Finite Math Examples

$3 x + 4 y + 2 z = 0$ , $4 x - y + 5 z = 1$ , $7 x + 3 y + 3 z = 1$

Step 1

Solve for $x$ in $3 x + 4 y + 2 z = 0$ .

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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

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

$3 x + 2 z = - 4 y$

$4 x - y + 5 z = 1$

$7 x + 3 y + 3 z = 1$

Step 1.1.2

Subtract $2 z$ from both sides of the equation.

$3 x = - 4 y - 2 z$

$4 x - y + 5 z = 1$

$7 x + 3 y + 3 z = 1$

$3 x = - 4 y - 2 z$

$4 x - y + 5 z = 1$

$7 x + 3 y + 3 z = 1$

Step 1.2

Divide each term in $3 x = - 4 y - 2 z$ by $3$ and simplify.

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Step 1.2.1

Divide each term in $3 x = - 4 y - 2 z$ by $3$ .

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

$4 x - y + 5 z = 1$

$7 x + 3 y + 3 z = 1$

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.

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

$4 x - y + 5 z = 1$

$7 x + 3 y + 3 z = 1$

Step 1.2.2.1.2

Divide $x$ by $1$ .

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

$4 x - y + 5 z = 1$

$7 x + 3 y + 3 z = 1$

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

$4 x - y + 5 z = 1$

$7 x + 3 y + 3 z = 1$

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

$4 x - y + 5 z = 1$

$7 x + 3 y + 3 z = 1$

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.

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

$4 x - y + 5 z = 1$

$7 x + 3 y + 3 z = 1$

Step 1.2.3.1.2

Move the negative in front of the fraction.

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

$4 x - y + 5 z = 1$

$7 x + 3 y + 3 z = 1$

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

$4 x - y + 5 z = 1$

$7 x + 3 y + 3 z = 1$

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

$4 x - y + 5 z = 1$

$7 x + 3 y + 3 z = 1$

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

$4 x - y + 5 z = 1$

$7 x + 3 y + 3 z = 1$

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

$4 x - y + 5 z = 1$

$7 x + 3 y + 3 z = 1$

Step 2

Replace all occurrences of $x$ with $- \frac{4 y}{3} - \frac{2 z}{3}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $4 x - y + 5 z = 1$ with $- \frac{4 y}{3} - \frac{2 z}{3}$ .

$4 (- \frac{4 y}{3} - \frac{2 z}{3}) - y + 5 z = 1$

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

$7 x + 3 y + 3 z = 1$

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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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.

$4 (- \frac{4 y}{3}) + 4 (- \frac{2 z}{3}) - y + 5 z = 1$

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

$7 x + 3 y + 3 z = 1$

Step 2.2.1.1.2

Multiply $4 (- \frac{4 y}{3})$ .

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Step 2.2.1.1.2.1

Multiply $- 1$ by $4$ .

$- 4 \frac{4 y}{3} + 4 (- \frac{2 z}{3}) - y + 5 z = 1$

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

$7 x + 3 y + 3 z = 1$

Step 2.2.1.1.2.2

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

$\frac{- 4 (4 y)}{3} + 4 (- \frac{2 z}{3}) - y + 5 z = 1$

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

$7 x + 3 y + 3 z = 1$

Step 2.2.1.1.2.3

Multiply $4$ by $- 4$ .

$\frac{- 16 y}{3} + 4 (- \frac{2 z}{3}) - y + 5 z = 1$

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

$7 x + 3 y + 3 z = 1$

$\frac{- 16 y}{3} + 4 (- \frac{2 z}{3}) - y + 5 z = 1$

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

$7 x + 3 y + 3 z = 1$

Step 2.2.1.1.3

Multiply $4 (- \frac{2 z}{3})$ .

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Step 2.2.1.1.3.1

Multiply $- 1$ by $4$ .

$\frac{- 16 y}{3} - 4 \frac{2 z}{3} - y + 5 z = 1$

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

$7 x + 3 y + 3 z = 1$

Step 2.2.1.1.3.2

Combine $- 4$ and $\frac{2 z}{3}$ .

$\frac{- 16 y}{3} + \frac{- 4 (2 z)}{3} - y + 5 z = 1$

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

$7 x + 3 y + 3 z = 1$

Step 2.2.1.1.3.3

Multiply $2$ by $- 4$ .

$\frac{- 16 y}{3} + \frac{- 8 z}{3} - y + 5 z = 1$

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

$7 x + 3 y + 3 z = 1$

$\frac{- 16 y}{3} + \frac{- 8 z}{3} - y + 5 z = 1$

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

$7 x + 3 y + 3 z = 1$

Step 2.2.1.1.4

Simplify each term.

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Step 2.2.1.1.4.1

Move the negative in front of the fraction.

$- \frac{16 y}{3} + \frac{- 8 z}{3} - y + 5 z = 1$

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

$7 x + 3 y + 3 z = 1$

Step 2.2.1.1.4.2

Move the negative in front of the fraction.

$- \frac{16 y}{3} - \frac{8 z}{3} - y + 5 z = 1$

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

$7 x + 3 y + 3 z = 1$

$- \frac{16 y}{3} - \frac{8 z}{3} - y + 5 z = 1$

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

$7 x + 3 y + 3 z = 1$

$- \frac{16 y}{3} - \frac{8 z}{3} - y + 5 z = 1$

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

$7 x + 3 y + 3 z = 1$

Step 2.2.1.2

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

$- \frac{8 z}{3} - \frac{16 y}{3} - y \cdot \frac{3}{3} + 5 z = 1$

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

$7 x + 3 y + 3 z = 1$

Step 2.2.1.3

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

$- \frac{8 z}{3} - \frac{16 y}{3} + \frac{- y \cdot 3}{3} + 5 z = 1$

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

$7 x + 3 y + 3 z = 1$

Step 2.2.1.4

Combine the numerators over the common denominator.

