Finite Math Examples

$35 x + 70 y + 175 z + 280 = 2625$ , $40 x + 40 y + 80 z + 160 = 1320$ , $40 x + 40 y + 90 z + 140 = 1410$

Step 1

Solve for $x$ in $35 x + 70 y + 175 z + 280 = 2625$ .

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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 $70 y$ from both sides of the equation.

$35 x + 175 z + 280 = 2625 - 70 y$

$40 x + 40 y + 80 z + 160 = 1320$

$40 x + 40 y + 90 z + 140 = 1410$

Step 1.1.2

Subtract $175 z$ from both sides of the equation.

$35 x + 280 = 2625 - 70 y - 175 z$

$40 x + 40 y + 80 z + 160 = 1320$

$40 x + 40 y + 90 z + 140 = 1410$

Step 1.1.3

Subtract $280$ from both sides of the equation.

$35 x = 2625 - 70 y - 175 z - 280$

$40 x + 40 y + 80 z + 160 = 1320$

$40 x + 40 y + 90 z + 140 = 1410$

Step 1.1.4

Subtract $280$ from $2625$ .

$35 x = - 70 y - 175 z + 2345$

$40 x + 40 y + 80 z + 160 = 1320$

$40 x + 40 y + 90 z + 140 = 1410$

$35 x = - 70 y - 175 z + 2345$

$40 x + 40 y + 80 z + 160 = 1320$

$40 x + 40 y + 90 z + 140 = 1410$

Step 1.2

Divide each term in $35 x = - 70 y - 175 z + 2345$ by $35$ and simplify.

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Step 1.2.1

Divide each term in $35 x = - 70 y - 175 z + 2345$ by $35$ .

$\frac{35 x}{35} = \frac{- 70 y}{35} + \frac{- 175 z}{35} + \frac{2345}{35}$

$40 x + 40 y + 80 z + 160 = 1320$

$40 x + 40 y + 90 z + 140 = 1410$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $35$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{35 x}{35} = \frac{- 70 y}{35} + \frac{- 175 z}{35} + \frac{2345}{35}$

$40 x + 40 y + 80 z + 160 = 1320$

$40 x + 40 y + 90 z + 140 = 1410$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 70 y}{35} + \frac{- 175 z}{35} + \frac{2345}{35}$

$40 x + 40 y + 80 z + 160 = 1320$

$40 x + 40 y + 90 z + 140 = 1410$

$x = \frac{- 70 y}{35} + \frac{- 175 z}{35} + \frac{2345}{35}$

$40 x + 40 y + 80 z + 160 = 1320$

$40 x + 40 y + 90 z + 140 = 1410$

$x = \frac{- 70 y}{35} + \frac{- 175 z}{35} + \frac{2345}{35}$

$40 x + 40 y + 80 z + 160 = 1320$

$40 x + 40 y + 90 z + 140 = 1410$

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

Cancel the common factor of $- 70$ and $35$ .

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Step 1.2.3.1.1.1

Factor $35$ out of $- 70 y$ .

$x = \frac{35 (- 2 y)}{35} + \frac{- 175 z}{35} + \frac{2345}{35}$

$40 x + 40 y + 80 z + 160 = 1320$

$40 x + 40 y + 90 z + 140 = 1410$

Step 1.2.3.1.1.2

Cancel the common factors.

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Step 1.2.3.1.1.2.1

Factor $35$ out of $35$ .

$x = \frac{35 (- 2 y)}{35 (1)} + \frac{- 175 z}{35} + \frac{2345}{35}$

$40 x + 40 y + 80 z + 160 = 1320$

$40 x + 40 y + 90 z + 140 = 1410$

Step 1.2.3.1.1.2.2

Cancel the common factor.

$x = \frac{35 (- 2 y)}{35 \cdot 1} + \frac{- 175 z}{35} + \frac{2345}{35}$

$40 x + 40 y + 80 z + 160 = 1320$

$40 x + 40 y + 90 z + 140 = 1410$

Step 1.2.3.1.1.2.3

Rewrite the expression.

$x = \frac{- 2 y}{1} + \frac{- 175 z}{35} + \frac{2345}{35}$

$40 x + 40 y + 80 z + 160 = 1320$

$40 x + 40 y + 90 z + 140 = 1410$

Step 1.2.3.1.1.2.4

Divide $- 2 y$ by $1$ .

