Finite Math Examples

$3 x - 2 y = 5$ , $3 y + 8 z = 1$ , $5 x - 2 z = 4$

Step 1

Solve for $x$ in $3 x - 2 y = 5$ .

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Step 1.1

Add $2 y$ to both sides of the equation.

$3 x = 5 + 2 y$

$3 y + 8 z = 1$

$5 x - 2 z = 4$

Step 1.2

Divide each term in $3 x = 5 + 2 y$ by $3$ and simplify.

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Step 1.2.1

Divide each term in $3 x = 5 + 2 y$ by $3$ .

$\frac{3 x}{3} = \frac{5}{3} + \frac{2 y}{3}$

$3 y + 8 z = 1$

$5 x - 2 z = 4$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $3$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{5}{3} + \frac{2 y}{3}$

$3 y + 8 z = 1$

$5 x - 2 z = 4$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{5}{3} + \frac{2 y}{3}$

$3 y + 8 z = 1$

$5 x - 2 z = 4$

$x = \frac{5}{3} + \frac{2 y}{3}$

$3 y + 8 z = 1$

$5 x - 2 z = 4$

$x = \frac{5}{3} + \frac{2 y}{3}$

$3 y + 8 z = 1$

$5 x - 2 z = 4$

$x = \frac{5}{3} + \frac{2 y}{3}$

$3 y + 8 z = 1$

$5 x - 2 z = 4$

$x = \frac{5}{3} + \frac{2 y}{3}$

$3 y + 8 z = 1$

$5 x - 2 z = 4$

Step 2

Replace all occurrences of $x$ with $\frac{5}{3} + \frac{2 y}{3}$ in each equation.

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Step 2.1

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

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

$x = \frac{5}{3} + \frac{2 y}{3}$

$3 y + 8 z = 1$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Apply the distributive property.

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

$x = \frac{5}{3} + \frac{2 y}{3}$

$3 y + 8 z = 1$

Step 2.2.1.2

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

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Step 2.2.1.2.1

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

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

$x = \frac{5}{3} + \frac{2 y}{3}$

$3 y + 8 z = 1$

Step 2.2.1.2.2

Multiply $5$ by $5$ .

$\frac{25}{3} + 5 (\frac{2 y}{3}) - 2 z = 4$

$x = \frac{5}{3} + \frac{2 y}{3}$

$3 y + 8 z = 1$

$\frac{25}{3} + 5 (\frac{2 y}{3}) - 2 z = 4$

$x = \frac{5}{3} + \frac{2 y}{3}$

$3 y + 8 z = 1$

Step 2.2.1.3

Multiply $5 \frac{2 y}{3}$ .

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Step 2.2.1.3.1

Combine $5$ and $\frac{2 y}{3}$ .

$\frac{25}{3} + \frac{5 (2 y)}{3} - 2 z = 4$

$x = \frac{5}{3} + \frac{2 y}{3}$

$3 y + 8 z = 1$

Step 2.2.1.3.2

Multiply $2$ by $5$ .

$\frac{25}{3} + \frac{10 y}{3} - 2 z = 4$

$x = \frac{5}{3} + \frac{2 y}{3}$

$3 y + 8 z = 1$

$\frac{25}{3} + \frac{10 y}{3} - 2 z = 4$

$x = \frac{5}{3} + \frac{2 y}{3}$

$3 y + 8 z = 1$

$\frac{25}{3} + \frac{10 y}{3} - 2 z = 4$

$x = \frac{5}{3} + \frac{2 y}{3}$

$3 y + 8 z = 1$

$\frac{25}{3} + \frac{10 y}{3} - 2 z = 4$

$x = \frac{5}{3} + \frac{2 y}{3}$

$3 y + 8 z = 1$

$\frac{25}{3} + \frac{10 y}{3} - 2 z = 4$

$x = \frac{5}{3} + \frac{2 y}{3}$

$3 y + 8 z = 1$

Step 3

Reorder $\frac{5}{3}$ and $\frac{2 y}{3}$ .

$x = \frac{2 y}{3} + \frac{5}{3}$

$\frac{25}{3} + \frac{10 y}{3} - 2 z = 4$

$3 y + 8 z = 1$

Step 4

Solve for $z$ in $\frac{25}{3} + \frac{10 y}{3} - 2 z = 4$ .