$- \frac{8 z}{3} + \frac{- 16 y - y \cdot 3}{3} + 5 z = 1$

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

$7 x + 3 y + 3 z = 1$

Step 2.2.1.5

Combine the numerators over the common denominator.

$5 z + \frac{- 8 z - 16 y - y \cdot 3}{3} = 1$

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

$7 x + 3 y + 3 z = 1$

Step 2.2.1.6

Multiply $3$ by $- 1$ .

$5 z + \frac{- 8 z - 16 y - 3 y}{3} = 1$

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

$7 x + 3 y + 3 z = 1$

Step 2.2.1.7

Subtract $3 y$ from $- 16 y$ .

$5 z + \frac{- 8 z - 19 y}{3} = 1$

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

$7 x + 3 y + 3 z = 1$

Step 2.2.1.8

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

$5 z \cdot \frac{3}{3} + \frac{- 8 z - 19 y}{3} = 1$

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

$7 x + 3 y + 3 z = 1$

Step 2.2.1.9

Simplify terms.

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Step 2.2.1.9.1

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

$\frac{5 z \cdot 3}{3} + \frac{- 8 z - 19 y}{3} = 1$

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

$7 x + 3 y + 3 z = 1$

Step 2.2.1.9.2

Combine the numerators over the common denominator.

$\frac{5 z \cdot 3 - 8 z - 19 y}{3} = 1$

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

$7 x + 3 y + 3 z = 1$

$\frac{5 z \cdot 3 - 8 z - 19 y}{3} = 1$

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

$7 x + 3 y + 3 z = 1$

Step 2.2.1.10

Simplify the numerator.

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Step 2.2.1.10.1

Multiply $3$ by $5$ .

$\frac{15 z - 8 z - 19 y}{3} = 1$

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

$7 x + 3 y + 3 z = 1$

Step 2.2.1.10.2

Subtract $8 z$ from $15 z$ .

$\frac{7 z - 19 y}{3} = 1$

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

$7 x + 3 y + 3 z = 1$

$\frac{7 z - 19 y}{3} = 1$

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

$7 x + 3 y + 3 z = 1$

$\frac{7 z - 19 y}{3} = 1$

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

$7 x + 3 y + 3 z = 1$

$\frac{7 z - 19 y}{3} = 1$

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

$7 x + 3 y + 3 z = 1$

Step 2.3

Replace all occurrences of $x$ in $7 x + 3 y + 3 z = 1$ with $- \frac{4 y}{3} - \frac{2 z}{3}$ .

$7 (- \frac{4 y}{3} - \frac{2 z}{3}) + 3 y + 3 z = 1$

$\frac{7 z - 19 y}{3} = 1$

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

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $7 (- \frac{4 y}{3} - \frac{2 z}{3}) + 3 y + 3 z$ .

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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.

$7 (- \frac{4 y}{3}) + 7 (- \frac{2 z}{3}) + 3 y + 3 z = 1$

$\frac{7 z - 19 y}{3} = 1$

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

Step 2.4.1.1.2

Multiply $7 (- \frac{4 y}{3})$ .

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Step 2.4.1.1.2.1

Multiply $- 1$ by $7$ .

$- 7 \frac{4 y}{3} + 7 (- \frac{2 z}{3}) + 3 y + 3 z = 1$

$\frac{7 z - 19 y}{3} = 1$

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

Step 2.4.1.1.2.2

Combine $- 7$ and $\frac{4 y}{3}$ .

$\frac{- 7 (4 y)}{3} + 7 (- \frac{2 z}{3}) + 3 y + 3 z = 1$

$\frac{7 z - 19 y}{3} = 1$

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

Step 2.4.1.1.2.3

Multiply $4$ by $- 7$ .

$\frac{- 28 y}{3} + 7 (- \frac{2 z}{3}) + 3 y + 3 z = 1$

$\frac{7 z - 19 y}{3} = 1$

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

$\frac{- 28 y}{3} + 7 (- \frac{2 z}{3}) + 3 y + 3 z = 1$

$\frac{7 z - 19 y}{3} = 1$

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

Step 2.4.1.1.3

Multiply $7 (- \frac{2 z}{3})$ .

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Step 2.4.1.1.3.1

Multiply $- 1$ by $7$ .

$\frac{- 28 y}{3} - 7 \frac{2 z}{3} + 3 y + 3 z = 1$

$\frac{7 z - 19 y}{3} = 1$

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

Step 2.4.1.1.3.2

Combine $- 7$ and $\frac{2 z}{3}$ .

$\frac{- 28 y}{3} + \frac{- 7 (2 z)}{3} + 3 y + 3 z = 1$

$\frac{7 z - 19 y}{3} = 1$

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

Step 2.4.1.1.3.3

Multiply $2$ by $- 7$ .

$\frac{- 28 y}{3} + \frac{- 14 z}{3} + 3 y + 3 z = 1$

$\frac{7 z - 19 y}{3} = 1$

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

$\frac{- 28 y}{3} + \frac{- 14 z}{3} + 3 y + 3 z = 1$

$\frac{7 z - 19 y}{3} = 1$

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

Step 2.4.1.1.4

Simplify each term.

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Step 2.4.1.1.4.1

Move the negative in front of the fraction.

$- \frac{28 y}{3} + \frac{- 14 z}{3} + 3 y + 3 z = 1$

$\frac{7 z - 19 y}{3} = 1$

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

Step 2.4.1.1.4.2

Move the negative in front of the fraction.

