) = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$- (5 (- y + 3)) = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 4.2.1.2.2

Apply the distributive property.

$- (5 (- y) + 5 \cdot 3) = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 4.2.1.2.3

Multiply.

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Step 4.2.1.2.3.1

Multiply $- 1$ by $5$ .

$- (- 5 y + 5 \cdot 3) = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 4.2.1.2.3.2

Multiply $5$ by $3$ .

$- (- 5 y + 15) = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$- (- 5 y + 15) = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 4.2.1.2.4

Apply the distributive property.

$- (- 5 y) - 1 \cdot 15 = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 4.2.1.2.5

Multiply.

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Step 4.2.1.2.5.1

Multiply $- 5$ by $- 1$ .

$5 y - 1 \cdot 15 = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 4.2.1.2.5.2

Multiply $- 1$ by $15$ .

$5 y - 15 = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$5 y - 15 = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$5 y - 15 = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$5 y - 15 = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

$5 y - 15 = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$

Step 4.3

Replace all occurrences of $z$ in $x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 z}{14}$ with $- \frac{y}{3} + \frac{5}{3}$ .

$x = - \frac{15}{14} + \frac{3 y}{14} - \frac{5 (- \frac{y}{3} + \frac{5}{3})}{14}$

$5 y - 15 = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $- \frac{15}{14} + \frac{3 y}{14} - \frac{5 (- \frac{y}{3} + \frac{5}{3})}{14}$ .

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Step 4.4.1.1

Combine the numerators over the common denominator.

$x = \frac{- 15 + 3 y - 5 (- \frac{y}{3} + \frac{5}{3})}{14}$

$5 y - 15 = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

Step 4.4.1.2

Simplify each term.

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Step 4.4.1.2.1

Apply the distributive property.

$x = \frac{- 15 + 3 y - 5 (- \frac{y}{3}) - 5 (\frac{5}{3})}{14}$

$5 y - 15 = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

Step 4.4.1.2.2

Multiply $- 5 (- \frac{y}{3})$ .

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Step 4.4.1.2.2.1

Multiply $- 1$ by $- 5$ .

$x = \frac{- 15 + 3 y + 5 (\frac{y}{3}) - 5 (\frac{5}{3})}{14}$

$5 y - 15 = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

Step 4.4.1.2.2.2

Combine $5$ and $\frac{y}{3}$ .

$x = \frac{- 15 + 3 y + \frac{5 y}{3} - 5 (\frac{5}{3})}{14}$

$5 y - 15 = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

$x = \frac{- 15 + 3 y + \frac{5 y}{3} - 5 (\frac{5}{3})}{14}$

$5 y - 15 = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

Step 4.4.1.2.3

Multiply $- 5 (\frac{5}{3})$ .

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Step 4.4.1.2.3.1

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

$x = \frac{- 15 + 3 y + \frac{5 y}{3} + \frac{- 5 \cdot 5}{3}}{14}$

$5 y - 15 = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

Step 4.4.1.2.3.2

Multiply $- 5$ by $5$ .

$x = \frac{- 15 + 3 y + \frac{5 y}{3} + \frac{- 25}{3}}{14}$

$5 y - 15 = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

$x = \frac{- 15 + 3 y + \frac{5 y}{3} + \frac{- 25}{3}}{14}$

$5 y - 15 = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

Step 4.4.1.2.4

Move the negative in front of the fraction.

$x = \frac{- 15 + 3 y + \frac{5 y}{3} - \frac{25}{3}}{14}$

$5 y - 15 = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

$x = \frac{- 15 + 3 y + \frac{5 y}{3} - \frac{25}{3}}{14}$

$5 y - 15 = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

Step 4.4.1.3

To write $- 15$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = \frac{3 y + \frac{5 y}{3} - 15 \cdot \frac{3}{3} - \frac{25}{3}}{14}$

$5 y - 15 = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

Step 4.4.1.4

Combine $- 15$ and $\frac{3}{3}$ .