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Step 4.1

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

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Step 4.1.1

Subtract $\frac{25}{3}$ from both sides of the equation.

$\frac{10 y}{3} - 2 z = 4 - \frac{25}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$3 y + 8 z = 1$

Step 4.1.2

Subtract $\frac{10 y}{3}$ from both sides of the equation.

$- 2 z = 4 - \frac{25}{3} - \frac{10 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$3 y + 8 z = 1$

Step 4.1.3

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

$- 2 z = 4 \cdot \frac{3}{3} - \frac{25}{3} - \frac{10 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$3 y + 8 z = 1$

Step 4.1.4

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

$- 2 z = \frac{4 \cdot 3}{3} - \frac{25}{3} - \frac{10 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$3 y + 8 z = 1$

Step 4.1.5

Combine the numerators over the common denominator.

$- 2 z = \frac{4 \cdot 3 - 25}{3} - \frac{10 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$3 y + 8 z = 1$

Step 4.1.6

Simplify the numerator.

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Step 4.1.6.1

Multiply $4$ by $3$ .

$- 2 z = \frac{12 - 25}{3} - \frac{10 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$3 y + 8 z = 1$

Step 4.1.6.2

Subtract $25$ from $12$ .

$- 2 z = \frac{- 13}{3} - \frac{10 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$3 y + 8 z = 1$

$- 2 z = \frac{- 13}{3} - \frac{10 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$3 y + 8 z = 1$

Step 4.1.7

Move the negative in front of the fraction.

$- 2 z = - \frac{13}{3} - \frac{10 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$3 y + 8 z = 1$

$- 2 z = - \frac{13}{3} - \frac{10 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$3 y + 8 z = 1$

Step 4.2

Divide each term in $- 2 z = - \frac{13}{3} - \frac{10 y}{3}$ by $- 2$ and simplify.

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Step 4.2.1

Divide each term in $- 2 z = - \frac{13}{3} - \frac{10 y}{3}$ by $- 2$ .

$\frac{- 2 z}{- 2} = \frac{- \frac{13}{3}}{- 2} + \frac{- \frac{10 y}{3}}{- 2}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$3 y + 8 z = 1$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $- 2$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{- 2 z}{- 2} = \frac{- \frac{13}{3}}{- 2} + \frac{- \frac{10 y}{3}}{- 2}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$3 y + 8 z = 1$

Step 4.2.2.1.2

Divide $z$ by $1$ .

$z = \frac{- \frac{13}{3}}{- 2} + \frac{- \frac{10 y}{3}}{- 2}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$3 y + 8 z = 1$

$z = \frac{- \frac{13}{3}}{- 2} + \frac{- \frac{10 y}{3}}{- 2}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$3 y + 8 z = 1$

$z = \frac{- \frac{13}{3}}{- 2} + \frac{- \frac{10 y}{3}}{- 2}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$3 y + 8 z = 1$

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

Multiply the numerator by the reciprocal of the denominator.

$z = - \frac{13}{3} \cdot \frac{1}{- 2} + \frac{- \frac{10 y}{3}}{- 2}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$3 y + 8 z = 1$

Step 4.2.3.1.2

Move the negative in front of the fraction.

$z = - \frac{13}{3} \cdot (- \frac{1}{2}) + \frac{- \frac{10 y}{3}}{- 2}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$3 y + 8 z = 1$

Step 4.2.3.1.3

Multiply $- \frac{13}{3} (- \frac{1}{2})$ .

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Step 4.2.3.1.3.1

Multiply $- 1$ by $- 1$ .

$z = 1 (\frac{13}{3}) \cdot \frac{1}{2} + \frac{- \frac{10 y}{3}}{- 2}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$3 y + 8 z = 1$

Step 4.2.3.1.3.2

Multiply $\frac{13}{3}$ by $1$ .

$z = \frac{13}{3} \cdot \frac{1}{2} + \frac{- \frac{10 y}{3}}{- 2}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$3 y + 8 z = 1$

Step 4.2.3.1.3.3

Multiply $\frac{13}{3}$ by $\frac{1}{2}$ .