$- \frac{28 y}{3} - \frac{14 z}{3} + 3 y + 3 z = 1$

$\frac{7 z - 19 y}{3} = 1$

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

$- \frac{28 y}{3} - \frac{14 z}{3} + 3 y + 3 z = 1$

$\frac{7 z - 19 y}{3} = 1$

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

$- \frac{28 y}{3} - \frac{14 z}{3} + 3 y + 3 z = 1$

$\frac{7 z - 19 y}{3} = 1$

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

Step 2.4.1.2

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

$- \frac{14 z}{3} - \frac{28 y}{3} + 3 y \cdot \frac{3}{3} + 3 z = 1$

$\frac{7 z - 19 y}{3} = 1$

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

Step 2.4.1.3

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

$- \frac{14 z}{3} - \frac{28 y}{3} + \frac{3 y \cdot 3}{3} + 3 z = 1$

$\frac{7 z - 19 y}{3} = 1$

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

Step 2.4.1.4

Combine the numerators over the common denominator.

$- \frac{14 z}{3} + \frac{- 28 y + 3 y \cdot 3}{3} + 3 z = 1$

$\frac{7 z - 19 y}{3} = 1$

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

Step 2.4.1.5

Combine the numerators over the common denominator.

$3 z + \frac{- 14 z - 28 y + 3 y \cdot 3}{3} = 1$

$\frac{7 z - 19 y}{3} = 1$

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

Step 2.4.1.6

Multiply $3$ by $3$ .

$3 z + \frac{- 14 z - 28 y + 9 y}{3} = 1$

$\frac{7 z - 19 y}{3} = 1$

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

Step 2.4.1.7

Add $- 28 y$ and $9 y$ .

$3 z + \frac{- 14 z - 19 y}{3} = 1$

$\frac{7 z - 19 y}{3} = 1$

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

Step 2.4.1.8

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

$3 z \cdot \frac{3}{3} + \frac{- 14 z - 19 y}{3} = 1$

$\frac{7 z - 19 y}{3} = 1$

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

Step 2.4.1.9

Simplify terms.

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Step 2.4.1.9.1

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

$\frac{3 z \cdot 3}{3} + \frac{- 14 z - 19 y}{3} = 1$

$\frac{7 z - 19 y}{3} = 1$

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

Step 2.4.1.9.2

Combine the numerators over the common denominator.

$\frac{3 z \cdot 3 - 14 z - 19 y}{3} = 1$

$\frac{7 z - 19 y}{3} = 1$

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

$\frac{3 z \cdot 3 - 14 z - 19 y}{3} = 1$

$\frac{7 z - 19 y}{3} = 1$

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

Step 2.4.1.10

Simplify the numerator.

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Step 2.4.1.10.1

Multiply $3$ by $3$ .

$\frac{9 z - 14 z - 19 y}{3} = 1$

$\frac{7 z - 19 y}{3} = 1$

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

Step 2.4.1.10.2

Subtract $14 z$ from $9 z$ .

$\frac{- 5 z - 19 y}{3} = 1$

$\frac{7 z - 19 y}{3} = 1$

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

$\frac{- 5 z - 19 y}{3} = 1$

$\frac{7 z - 19 y}{3} = 1$

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

Step 2.4.1.11

Simplify with factoring out.

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Step 2.4.1.11.1

Factor $- 1$ out of $- 5 z$ .

$\frac{- (5 z) - 19 y}{3} = 1$

$\frac{7 z - 19 y}{3} = 1$

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

Step 2.4.1.11.2

Factor $- 1$ out of $- 19 y$ .

$\frac{- (5 z) - (19 y)}{3} = 1$

$\frac{7 z - 19 y}{3} = 1$

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

Step 2.4.1.11.3

Factor $- 1$ out of $- (5 z) - (19 y)$ .

$\frac{- (5 z + 19 y)}{3} = 1$

$\frac{7 z - 19 y}{3} = 1$

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

Step 2.4.1.11.4

Simplify the expression.

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Step 2.4.1.11.4.1

Rewrite $- (5 z + 19 y)$ as $- 1 (5 z + 19 y)$ .

$\frac{- 1 (5 z + 19 y)}{3} = 1$

$\frac{7 z - 19 y}{3} = 1$

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

Step 2.4.1.11.4.2

Move the negative in front of the fraction.

$- \frac{5 z + 19 y}{3} = 1$

$\frac{7 z - 19 y}{3} = 1$

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

$- \frac{5 z + 19 y}{3} = 1$

$\frac{7 z - 19 y}{3} = 1$

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

$- \frac{5 z + 19 y}{3} = 1$

$\frac{7 z - 19 y}{3} = 1$

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

$- \frac{5 z + 19 y}{3} = 1$

$\frac{7 z - 19 y}{3} = 1$

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

$- \frac{5 z + 19 y}{3} = 1$

$\frac{7 z - 19 y}{3} = 1$

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

$- \frac{5 z + 19 y}{3} = 1$

$\frac{7 z - 19 y}{3} = 1$

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

Step 3

Solve for $z$ in $\frac{7 z - 19 y}{3} = 1$ .

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Step 3.1

Multiply both sides by $3$ .

$\frac{7 z - 19 y}{3} \cdot 3 = 1 \cdot 3$

$- \frac{5 z + 19 y}{3} = 1$

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

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 $\frac{7 z - 19 y}{3} \cdot 3$ .

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Step 3.2.1.1.1

Cancel the common factor of $3$ .

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Step 3.2.1.1.1.1

Cancel the common factor.

$\frac{7 z - 19 y}{3} \cdot 3 = 1 \cdot 3$

$- \frac{5 z + 19 y}{3} = 1$

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

Step 3.2.1.1.1.2

Rewrite the expression.

$7 z - 19 y = 1 \cdot 3$

$- \frac{5 z + 19 y}{3} = 1$

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

$7 z - 19 y = 1 \cdot 3$

$- \frac{5 z + 19 y}{3} = 1$

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

Step 3.2.1.1.2

Reorder $7 z$ and $- 19 y$ .