$x = \frac{3 y + \frac{5 y}{3} + \frac{- 15 \cdot 3}{3} - \frac{25}{3}}{14}$

$5 y - 15 = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

Step 4.4.1.5

Combine the numerators over the common denominator.

$x = \frac{3 y + \frac{5 y}{3} + \frac{- 15 \cdot 3 - 25}{3}}{14}$

$5 y - 15 = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

Step 4.4.1.6

Simplify the numerator.

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Step 4.4.1.6.1

Multiply $- 15$ by $3$ .

$x = \frac{3 y + \frac{5 y}{3} + \frac{- 45 - 25}{3}}{14}$

$5 y - 15 = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

Step 4.4.1.6.2

Subtract $25$ from $- 45$ .

$x = \frac{3 y + \frac{5 y}{3} + \frac{- 70}{3}}{14}$

$5 y - 15 = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

$x = \frac{3 y + \frac{5 y}{3} + \frac{- 70}{3}}{14}$

$5 y - 15 = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

Step 4.4.1.7

Move the negative in front of the fraction.

$x = \frac{3 y + \frac{5 y}{3} - \frac{70}{3}}{14}$

$5 y - 15 = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

Step 4.4.1.8

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

$x = \frac{3 y \cdot \frac{3}{3} + \frac{5 y}{3} - \frac{70}{3}}{14}$

$5 y - 15 = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

Step 4.4.1.9

Combine $3 y$ and $\frac{3}{3}$ .

$x = \frac{\frac{3 y \cdot 3}{3} + \frac{5 y}{3} - \frac{70}{3}}{14}$

$5 y - 15 = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

Step 4.4.1.10

Combine the numerators over the common denominator.

$x = \frac{\frac{3 y \cdot 3 + 5 y}{3} - \frac{70}{3}}{14}$

$5 y - 15 = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

Step 4.4.1.11

Combine the numerators over the common denominator.

$x = \frac{\frac{3 y \cdot 3 + 5 y - 70}{3}}{14}$

$5 y - 15 = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

Step 4.4.1.12

Multiply $3$ by $3$ .

$x = \frac{\frac{9 y + 5 y - 70}{3}}{14}$

$5 y - 15 = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

Step 4.4.1.13

Add $9 y$ and $5 y$ .

$x = \frac{\frac{14 y - 70}{3}}{14}$

$5 y - 15 = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

Step 4.4.1.14

Factor $14$ out of $14 y - 70$ .

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Step 4.4.1.14.1

Factor $14$ out of $14 y$ .

$x = \frac{\frac{14 (y) - 70}{3}}{14}$

$5 y - 15 = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

Step 4.4.1.14.2

Factor $14$ out of $- 70$ .

$x = \frac{\frac{14 y + 14 \cdot - 5}{3}}{14}$

$5 y - 15 = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

Step 4.4.1.14.3

Factor $14$ out of $14 y + 14 \cdot - 5$ .

$x = \frac{\frac{14 (y - 5)}{3}}{14}$

$5 y - 15 = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

$x = \frac{\frac{14 (y - 5)}{3}}{14}$

$5 y - 15 = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

Step 4.4.1.15

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{14 (y - 5)}{3} \cdot \frac{1}{14}$

$5 y - 15 = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

Step 4.4.1.16

Cancel the common factor of $14$ .

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Step 4.4.1.16.1

Cancel the common factor.

$x = \frac{14 (y - 5)}{3} \cdot \frac{1}{14}$

$5 y - 15 = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

Step 4.4.1.16.2

Rewrite the expression.

$x = \frac{y - 5}{3}$

$5 y - 15 = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

$x = \frac{y - 5}{3}$

$5 y - 15 = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

$x = \frac{y - 5}{3}$

$5 y - 15 = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

$x = \frac{y - 5}{3}$

$5 y - 15 = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

$x = \frac{y - 5}{3}$

$5 y - 15 = 10$

$z = - \frac{y}{3} + \frac{5}{3}$

Step 5

Solve for $y$ in $5 y - 15 = 10$ .

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Step 5.1

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

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Step 5.1.1

Add $15$ to both sides of the equation.