$x = - 2 y + \frac{- 175 z}{35} + \frac{2345}{35}$

$40 x + 40 y + 80 z + 160 = 1320$

$40 x + 40 y + 90 z + 140 = 1410$

$x = - 2 y + \frac{- 175 z}{35} + \frac{2345}{35}$

$40 x + 40 y + 80 z + 160 = 1320$

$40 x + 40 y + 90 z + 140 = 1410$

$x = - 2 y + \frac{- 175 z}{35} + \frac{2345}{35}$

$40 x + 40 y + 80 z + 160 = 1320$

$40 x + 40 y + 90 z + 140 = 1410$

Step 1.2.3.1.2

Cancel the common factor of $- 175$ and $35$ .

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Step 1.2.3.1.2.1

Factor $35$ out of $- 175 z$ .

$x = - 2 y + \frac{35 (- 5 z)}{35} + \frac{2345}{35}$

$40 x + 40 y + 80 z + 160 = 1320$

$40 x + 40 y + 90 z + 140 = 1410$

Step 1.2.3.1.2.2

Cancel the common factors.

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Step 1.2.3.1.2.2.1

Factor $35$ out of $35$ .

$x = - 2 y + \frac{35 (- 5 z)}{35 (1)} + \frac{2345}{35}$

$40 x + 40 y + 80 z + 160 = 1320$

$40 x + 40 y + 90 z + 140 = 1410$

Step 1.2.3.1.2.2.2

Cancel the common factor.

$x = - 2 y + \frac{35 (- 5 z)}{35 \cdot 1} + \frac{2345}{35}$

$40 x + 40 y + 80 z + 160 = 1320$

$40 x + 40 y + 90 z + 140 = 1410$

Step 1.2.3.1.2.2.3

Rewrite the expression.

$x = - 2 y + \frac{- 5 z}{1} + \frac{2345}{35}$

$40 x + 40 y + 80 z + 160 = 1320$

$40 x + 40 y + 90 z + 140 = 1410$

Step 1.2.3.1.2.2.4

Divide $- 5 z$ by $1$ .

$x = - 2 y - 5 z + \frac{2345}{35}$

$40 x + 40 y + 80 z + 160 = 1320$

$40 x + 40 y + 90 z + 140 = 1410$

$x = - 2 y - 5 z + \frac{2345}{35}$

$40 x + 40 y + 80 z + 160 = 1320$

$40 x + 40 y + 90 z + 140 = 1410$

$x = - 2 y - 5 z + \frac{2345}{35}$

$40 x + 40 y + 80 z + 160 = 1320$

$40 x + 40 y + 90 z + 140 = 1410$

Step 1.2.3.1.3

Divide $2345$ by $35$ .

$x = - 2 y - 5 z + 67$

$40 x + 40 y + 80 z + 160 = 1320$

$40 x + 40 y + 90 z + 140 = 1410$

$x = - 2 y - 5 z + 67$

$40 x + 40 y + 80 z + 160 = 1320$

$40 x + 40 y + 90 z + 140 = 1410$

$x = - 2 y - 5 z + 67$

$40 x + 40 y + 80 z + 160 = 1320$

$40 x + 40 y + 90 z + 140 = 1410$

$x = - 2 y - 5 z + 67$

$40 x + 40 y + 80 z + 160 = 1320$

$40 x + 40 y + 90 z + 140 = 1410$

$x = - 2 y - 5 z + 67$

$40 x + 40 y + 80 z + 160 = 1320$

$40 x + 40 y + 90 z + 140 = 1410$

Step 2

Replace all occurrences of $x$ with $- 2 y - 5 z + 67$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $40 x + 40 y + 80 z + 160 = 1320$ with $- 2 y - 5 z + 67$ .

$40 (- 2 y - 5 z + 67) + 40 y + 80 z + 160 = 1320$

$x = - 2 y - 5 z + 67$

$40 x + 40 y + 90 z + 140 = 1410$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $40 (- 2 y - 5 z + 67) + 40 y + 80 z + 160$ .

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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.

$40 (- 2 y) + 40 (- 5 z) + 40 \cdot 67 + 40 y + 80 z + 160 = 1320$

$x = - 2 y - 5 z + 67$

$40 x + 40 y + 90 z + 140 = 1410$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $- 2$ by $40$ .