$- 19 y + 7 z = 1 \cdot 3$

$- \frac{5 z + 19 y}{3} = 1$

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

$- 19 y + 7 z = 1 \cdot 3$

$- \frac{5 z + 19 y}{3} = 1$

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

$- 19 y + 7 z = 1 \cdot 3$

$- \frac{5 z + 19 y}{3} = 1$

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Multiply $3$ by $1$ .

$- 19 y + 7 z = 3$

$- \frac{5 z + 19 y}{3} = 1$

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

$- 19 y + 7 z = 3$

$- \frac{5 z + 19 y}{3} = 1$

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

$- 19 y + 7 z = 3$

$- \frac{5 z + 19 y}{3} = 1$

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

Step 3.3

Solve for $z$ .

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Step 3.3.1

Add $19 y$ to both sides of the equation.

$7 z = 3 + 19 y$

$- \frac{5 z + 19 y}{3} = 1$

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

Step 3.3.2

Divide each term in $7 z = 3 + 19 y$ by $7$ and simplify.

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Step 3.3.2.1

Divide each term in $7 z = 3 + 19 y$ by $7$ .

$\frac{7 z}{7} = \frac{3}{7} + \frac{19 y}{7}$

$- \frac{5 z + 19 y}{3} = 1$

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

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Cancel the common factor of $7$ .

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Step 3.3.2.2.1.1

Cancel the common factor.

$\frac{7 z}{7} = \frac{3}{7} + \frac{19 y}{7}$

$- \frac{5 z + 19 y}{3} = 1$

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

Step 3.3.2.2.1.2

Divide $z$ by $1$ .

$z = \frac{3}{7} + \frac{19 y}{7}$

$- \frac{5 z + 19 y}{3} = 1$

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

$z = \frac{3}{7} + \frac{19 y}{7}$

$- \frac{5 z + 19 y}{3} = 1$

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

$z = \frac{3}{7} + \frac{19 y}{7}$

$- \frac{5 z + 19 y}{3} = 1$

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

$z = \frac{3}{7} + \frac{19 y}{7}$

$- \frac{5 z + 19 y}{3} = 1$

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

$z = \frac{3}{7} + \frac{19 y}{7}$

$- \frac{5 z + 19 y}{3} = 1$

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

$z = \frac{3}{7} + \frac{19 y}{7}$

$- \frac{5 z + 19 y}{3} = 1$

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

Step 4

Replace all occurrences of $z$ with $\frac{3}{7} + \frac{19 y}{7}$ in each equation.

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Step 4.1

Replace all occurrences of $z$ in $- \frac{5 z + 19 y}{3} = 1$ with $\frac{3}{7} + \frac{19 y}{7}$ .

$- \frac{5 (\frac{3}{7} + \frac{19 y}{7}) + 19 y}{3} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

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

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $- \frac{5 (\frac{3}{7} + \frac{19 y}{7}) + 19 y}{3}$ .

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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.

$- \frac{5 (\frac{3}{7}) + 5 (\frac{19 y}{7}) + 19 y}{3} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

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

Step 4.2.1.1.2

Multiply $5 (\frac{3}{7})$ .

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Step 4.2.1.1.2.1

Combine $5$ and $\frac{3}{7}$ .

$- \frac{\frac{5 \cdot 3}{7} + 5 (\frac{19 y}{7}) + 19 y}{3} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

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

Step 4.2.1.1.2.2

Multiply $5$ by $3$ .

$- \frac{\frac{15}{7} + 5 (\frac{19 y}{7}) + 19 y}{3} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

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

$- \frac{\frac{15}{7} + 5 (\frac{19 y}{7}) + 19 y}{3} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

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

Step 4.2.1.1.3

Multiply $5 \frac{19 y}{7}$ .

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Step 4.2.1.1.3.1

Combine $5$ and $\frac{19 y}{7}$ .

$- \frac{\frac{15}{7} + \frac{5 (19 y)}{7} + 19 y}{3} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

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

Step 4.2.1.1.3.2

Multiply $19$ by $5$ .

$- \frac{\frac{15}{7} + \frac{95 y}{7} + 19 y}{3} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

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

$- \frac{\frac{15}{7} + \frac{95 y}{7} + 19 y}{3} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

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

Step 4.2.1.1.4

To write $19 y$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$- \frac{\frac{15}{7} + \frac{95 y}{7} + 19 y \cdot \frac{7}{7}}{3} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

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

Step 4.2.1.1.5

Combine $19 y$ and $\frac{7}{7}$ .

$- \frac{\frac{15}{7} + \frac{95 y}{7} + \frac{19 y \cdot 7}{7}}{3} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

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

Step 4.2.1.1.6

Combine the numerators over the common denominator.

$- \frac{\frac{15}{7} + \frac{95 y + 19 y \cdot 7}{7}}{3} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

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

Step 4.2.1.1.7

Combine the numerators over the common denominator.

$- \frac{\frac{15 + 95 y + 19 y \cdot 7}{7}}{3} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

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

Step 4.2.1.1.8

Reorder terms.

$- \frac{\frac{7 \cdot (19 y) + 95 y + 15}{7}}{3} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

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

Step 4.2.1.1.9

Rewrite $\frac{7 \cdot 19 y + 95 y + 15}{7}$ in a factored form.

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Step 4.2.1.1.9.1

Multiply $7$ by $19$ .

$- \frac{\frac{133 y + 95 y + 15}{7}}{3} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

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

Step 4.2.1.1.9.2

Add $133 y$ and $95 y$ .

$- \frac{\frac{228 y + 15}{7}}{3} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

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

Step 4.2.1.1.9.3

Factor $3$ out of $228 y + 15$ .

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Step 4.2.1.1.9.3.1

Factor $3$ out of $228 y$ .

$- \frac{\frac{3 (76 y) + 15}{7}}{3} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

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

Step 4.2.1.1.9.3.2

Factor $3$ out of $15$ .