$z = \frac{13}{3 \cdot 2} + \frac{- \frac{10 y}{3}}{- 2}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$3 y + 8 z = 1$

Step 4.2.3.1.3.4

Multiply $3$ by $2$ .

$z = \frac{13}{6} + \frac{- \frac{10 y}{3}}{- 2}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$3 y + 8 z = 1$

$z = \frac{13}{6} + \frac{- \frac{10 y}{3}}{- 2}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$3 y + 8 z = 1$

Step 4.2.3.1.4

Multiply the numerator by the reciprocal of the denominator.

$z = \frac{13}{6} - \frac{10 y}{3} \cdot \frac{1}{- 2}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$3 y + 8 z = 1$

Step 4.2.3.1.5

Cancel the common factor of $2$ .

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Step 4.2.3.1.5.1

Move the leading negative in $- \frac{10 y}{3}$ into the numerator.

$z = \frac{13}{6} + \frac{- 10 y}{3} \cdot \frac{1}{- 2}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$3 y + 8 z = 1$

Step 4.2.3.1.5.2

Factor $2$ out of $- 10 y$ .

$z = \frac{13}{6} + \frac{2 (- 5 y)}{3} \cdot \frac{1}{- 2}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$3 y + 8 z = 1$

Step 4.2.3.1.5.3

Factor $2$ out of $- 2$ .

$z = \frac{13}{6} + \frac{2 (- 5 y)}{3} \cdot \frac{1}{2 (- 1)}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$3 y + 8 z = 1$

Step 4.2.3.1.5.4

Cancel the common factor.

$z = \frac{13}{6} + \frac{2 (- 5 y)}{3} \cdot \frac{1}{2 \cdot - 1}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$3 y + 8 z = 1$

Step 4.2.3.1.5.5

Rewrite the expression.

$z = \frac{13}{6} + \frac{- 5 y}{3} \cdot \frac{1}{- 1}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$3 y + 8 z = 1$

$z = \frac{13}{6} + \frac{- 5 y}{3} \cdot \frac{1}{- 1}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$3 y + 8 z = 1$

Step 4.2.3.1.6

Multiply $\frac{- 5 y}{3}$ by $\frac{1}{- 1}$ .

$z = \frac{13}{6} + \frac{- 5 y}{3 \cdot - 1}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$3 y + 8 z = 1$

Step 4.2.3.1.7

Multiply $3$ by $- 1$ .

$z = \frac{13}{6} + \frac{- 5 y}{- 3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$3 y + 8 z = 1$

Step 4.2.3.1.8

Dividing two negative values results in a positive value.

$z = \frac{13}{6} + \frac{5 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$3 y + 8 z = 1$

$z = \frac{13}{6} + \frac{5 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$3 y + 8 z = 1$

$z = \frac{13}{6} + \frac{5 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$3 y + 8 z = 1$

$z = \frac{13}{6} + \frac{5 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$3 y + 8 z = 1$

$z = \frac{13}{6} + \frac{5 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$3 y + 8 z = 1$

Step 5

Replace all occurrences of $z$ with $\frac{13}{6} + \frac{5 y}{3}$ in each equation.

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Step 5.1

Replace all occurrences of $z$ in $3 y + 8 z = 1$ with $\frac{13}{6} + \frac{5 y}{3}$ .

$3 y + 8 (\frac{13}{6} + \frac{5 y}{3}) = 1$

$z = \frac{13}{6} + \frac{5 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Simplify $3 y + 8 (\frac{13}{6} + \frac{5 y}{3})$ .

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Step 5.2.1.1

Simplify each term.

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Step 5.2.1.1.1

Apply the distributive property.

$3 y + 8 (\frac{13}{6}) + 8 (\frac{5 y}{3}) = 1$

$z = \frac{13}{6} + \frac{5 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

Step 5.2.1.1.2

Cancel the common factor of $2$ .

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Step 5.2.1.1.2.1

Factor $2$ out of $8$ .

$3 y + 2 (4) (\frac{13}{6}) + 8 (\frac{5 y}{3}) = 1$

$z = \frac{13}{6} + \frac{5 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

Step 5.2.1.1.2.2

Factor $2$ out of $6$ .