$- 80 y + 40 (- 5 z) + 40 \cdot 67 + 40 y + 80 z + 160 = 1320$

$x = - 2 y - 5 z + 67$

$40 x + 40 y + 90 z + 140 = 1410$

Step 2.2.1.1.2.2

Multiply $- 5$ by $40$ .

$- 80 y - 200 z + 40 \cdot 67 + 40 y + 80 z + 160 = 1320$

$x = - 2 y - 5 z + 67$

$40 x + 40 y + 90 z + 140 = 1410$

Step 2.2.1.1.2.3

Multiply $40$ by $67$ .

$- 80 y - 200 z + 2680 + 40 y + 80 z + 160 = 1320$

$x = - 2 y - 5 z + 67$

$40 x + 40 y + 90 z + 140 = 1410$

$- 80 y - 200 z + 2680 + 40 y + 80 z + 160 = 1320$

$x = - 2 y - 5 z + 67$

$40 x + 40 y + 90 z + 140 = 1410$

$- 80 y - 200 z + 2680 + 40 y + 80 z + 160 = 1320$

$x = - 2 y - 5 z + 67$

$40 x + 40 y + 90 z + 140 = 1410$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Add $- 80 y$ and $40 y$ .

$- 40 y - 200 z + 2680 + 80 z + 160 = 1320$

$x = - 2 y - 5 z + 67$

$40 x + 40 y + 90 z + 140 = 1410$

Step 2.2.1.2.2

Add $- 200 z$ and $80 z$ .

$- 40 y - 120 z + 2680 + 160 = 1320$

$x = - 2 y - 5 z + 67$

$40 x + 40 y + 90 z + 140 = 1410$

Step 2.2.1.2.3

Add $2680$ and $160$ .

$- 40 y - 120 z + 2840 = 1320$

$x = - 2 y - 5 z + 67$

$40 x + 40 y + 90 z + 140 = 1410$

$- 40 y - 120 z + 2840 = 1320$

$x = - 2 y - 5 z + 67$

$40 x + 40 y + 90 z + 140 = 1410$

$- 40 y - 120 z + 2840 = 1320$

$x = - 2 y - 5 z + 67$

$40 x + 40 y + 90 z + 140 = 1410$

$- 40 y - 120 z + 2840 = 1320$

$x = - 2 y - 5 z + 67$

$40 x + 40 y + 90 z + 140 = 1410$

Step 2.3

Replace all occurrences of $x$ in $40 x + 40 y + 90 z + 140 = 1410$ with $- 2 y - 5 z + 67$ .

$40 (- 2 y - 5 z + 67) + 40 y + 90 z + 140 = 1410$

$- 40 y - 120 z + 2840 = 1320$

$x = - 2 y - 5 z + 67$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $40 (- 2 y - 5 z + 67) + 40 y + 90 z + 140$ .

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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.

$40 (- 2 y) + 40 (- 5 z) + 40 \cdot 67 + 40 y + 90 z + 140 = 1410$

$- 40 y - 120 z + 2840 = 1320$

$x = - 2 y - 5 z + 67$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $- 2$ by $40$ .

$- 80 y + 40 (- 5 z) + 40 \cdot 67 + 40 y + 90 z + 140 = 1410$

$- 40 y - 120 z + 2840 = 1320$

$x = - 2 y - 5 z + 67$

Step 2.4.1.1.2.2

Multiply $- 5$ by $40$ .

$- 80 y - 200 z + 40 \cdot 67 + 40 y + 90 z + 140 = 1410$

$- 40 y - 120 z + 2840 = 1320$

$x = - 2 y - 5 z + 67$

Step 2.4.1.1.2.3

Multiply $40$ by $67$ .

$- 80 y - 200 z + 2680 + 40 y + 90 z + 140 = 1410$

$- 40 y - 120 z + 2840 = 1320$

$x = - 2 y - 5 z + 67$

$- 80 y - 200 z + 2680 + 40 y + 90 z + 140 = 1410$

$- 40 y - 120 z + 2840 = 1320$

$x = - 2 y - 5 z + 67$

$- 80 y - 200 z + 2680 + 40 y + 90 z + 140 = 1410$

$- 40 y - 120 z + 2840 = 1320$

$x = - 2 y - 5 z + 67$

Step 2.4.1.2

Simplify by adding terms.

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Step 2.4.1.2.1

Add $- 80 y$ and $40 y$ .