$- \frac{\frac{3 (76 y) + 3 (5)}{7}}{3} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

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

Step 4.2.1.1.9.3.3

Factor $3$ out of $3 (76 y) + 3 (5)$ .

$- \frac{\frac{3 (76 y + 5)}{7}}{3} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

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

$- \frac{\frac{3 (76 y + 5)}{7}}{3} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

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

$- \frac{\frac{3 (76 y + 5)}{7}}{3} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

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

$- \frac{\frac{3 (76 y + 5)}{7}}{3} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

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

Step 4.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$- (\frac{3 (76 y + 5)}{7} \cdot \frac{1}{3}) = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

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

Step 4.2.1.3

Cancel the common factor of $3$ .

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Step 4.2.1.3.1

Cancel the common factor.

$- (\frac{3 (76 y + 5)}{7} \cdot \frac{1}{3}) = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

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

Step 4.2.1.3.2

Rewrite the expression.

$- \frac{76 y + 5}{7} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

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

$- \frac{76 y + 5}{7} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

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

$- \frac{76 y + 5}{7} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

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

$- \frac{76 y + 5}{7} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

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

Step 4.3

Replace all occurrences of $z$ in $x = - \frac{4 y}{3} - \frac{2 z}{3}$ with $\frac{3}{7} + \frac{19 y}{7}$ .

$x = - \frac{4 y}{3} - \frac{2 (\frac{3}{7} + \frac{19 y}{7})}{3}$

$- \frac{76 y + 5}{7} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $- \frac{4 y}{3} - \frac{2 (\frac{3}{7} + \frac{19 y}{7})}{3}$ .

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Step 4.4.1.1

Combine the numerators over the common denominator.

$x = \frac{- 4 y - 2 (\frac{3}{7} + \frac{19 y}{7})}{3}$

$- \frac{76 y + 5}{7} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 4.4.1.2

Simplify each term.

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Step 4.4.1.2.1

Apply the distributive property.

$x = \frac{- 4 y - 2 (\frac{3}{7}) - 2 \frac{19 y}{7}}{3}$

$- \frac{76 y + 5}{7} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 4.4.1.2.2

Multiply $- 2 (\frac{3}{7})$ .

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Step 4.4.1.2.2.1

Combine $- 2$ and $\frac{3}{7}$ .

$x = \frac{- 4 y + \frac{- 2 \cdot 3}{7} - 2 \frac{19 y}{7}}{3}$

$- \frac{76 y + 5}{7} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 4.4.1.2.2.2

Multiply $- 2$ by $3$ .

$x = \frac{- 4 y + \frac{- 6}{7} - 2 \frac{19 y}{7}}{3}$

$- \frac{76 y + 5}{7} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

$x = \frac{- 4 y + \frac{- 6}{7} - 2 \frac{19 y}{7}}{3}$

$- \frac{76 y + 5}{7} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 4.4.1.2.3

Multiply $- 2 \frac{19 y}{7}$ .

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Step 4.4.1.2.3.1

Combine $- 2$ and $\frac{19 y}{7}$ .

$x = \frac{- 4 y + \frac{- 6}{7} + \frac{- 2 (19 y)}{7}}{3}$

$- \frac{76 y + 5}{7} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 4.4.1.2.3.2

Multiply $19$ by $- 2$ .

$x = \frac{- 4 y + \frac{- 6}{7} + \frac{- 38 y}{7}}{3}$

$- \frac{76 y + 5}{7} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

$x = \frac{- 4 y + \frac{- 6}{7} + \frac{- 38 y}{7}}{3}$

$- \frac{76 y + 5}{7} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 4.4.1.2.4

Simplify each term.

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Step 4.4.1.2.4.1

Move the negative in front of the fraction.

$x = \frac{- 4 y - \frac{6}{7} + \frac{- 38 y}{7}}{3}$

$- \frac{76 y + 5}{7} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 4.4.1.2.4.2

Move the negative in front of the fraction.

$x = \frac{- 4 y - \frac{6}{7} - \frac{38 y}{7}}{3}$

$- \frac{76 y + 5}{7} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

$x = \frac{- 4 y - \frac{6}{7} - \frac{38 y}{7}}{3}$

$- \frac{76 y + 5}{7} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

$x = \frac{- 4 y - \frac{6}{7} - \frac{38 y}{7}}{3}$

$- \frac{76 y + 5}{7} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 4.4.1.3

To write $- 4 y$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$x = \frac{- 4 y \cdot \frac{7}{7} - \frac{38 y}{7} - \frac{6}{7}}{3}$

$- \frac{76 y + 5}{7} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 4.4.1.4

Combine $- 4 y$ and $\frac{7}{7}$ .

$x = \frac{\frac{- 4 y \cdot 7}{7} - \frac{38 y}{7} - \frac{6}{7}}{3}$

$- \frac{76 y + 5}{7} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 4.4.1.5

Combine the numerators over the common denominator.

$x = \frac{\frac{- 4 y \cdot 7 - 38 y}{7} - \frac{6}{7}}{3}$

$- \frac{76 y + 5}{7} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 4.4.1.6

Combine the numerators over the common denominator.

$x = \frac{\frac{- 4 y \cdot 7 - 38 y - 6}{7}}{3}$

$- \frac{76 y + 5}{7} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 4.4.1.7

Multiply $7$ by $- 4$ .

$x = \frac{\frac{- 28 y - 38 y - 6}{7}}{3}$

$- \frac{76 y + 5}{7} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 4.4.1.8

Subtract $38 y$ from $- 28 y$ .

$x = \frac{\frac{- 66 y - 6}{7}}{3}$

$- \frac{76 y + 5}{7} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 4.4.1.9

Factor $6$ out of $- 66 y - 6$ .

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Step 4.4.1.9.1

Factor $6$ out of $- 66 y$ .