$3 y + 2 \cdot (4 (\frac{13}{2 \cdot 3})) + 8 (\frac{5 y}{3}) = 1$

$z = \frac{13}{6} + \frac{5 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

Step 5.2.1.1.2.3

Cancel the common factor.

$3 y + 2 \cdot (4 (\frac{13}{2 \cdot 3})) + 8 (\frac{5 y}{3}) = 1$

$z = \frac{13}{6} + \frac{5 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

Step 5.2.1.1.2.4

Rewrite the expression.

$3 y + 4 (\frac{13}{3}) + 8 (\frac{5 y}{3}) = 1$

$z = \frac{13}{6} + \frac{5 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$3 y + 4 (\frac{13}{3}) + 8 (\frac{5 y}{3}) = 1$

$z = \frac{13}{6} + \frac{5 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

Step 5.2.1.1.3

Combine $4$ and $\frac{13}{3}$ .

$3 y + \frac{4 \cdot 13}{3} + 8 (\frac{5 y}{3}) = 1$

$z = \frac{13}{6} + \frac{5 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

Step 5.2.1.1.4

Multiply $4$ by $13$ .

$3 y + \frac{52}{3} + 8 (\frac{5 y}{3}) = 1$

$z = \frac{13}{6} + \frac{5 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

Step 5.2.1.1.5

Multiply $8 \frac{5 y}{3}$ .

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Step 5.2.1.1.5.1

Combine $8$ and $\frac{5 y}{3}$ .

$3 y + \frac{52}{3} + \frac{8 (5 y)}{3} = 1$

$z = \frac{13}{6} + \frac{5 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

Step 5.2.1.1.5.2

Multiply $5$ by $8$ .

$3 y + \frac{52}{3} + \frac{40 y}{3} = 1$

$z = \frac{13}{6} + \frac{5 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$3 y + \frac{52}{3} + \frac{40 y}{3} = 1$

$z = \frac{13}{6} + \frac{5 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$3 y + \frac{52}{3} + \frac{40 y}{3} = 1$

$z = \frac{13}{6} + \frac{5 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

Step 5.2.1.2

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

$3 y \cdot \frac{3}{3} + \frac{40 y}{3} + \frac{52}{3} = 1$

$z = \frac{13}{6} + \frac{5 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

Step 5.2.1.3

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

$\frac{3 y \cdot 3}{3} + \frac{40 y}{3} + \frac{52}{3} = 1$

$z = \frac{13}{6} + \frac{5 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

Step 5.2.1.4

Combine the numerators over the common denominator.

$\frac{3 y \cdot 3 + 40 y}{3} + \frac{52}{3} = 1$

$z = \frac{13}{6} + \frac{5 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

Step 5.2.1.5

Combine the numerators over the common denominator.

$\frac{3 y \cdot 3 + 40 y + 52}{3} = 1$

$z = \frac{13}{6} + \frac{5 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

Step 5.2.1.6

Multiply $3$ by $3$ .

$\frac{9 y + 40 y + 52}{3} = 1$

$z = \frac{13}{6} + \frac{5 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

Step 5.2.1.7

Add $9 y$ and $40 y$ .

$\frac{49 y + 52}{3} = 1$

$z = \frac{13}{6} + \frac{5 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$\frac{49 y + 52}{3} = 1$

$z = \frac{13}{6} + \frac{5 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$\frac{49 y + 52}{3} = 1$

$z = \frac{13}{6} + \frac{5 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$\frac{49 y + 52}{3} = 1$

$z = \frac{13}{6} + \frac{5 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

Step 6

Solve for $y$ in $\frac{49 y + 52}{3} = 1$ .

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Step 6.1

Multiply both sides by $3$ .

$\frac{49 y + 52}{3} \cdot 3 = 1 \cdot 3$

$z = \frac{13}{6} + \frac{5 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

Step 6.2

Simplify.

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Step 6.2.1

Simplify the left side.

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Step 6.2.1.1

Cancel the common factor of $3$ .

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Step 6.2.1.1.1

Cancel the common factor.

$\frac{49 y + 52}{3} \cdot 3 = 1 \cdot 3$

$z = \frac{13}{6} + \frac{5 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

Step 6.2.1.1.2

Rewrite the expression.