$- 40 y - 200 z + 2680 + 90 z + 140 = 1410$

$- 40 y - 120 z + 2840 = 1320$

$x = - 2 y - 5 z + 67$

Step 2.4.1.2.2

Add $- 200 z$ and $90 z$ .

$- 40 y - 110 z + 2680 + 140 = 1410$

$- 40 y - 120 z + 2840 = 1320$

$x = - 2 y - 5 z + 67$

Step 2.4.1.2.3

Add $2680$ and $140$ .

$- 40 y - 110 z + 2820 = 1410$

$- 40 y - 120 z + 2840 = 1320$

$x = - 2 y - 5 z + 67$

$- 40 y - 110 z + 2820 = 1410$

$- 40 y - 120 z + 2840 = 1320$

$x = - 2 y - 5 z + 67$

$- 40 y - 110 z + 2820 = 1410$

$- 40 y - 120 z + 2840 = 1320$

$x = - 2 y - 5 z + 67$

$- 40 y - 110 z + 2820 = 1410$

$- 40 y - 120 z + 2840 = 1320$

$x = - 2 y - 5 z + 67$

$- 40 y - 110 z + 2820 = 1410$

$- 40 y - 120 z + 2840 = 1320$

$x = - 2 y - 5 z + 67$

Step 3

Solve for $y$ in $- 40 y - 110 z + 2820 = 1410$ .

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Step 3.1

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

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Step 3.1.1

Add $110 z$ to both sides of the equation.

$- 40 y + 2820 = 1410 + 110 z$

$- 40 y - 120 z + 2840 = 1320$

$x = - 2 y - 5 z + 67$

Step 3.1.2

Subtract $2820$ from both sides of the equation.

$- 40 y = 1410 + 110 z - 2820$

$- 40 y - 120 z + 2840 = 1320$

$x = - 2 y - 5 z + 67$

Step 3.1.3

Subtract $2820$ from $1410$ .

$- 40 y = 110 z - 1410$

$- 40 y - 120 z + 2840 = 1320$

$x = - 2 y - 5 z + 67$

$- 40 y = 110 z - 1410$

$- 40 y - 120 z + 2840 = 1320$

$x = - 2 y - 5 z + 67$

Step 3.2

Divide each term in $- 40 y = 110 z - 1410$ by $- 40$ and simplify.

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Step 3.2.1

Divide each term in $- 40 y = 110 z - 1410$ by $- 40$ .

$\frac{- 40 y}{- 40} = \frac{110 z}{- 40} + \frac{- 1410}{- 40}$

$- 40 y - 120 z + 2840 = 1320$

$x = - 2 y - 5 z + 67$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $- 40$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{- 40 y}{- 40} = \frac{110 z}{- 40} + \frac{- 1410}{- 40}$

$- 40 y - 120 z + 2840 = 1320$

$x = - 2 y - 5 z + 67$

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{110 z}{- 40} + \frac{- 1410}{- 40}$

$- 40 y - 120 z + 2840 = 1320$

$x = - 2 y - 5 z + 67$

$y = \frac{110 z}{- 40} + \frac{- 1410}{- 40}$

$- 40 y - 120 z + 2840 = 1320$

$x = - 2 y - 5 z + 67$

$y = \frac{110 z}{- 40} + \frac{- 1410}{- 40}$

$- 40 y - 120 z + 2840 = 1320$

$x = - 2 y - 5 z + 67$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

Cancel the common factor of $110$ and $- 40$ .

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Step 3.2.3.1.1.1

Factor $10$ out of $110 z$ .

$y = \frac{10 (11 z)}{- 40} + \frac{- 1410}{- 40}$

$- 40 y - 120 z + 2840 = 1320$

$x = - 2 y - 5 z + 67$

Step 3.2.3.1.1.2

Cancel the common factors.

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Step 3.2.3.1.1.2.1

Factor $10$ out of $- 40$ .

$y = \frac{10 (11 z)}{10 (- 4)} + \frac{- 1410}{- 40}$

$- 40 y - 120 z + 2840 = 1320$

$x = - 2 y - 5 z + 67$

Step 3.2.3.1.1.2.2

Cancel the common factor.

$y = \frac{10 (11 z)}{10 \cdot - 4} + \frac{- 1410}{- 40}$

$- 40 y - 120 z + 2840 = 1320$

$x = - 2 y - 5 z + 67$

Step 3.2.3.1.1.2.3

Rewrite the expression.