$x = \frac{\frac{6 (- 11 y) - 6}{7}}{3}$

$- \frac{76 y + 5}{7} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 4.4.1.9.2

Factor $6$ out of $- 6$ .

$x = \frac{\frac{6 (- 11 y) + 6 (- 1)}{7}}{3}$

$- \frac{76 y + 5}{7} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 4.4.1.9.3

Factor $6$ out of $6 (- 11 y) + 6 (- 1)$ .

$x = \frac{\frac{6 (- 11 y - 1)}{7}}{3}$

$- \frac{76 y + 5}{7} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

$x = \frac{\frac{6 (- 11 y - 1)}{7}}{3}$

$- \frac{76 y + 5}{7} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 4.4.1.10

Factor $- 1$ out of $- 11 y$ .

$x = \frac{\frac{6 (- (11 y) - 1)}{7}}{3}$

$- \frac{76 y + 5}{7} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 4.4.1.11

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

$x = \frac{\frac{6 (- (11 y) - 1 \cdot 1)}{7}}{3}$

$- \frac{76 y + 5}{7} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 4.4.1.12

Factor $- 1$ out of $- (11 y) - 1 (1)$ .

$x = \frac{\frac{6 (- (11 y + 1))}{7}}{3}$

$- \frac{76 y + 5}{7} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 4.4.1.13

Simplify the expression.

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Step 4.4.1.13.1

Rewrite $- (11 y + 1)$ as $- 1 (11 y + 1)$ .

$x = \frac{\frac{6 (- 1 (11 y + 1))}{7}}{3}$

$- \frac{76 y + 5}{7} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 4.4.1.13.2

Move the negative in front of the fraction.

$x = \frac{- \frac{6 (11 y + 1)}{7}}{3}$

$- \frac{76 y + 5}{7} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

$x = \frac{- \frac{6 (11 y + 1)}{7}}{3}$

$- \frac{76 y + 5}{7} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 4.4.1.14

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{6 (11 y + 1)}{7} \cdot \frac{1}{3}$

$- \frac{76 y + 5}{7} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 4.4.1.15

Cancel the common factor of $3$ .

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Step 4.4.1.15.1

Move the leading negative in $- \frac{6 (11 y + 1)}{7}$ into the numerator.

$x = \frac{- 6 (11 y + 1)}{7} \cdot \frac{1}{3}$

$- \frac{76 y + 5}{7} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 4.4.1.15.2

Factor $3$ out of $- 6 (11 y + 1)$ .

$x = \frac{3 (- 2 (11 y + 1))}{7} \cdot \frac{1}{3}$

$- \frac{76 y + 5}{7} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 4.4.1.15.3

Cancel the common factor.

$x = \frac{3 (- 2 (11 y + 1))}{7} \cdot \frac{1}{3}$

$- \frac{76 y + 5}{7} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 4.4.1.15.4

Rewrite the expression.

$x = \frac{- 2 (11 y + 1)}{7}$

$- \frac{76 y + 5}{7} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

$x = \frac{- 2 (11 y + 1)}{7}$

$- \frac{76 y + 5}{7} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 4.4.1.16

Move the negative in front of the fraction.

$x = - \frac{2 (11 y + 1)}{7}$

$- \frac{76 y + 5}{7} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

$x = - \frac{2 (11 y + 1)}{7}$

$- \frac{76 y + 5}{7} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

$x = - \frac{2 (11 y + 1)}{7}$

$- \frac{76 y + 5}{7} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

$x = - \frac{2 (11 y + 1)}{7}$

$- \frac{76 y + 5}{7} = 1$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 5

Solve for $y$ in $- \frac{76 y + 5}{7} = 1$ .

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Step 5.1

Multiply both sides of the equation by $- 7$ .

$- 7 (- \frac{76 y + 5}{7}) = - 7 \cdot 1$

$x = - \frac{2 (11 y + 1)}{7}$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 5.2

Simplify both sides of the equation.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

Simplify $- 7 (- \frac{76 y + 5}{7})$ .

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Step 5.2.1.1.1

Cancel the common factor of $7$ .

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Step 5.2.1.1.1.1

Move the leading negative in $- \frac{76 y + 5}{7}$ into the numerator.

$- 7 \frac{- (76 y + 5)}{7} = - 7 \cdot 1$

$x = - \frac{2 (11 y + 1)}{7}$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 5.2.1.1.1.2

Factor $7$ out of $- 7$ .

$7 (- 1) (\frac{- (76 y + 5)}{7}) = - 7 \cdot 1$

$x = - \frac{2 (11 y + 1)}{7}$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 5.2.1.1.1.3

Cancel the common factor.

$7 \cdot (- 1 \frac{- (76 y + 5)}{7}) = - 7 \cdot 1$

$x = - \frac{2 (11 y + 1)}{7}$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 5.2.1.1.1.4

Rewrite the expression.

$76 y + 5 = - 7 \cdot 1$

$x = - \frac{2 (11 y + 1)}{7}$

$z = \frac{3}{7} + \frac{19 y}{7}$

$76 y + 5 = - 7 \cdot 1$

$x = - \frac{2 (11 y + 1)}{7}$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 5.2.1.1.2

Multiply.

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Step 5.2.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 (76 y + 5) = - 7 \cdot 1$

$x = - \frac{2 (11 y + 1)}{7}$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 5.2.1.1.2.2

Multiply $76 y + 5$ by $1$ .

$76 y + 5 = - 7 \cdot 1$

$x = - \frac{2 (11 y + 1)}{7}$

$z = \frac{3}{7} + \frac{19 y}{7}$

$76 y + 5 = - 7 \cdot 1$

$x = - \frac{2 (11 y + 1)}{7}$

$z = \frac{3}{7} + \frac{19 y}{7}$

$76 y + 5 = - 7 \cdot 1$

$x = - \frac{2 (11 y + 1)}{7}$

$z = \frac{3}{7} + \frac{19 y}{7}$

$76 y + 5 = - 7 \cdot 1$

$x = - \frac{2 (11 y + 1)}{7}$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Multiply $- 7$ by $1$ .