$49 y + 52 = 1 \cdot 3$

$z = \frac{13}{6} + \frac{5 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$49 y + 52 = 1 \cdot 3$

$z = \frac{13}{6} + \frac{5 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$49 y + 52 = 1 \cdot 3$

$z = \frac{13}{6} + \frac{5 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

Step 6.2.2

Simplify the right side.

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Step 6.2.2.1

Multiply $3$ by $1$ .

$49 y + 52 = 3$

$z = \frac{13}{6} + \frac{5 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$49 y + 52 = 3$

$z = \frac{13}{6} + \frac{5 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$49 y + 52 = 3$

$z = \frac{13}{6} + \frac{5 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

Step 6.3

Solve for $y$ .

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Step 6.3.1

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

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Step 6.3.1.1

Subtract $52$ from both sides of the equation.

$49 y = 3 - 52$

$z = \frac{13}{6} + \frac{5 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

Step 6.3.1.2

Subtract $52$ from $3$ .

$49 y = - 49$

$z = \frac{13}{6} + \frac{5 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$49 y = - 49$

$z = \frac{13}{6} + \frac{5 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

Step 6.3.2

Divide each term in $49 y = - 49$ by $49$ and simplify.

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Step 6.3.2.1

Divide each term in $49 y = - 49$ by $49$ .

$\frac{49 y}{49} = \frac{- 49}{49}$

$z = \frac{13}{6} + \frac{5 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

Step 6.3.2.2

Simplify the left side.

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Step 6.3.2.2.1

Cancel the common factor of $49$ .

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Step 6.3.2.2.1.1

Cancel the common factor.

$\frac{49 y}{49} = \frac{- 49}{49}$

$z = \frac{13}{6} + \frac{5 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

Step 6.3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 49}{49}$

$z = \frac{13}{6} + \frac{5 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$y = \frac{- 49}{49}$

$z = \frac{13}{6} + \frac{5 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$y = \frac{- 49}{49}$

$z = \frac{13}{6} + \frac{5 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

Step 6.3.2.3

Simplify the right side.

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Step 6.3.2.3.1

Divide $- 49$ by $49$ .

$y = - 1$

$z = \frac{13}{6} + \frac{5 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$y = - 1$

$z = \frac{13}{6} + \frac{5 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$y = - 1$

$z = \frac{13}{6} + \frac{5 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$y = - 1$

$z = \frac{13}{6} + \frac{5 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

$y = - 1$

$z = \frac{13}{6} + \frac{5 y}{3}$

$x = \frac{2 y}{3} + \frac{5}{3}$

Step 7

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

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Step 7.1

Replace all occurrences of $y$ in $z = \frac{13}{6} + \frac{5 y}{3}$ with $- 1$ .

$z = \frac{13}{6} + \frac{5 (- 1)}{3}$

$y = - 1$

$x = \frac{2 y}{3} + \frac{5}{3}$

Step 7.2

Simplify the right side.

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Step 7.2.1

Simplify $\frac{13}{6} + \frac{5 (- 1)}{3}$ .

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Step 7.2.1.1

Multiply $5$ by $- 1$ .

$z = \frac{13}{6} + \frac{- 5}{3}$

$y = - 1$

$x = \frac{2 y}{3} + \frac{5}{3}$

Step 7.2.1.2

Move the negative in front of the fraction.

$z = \frac{13}{6} - \frac{5}{3}$

$y = - 1$

$x = \frac{2 y}{3} + \frac{5}{3}$

Step 7.2.1.3

To write $- \frac{5}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$z = \frac{13}{6} - \frac{5}{3} \cdot \frac{2}{2}$

$y = - 1$

$x = \frac{2 y}{3} + \frac{5}{3}$

Step 7.2.1.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.2.1.4.1

Multiply $\frac{5}{3}$ by $\frac{2}{2}$ .

$z = \frac{13}{6} - \frac{5 \cdot 2}{3 \cdot 2}$

$y = - 1$

$x = \frac{2 y}{3} + \frac{5}{3}$

Step 7.2.1.4.2

Multiply $3$ by $2$ .