$5 y = 10 + 15$

$x = \frac{y - 5}{3}$

$z = - \frac{y}{3} + \frac{5}{3}$

Step 5.1.2

Add $10$ and $15$ .

$5 y = 25$

$x = \frac{y - 5}{3}$

$z = - \frac{y}{3} + \frac{5}{3}$

$5 y = 25$

$x = \frac{y - 5}{3}$

$z = - \frac{y}{3} + \frac{5}{3}$

Step 5.2

Divide each term in $5 y = 25$ by $5$ and simplify.

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Step 5.2.1

Divide each term in $5 y = 25$ by $5$ .

$\frac{5 y}{5} = \frac{25}{5}$

$x = \frac{y - 5}{3}$

$z = - \frac{y}{3} + \frac{5}{3}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $5$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{5 y}{5} = \frac{25}{5}$

$x = \frac{y - 5}{3}$

$z = - \frac{y}{3} + \frac{5}{3}$

Step 5.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{25}{5}$

$x = \frac{y - 5}{3}$

$z = - \frac{y}{3} + \frac{5}{3}$

$y = \frac{25}{5}$

$x = \frac{y - 5}{3}$

$z = - \frac{y}{3} + \frac{5}{3}$

$y = \frac{25}{5}$

$x = \frac{y - 5}{3}$

$z = - \frac{y}{3} + \frac{5}{3}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $25$ by $5$ .

$y = 5$

$x = \frac{y - 5}{3}$

$z = - \frac{y}{3} + \frac{5}{3}$

$y = 5$

$x = \frac{y - 5}{3}$

$z = - \frac{y}{3} + \frac{5}{3}$

$y = 5$

$x = \frac{y - 5}{3}$

$z = - \frac{y}{3} + \frac{5}{3}$

$y = 5$

$x = \frac{y - 5}{3}$

$z = - \frac{y}{3} + \frac{5}{3}$

Step 6

Replace all occurrences of $y$ with $5$ in each equation.

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Step 6.1

Replace all occurrences of $y$ in $x = \frac{y - 5}{3}$ with $5$ .

$x = \frac{(5) - 5}{3}$

$y = 5$

$z = - \frac{y}{3} + \frac{5}{3}$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $\frac{(5) - 5}{3}$ .

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Step 6.2.1.1

Subtract $5$ from $5$ .

$x = \frac{0}{3}$

$y = 5$

$z = - \frac{y}{3} + \frac{5}{3}$

Step 6.2.1.2

Divide $0$ by $3$ .

$x = 0$

$y = 5$

$z = - \frac{y}{3} + \frac{5}{3}$

$x = 0$

$y = 5$

$z = - \frac{y}{3} + \frac{5}{3}$

$x = 0$

$y = 5$

$z = - \frac{y}{3} + \frac{5}{3}$

Step 6.3

Replace all occurrences of $y$ in $z = - \frac{y}{3} + \frac{5}{3}$ with $5$ .

$z = - \frac{5}{3} + \frac{5}{3}$

$x = 0$

$y = 5$

Step 6.4

Simplify the right side.

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Step 6.4.1

Simplify $- \frac{5}{3} + \frac{5}{3}$ .

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Step 6.4.1.1

Combine the numerators over the common denominator.

$z = \frac{- 5 + 5}{3}$

$x = 0$

$y = 5$

Step 6.4.1.2

Simplify the expression.

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Step 6.4.1.2.1

Add $- 5$ and $5$ .

$z = \frac{0}{3}$

$x = 0$

$y = 5$

Step 6.4.1.2.2

Divide $0$ by $3$ .

$z = 0$

$x = 0$

$y = 5$

$z = 0$

$x = 0$

$y = 5$

$z = 0$

$x = 0$

$y = 5$

$z = 0$

$x = 0$

$y = 5$

$z = 0$

$x = 0$

$y = 5$

Step 7

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

$(0, 5, 0)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(0, 5, 0)$

Equation Form:

$x = 0, y = 5, z = 0$