$76 y + 5 = - 7$

$x = - \frac{2 (11 y + 1)}{7}$

$z = \frac{3}{7} + \frac{19 y}{7}$

$76 y + 5 = - 7$

$x = - \frac{2 (11 y + 1)}{7}$

$z = \frac{3}{7} + \frac{19 y}{7}$

$76 y + 5 = - 7$

$x = - \frac{2 (11 y + 1)}{7}$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 5.3

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

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Step 5.3.1

Subtract $5$ from both sides of the equation.

$76 y = - 7 - 5$

$x = - \frac{2 (11 y + 1)}{7}$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 5.3.2

Subtract $5$ from $- 7$ .

$76 y = - 12$

$x = - \frac{2 (11 y + 1)}{7}$

$z = \frac{3}{7} + \frac{19 y}{7}$

$76 y = - 12$

$x = - \frac{2 (11 y + 1)}{7}$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 5.4

Divide each term in $76 y = - 12$ by $76$ and simplify.

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Step 5.4.1

Divide each term in $76 y = - 12$ by $76$ .

$\frac{76 y}{76} = \frac{- 12}{76}$

$x = - \frac{2 (11 y + 1)}{7}$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $76$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{76 y}{76} = \frac{- 12}{76}$

$x = - \frac{2 (11 y + 1)}{7}$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 5.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 12}{76}$

$x = - \frac{2 (11 y + 1)}{7}$

$z = \frac{3}{7} + \frac{19 y}{7}$

$y = \frac{- 12}{76}$

$x = - \frac{2 (11 y + 1)}{7}$

$z = \frac{3}{7} + \frac{19 y}{7}$

$y = \frac{- 12}{76}$

$x = - \frac{2 (11 y + 1)}{7}$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Cancel the common factor of $- 12$ and $76$ .

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Step 5.4.3.1.1

Factor $4$ out of $- 12$ .

$y = \frac{4 (- 3)}{76}$

$x = - \frac{2 (11 y + 1)}{7}$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 5.4.3.1.2

Cancel the common factors.

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Step 5.4.3.1.2.1

Factor $4$ out of $76$ .

$y = \frac{4 \cdot - 3}{4 \cdot 19}$

$x = - \frac{2 (11 y + 1)}{7}$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 5.4.3.1.2.2

Cancel the common factor.

$y = \frac{4 \cdot - 3}{4 \cdot 19}$

$x = - \frac{2 (11 y + 1)}{7}$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 5.4.3.1.2.3

Rewrite the expression.

$y = \frac{- 3}{19}$

$x = - \frac{2 (11 y + 1)}{7}$

$z = \frac{3}{7} + \frac{19 y}{7}$

$y = \frac{- 3}{19}$

$x = - \frac{2 (11 y + 1)}{7}$

$z = \frac{3}{7} + \frac{19 y}{7}$

$y = \frac{- 3}{19}$

$x = - \frac{2 (11 y + 1)}{7}$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 5.4.3.2

Move the negative in front of the fraction.

$y = - \frac{3}{19}$

$x = - \frac{2 (11 y + 1)}{7}$

$z = \frac{3}{7} + \frac{19 y}{7}$

$y = - \frac{3}{19}$

$x = - \frac{2 (11 y + 1)}{7}$

$z = \frac{3}{7} + \frac{19 y}{7}$

$y = - \frac{3}{19}$

$x = - \frac{2 (11 y + 1)}{7}$

$z = \frac{3}{7} + \frac{19 y}{7}$

$y = - \frac{3}{19}$

$x = - \frac{2 (11 y + 1)}{7}$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 6

Replace all occurrences of $y$ with $- \frac{3}{19}$ in each equation.

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Step 6.1

Replace all occurrences of $y$ in $x = - \frac{2 (11 y + 1)}{7}$ with $- \frac{3}{19}$ .

$x = - \frac{2 (11 (- \frac{3}{19}) + 1)}{7}$

$y = - \frac{3}{19}$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $- \frac{2 (11 (- \frac{3}{19}) + 1)}{7}$ .

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Step 6.2.1.1

Simplify the numerator.

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Step 6.2.1.1.1

Multiply $11 (- \frac{3}{19})$ .

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Step 6.2.1.1.1.1

Multiply $- 1$ by $11$ .

$x = - \frac{2 (- 11 (\frac{3}{19}) + 1)}{7}$

$y = - \frac{3}{19}$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 6.2.1.1.1.2

Combine $- 11$ and $\frac{3}{19}$ .

$x = - \frac{2 (\frac{- 11 \cdot 3}{19} + 1)}{7}$

$y = - \frac{3}{19}$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 6.2.1.1.1.3

Multiply $- 11$ by $3$ .

$x = - \frac{2 (\frac{- 33}{19} + 1)}{7}$

$y = - \frac{3}{19}$

$z = \frac{3}{7} + \frac{19 y}{7}$

$x = - \frac{2 (\frac{- 33}{19} + 1)}{7}$

$y = - \frac{3}{19}$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 6.2.1.1.2

Move the negative in front of the fraction.

$x = - \frac{2 (- \frac{33}{19} + 1)}{7}$

$y = - \frac{3}{19}$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 6.2.1.1.3

Write $1$ as a fraction with a common denominator.

$x = - \frac{2 (- \frac{33}{19} + \frac{19}{19})}{7}$

$y = - \frac{3}{19}$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 6.2.1.1.4

Combine the numerators over the common denominator.

$x = - \frac{2 (\frac{- 33 + 19}{19})}{7}$

$y = - \frac{3}{19}$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 6.2.1.1.5

Add $- 33$ and $19$ .