$z = \frac{13}{6} - \frac{5 \cdot 2}{6}$

$y = - 1$

$x = \frac{2 y}{3} + \frac{5}{3}$

$z = \frac{13}{6} - \frac{5 \cdot 2}{6}$

$y = - 1$

$x = \frac{2 y}{3} + \frac{5}{3}$

Step 7.2.1.5

Combine the numerators over the common denominator.

$z = \frac{13 - 5 \cdot 2}{6}$

$y = - 1$

$x = \frac{2 y}{3} + \frac{5}{3}$

Step 7.2.1.6

Simplify the numerator.

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Step 7.2.1.6.1

Multiply $- 5$ by $2$ .

$z = \frac{13 - 10}{6}$

$y = - 1$

$x = \frac{2 y}{3} + \frac{5}{3}$

Step 7.2.1.6.2

Subtract $10$ from $13$ .

$z = \frac{3}{6}$

$y = - 1$

$x = \frac{2 y}{3} + \frac{5}{3}$

$z = \frac{3}{6}$

$y = - 1$

$x = \frac{2 y}{3} + \frac{5}{3}$

Step 7.2.1.7

Cancel the common factor of $3$ and $6$ .

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Step 7.2.1.7.1

Factor $3$ out of $3$ .

$z = \frac{3 (1)}{6}$

$y = - 1$

$x = \frac{2 y}{3} + \frac{5}{3}$

Step 7.2.1.7.2

Cancel the common factors.

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Step 7.2.1.7.2.1

Factor $3$ out of $6$ .

$z = \frac{3 \cdot 1}{3 \cdot 2}$

$y = - 1$

$x = \frac{2 y}{3} + \frac{5}{3}$

Step 7.2.1.7.2.2

Cancel the common factor.

$z = \frac{3 \cdot 1}{3 \cdot 2}$

$y = - 1$

$x = \frac{2 y}{3} + \frac{5}{3}$

Step 7.2.1.7.2.3

Rewrite the expression.

$z = \frac{1}{2}$

$y = - 1$

$x = \frac{2 y}{3} + \frac{5}{3}$

$z = \frac{1}{2}$

$y = - 1$

$x = \frac{2 y}{3} + \frac{5}{3}$

$z = \frac{1}{2}$

$y = - 1$

$x = \frac{2 y}{3} + \frac{5}{3}$

$z = \frac{1}{2}$

$y = - 1$

$x = \frac{2 y}{3} + \frac{5}{3}$

$z = \frac{1}{2}$

$y = - 1$

$x = \frac{2 y}{3} + \frac{5}{3}$

Step 7.3

Replace all occurrences of $y$ in $x = \frac{2 y}{3} + \frac{5}{3}$ with $- 1$ .

$x = \frac{2 (- 1)}{3} + \frac{5}{3}$

$z = \frac{1}{2}$

$y = - 1$

Step 7.4

Simplify the right side.

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Step 7.4.1

Simplify $\frac{2 (- 1)}{3} + \frac{5}{3}$ .

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Step 7.4.1.1

Combine the numerators over the common denominator.

$x = \frac{2 (- 1) + 5}{3}$

$z = \frac{1}{2}$

$y = - 1$

Step 7.4.1.2

Simplify the expression.

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Step 7.4.1.2.1

Multiply $2$ by $- 1$ .

$x = \frac{- 2 + 5}{3}$

$z = \frac{1}{2}$

$y = - 1$

Step 7.4.1.2.2

Add $- 2$ and $5$ .

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

$z = \frac{1}{2}$

$y = - 1$

Step 7.4.1.2.3

Divide $3$ by $3$ .

$x = 1$

$z = \frac{1}{2}$

$y = - 1$

$x = 1$

$z = \frac{1}{2}$

$y = - 1$

$x = 1$

$z = \frac{1}{2}$

$y = - 1$

$x = 1$

$z = \frac{1}{2}$

$y = - 1$

$x = 1$

$z = \frac{1}{2}$

$y = - 1$

Step 8

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

$(1, - 1, \frac{1}{2})$

Step 9

The result can be shown in multiple forms.

Point Form:

$(1, - 1, \frac{1}{2})$

Equation Form:

$x = 1, y = - 1, z = \frac{1}{2}$