$x = - \frac{2 (\frac{- 14}{19})}{7}$

$y = - \frac{3}{19}$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 6.2.1.1.6

Move the negative in front of the fraction.

$x = - \frac{2 \cdot (- 1 (\frac{14}{19}))}{7}$

$y = - \frac{3}{19}$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 6.2.1.1.7

Combine exponents.

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Step 6.2.1.1.7.1

Factor out negative.

$x = - \frac{- (2 (\frac{14}{19}))}{7}$

$y = - \frac{3}{19}$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 6.2.1.1.7.2

Combine $2$ and $\frac{14}{19}$ .

$x = - \frac{- \frac{2 \cdot 14}{19}}{7}$

$y = - \frac{3}{19}$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 6.2.1.1.7.3

Multiply $2$ by $14$ .

$x = - \frac{- \frac{28}{19}}{7}$

$y = - \frac{3}{19}$

$z = \frac{3}{7} + \frac{19 y}{7}$

$x = - \frac{- \frac{28}{19}}{7}$

$y = - \frac{3}{19}$

$z = \frac{3}{7} + \frac{19 y}{7}$

$x = - \frac{- \frac{28}{19}}{7}$

$y = - \frac{3}{19}$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 6.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$x = - (- \frac{28}{19} \cdot \frac{1}{7})$

$y = - \frac{3}{19}$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 6.2.1.3

Cancel the common factor of $7$ .

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Step 6.2.1.3.1

Move the leading negative in $- \frac{28}{19}$ into the numerator.

$x = - (\frac{- 28}{19} \cdot \frac{1}{7})$

$y = - \frac{3}{19}$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 6.2.1.3.2

Factor $7$ out of $- 28$ .

$x = - (\frac{7 (- 4)}{19} \cdot \frac{1}{7})$

$y = - \frac{3}{19}$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 6.2.1.3.3

Cancel the common factor.

$x = - (\frac{7 \cdot - 4}{19} \cdot \frac{1}{7})$

$y = - \frac{3}{19}$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 6.2.1.3.4

Rewrite the expression.

$x = - \frac{- 4}{19}$

$y = - \frac{3}{19}$

$z = \frac{3}{7} + \frac{19 y}{7}$

$x = - \frac{- 4}{19}$

$y = - \frac{3}{19}$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 6.2.1.4

Move the negative in front of the fraction.

$x = \frac{4}{19}$

$y = - \frac{3}{19}$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 6.2.1.5

Multiply $- - \frac{4}{19}$ .

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Step 6.2.1.5.1

Multiply $- 1$ by $- 1$ .

$x = 1 (\frac{4}{19})$

$y = - \frac{3}{19}$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 6.2.1.5.2

Multiply $\frac{4}{19}$ by $1$ .

$x = \frac{4}{19}$

$y = - \frac{3}{19}$

$z = \frac{3}{7} + \frac{19 y}{7}$

$x = \frac{4}{19}$

$y = - \frac{3}{19}$

$z = \frac{3}{7} + \frac{19 y}{7}$

$x = \frac{4}{19}$

$y = - \frac{3}{19}$

$z = \frac{3}{7} + \frac{19 y}{7}$

$x = \frac{4}{19}$

$y = - \frac{3}{19}$

$z = \frac{3}{7} + \frac{19 y}{7}$

Step 6.3

Replace all occurrences of $y$ in $z = \frac{3}{7} + \frac{19 y}{7}$ with $- \frac{3}{19}$ .

$z = \frac{3}{7} + \frac{19 (- \frac{3}{19})}{7}$

$x = \frac{4}{19}$

$y = - \frac{3}{19}$

Step 6.4

Simplify the right side.

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Step 6.4.1

Simplify $\frac{3}{7} + \frac{19 (- \frac{3}{19})}{7}$ .

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Step 6.4.1.1

Combine the numerators over the common denominator.

$z = \frac{3 + 19 (- \frac{3}{19})}{7}$

$x = \frac{4}{19}$

$y = - \frac{3}{19}$

Step 6.4.1.2

Cancel the common factor of $19$ .

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Step 6.4.1.2.1

Move the leading negative in $- \frac{3}{19}$ into the numerator.

$z = \frac{3 + 19 (\frac{- 3}{19})}{7}$

$x = \frac{4}{19}$

$y = - \frac{3}{19}$

Step 6.4.1.2.2

Cancel the common factor.

$z = \frac{3 + 19 (\frac{- 3}{19})}{7}$

$x = \frac{4}{19}$

$y = - \frac{3}{19}$

Step 6.4.1.2.3

Rewrite the expression.

$z = \frac{3 - 3}{7}$

$x = \frac{4}{19}$

$y = - \frac{3}{19}$

$z = \frac{3 - 3}{7}$

$x = \frac{4}{19}$

$y = - \frac{3}{19}$

Step 6.4.1.3

Simplify the expression.

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Step 6.4.1.3.1

Subtract $3$ from $3$ .

$z = \frac{0}{7}$

$x = \frac{4}{19}$

$y = - \frac{3}{19}$

Step 6.4.1.3.2

Divide $0$ by $7$ .

$z = 0$

$x = \frac{4}{19}$

$y = - \frac{3}{19}$

$z = 0$

$x = \frac{4}{19}$

$y = - \frac{3}{19}$

$z = 0$

$x = \frac{4}{19}$

$y = - \frac{3}{19}$

$z = 0$

$x = \frac{4}{19}$

$y = - \frac{3}{19}$

$z = 0$

$x = \frac{4}{19}$

$y = - \frac{3}{19}$

Step 7

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

$(\frac{4}{19}, - \frac{3}{19}, 0)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(\frac{4}{19}, - \frac{3}{19}, 0)$

Equation Form:

$x = \frac{4}{19}, y = - \frac{3}{19}, z = 0$