$y = \frac{11 z}{- 4} + \frac{- 1410}{- 40}$

$- 40 y - 120 z + 2840 = 1320$

$x = - 2 y - 5 z + 67$

$y = \frac{11 z}{- 4} + \frac{- 1410}{- 40}$

$- 40 y - 120 z + 2840 = 1320$

$x = - 2 y - 5 z + 67$

$y = \frac{11 z}{- 4} + \frac{- 1410}{- 40}$

$- 40 y - 120 z + 2840 = 1320$

$x = - 2 y - 5 z + 67$

Step 3.2.3.1.2

Move the negative in front of the fraction.

$y = - \frac{11 z}{4} + \frac{- 1410}{- 40}$

$- 40 y - 120 z + 2840 = 1320$

$x = - 2 y - 5 z + 67$

Step 3.2.3.1.3

Cancel the common factor of $- 1410$ and $- 40$ .

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Step 3.2.3.1.3.1

Factor $- 10$ out of $- 1410$ .

$y = - \frac{11 z}{4} + \frac{- 10 \cdot 141}{- 40}$

$- 40 y - 120 z + 2840 = 1320$

$x = - 2 y - 5 z + 67$

Step 3.2.3.1.3.2

Cancel the common factors.

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Step 3.2.3.1.3.2.1

Factor $- 10$ out of $- 40$ .

$y = - \frac{11 z}{4} + \frac{- 10 \cdot 141}{- 10 \cdot 4}$

$- 40 y - 120 z + 2840 = 1320$

$x = - 2 y - 5 z + 67$

Step 3.2.3.1.3.2.2

Cancel the common factor.

$y = - \frac{11 z}{4} + \frac{- 10 \cdot 141}{- 10 \cdot 4}$

$- 40 y - 120 z + 2840 = 1320$

$x = - 2 y - 5 z + 67$

Step 3.2.3.1.3.2.3

Rewrite the expression.

$y = - \frac{11 z}{4} + \frac{141}{4}$

$- 40 y - 120 z + 2840 = 1320$

$x = - 2 y - 5 z + 67$

$y = - \frac{11 z}{4} + \frac{141}{4}$

$- 40 y - 120 z + 2840 = 1320$

$x = - 2 y - 5 z + 67$

$y = - \frac{11 z}{4} + \frac{141}{4}$

$- 40 y - 120 z + 2840 = 1320$

$x = - 2 y - 5 z + 67$

$y = - \frac{11 z}{4} + \frac{141}{4}$

$- 40 y - 120 z + 2840 = 1320$

$x = - 2 y - 5 z + 67$

$y = - \frac{11 z}{4} + \frac{141}{4}$

$- 40 y - 120 z + 2840 = 1320$

$x = - 2 y - 5 z + 67$

$y = - \frac{11 z}{4} + \frac{141}{4}$

$- 40 y - 120 z + 2840 = 1320$

$x = - 2 y - 5 z + 67$

$y = - \frac{11 z}{4} + \frac{141}{4}$

$- 40 y - 120 z + 2840 = 1320$

$x = - 2 y - 5 z + 67$

Step 4

Replace all occurrences of $y$ with $- \frac{11 z}{4} + \frac{141}{4}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $- 40 y - 120 z + 2840 = 1320$ with $- \frac{11 z}{4} + \frac{141}{4}$ .

$- 40 (- \frac{11 z}{4} + \frac{141}{4}) - 120 z + 2840 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

$x = - 2 y - 5 z + 67$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $- 40 (- \frac{11 z}{4} + \frac{141}{4}) - 120 z + 2840$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Apply the distributive property.

$- 40 (- \frac{11 z}{4}) - 40 (\frac{141}{4}) - 120 z + 2840 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

$x = - 2 y - 5 z + 67$

Step 4.2.1.1.2

Cancel the common factor of $4$ .

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Step 4.2.1.1.2.1

Move the leading negative in $- \frac{11 z}{4}$ into the numerator.

$- 40 \frac{- 11 z}{4} - 40 (\frac{141}{4}) - 120 z + 2840 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

$x = - 2 y - 5 z + 67$

Step 4.2.1.1.2.2

Factor $4$ out of $- 40$ .

$4 (- 10) (\frac{- 11 z}{4}) - 40 (\frac{141}{4}) - 120 z + 2840 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

$x = - 2 y - 5 z + 67$

Step 4.2.1.1.2.3

Cancel the common factor.

$4 \cdot (- 10 \frac{- 11 z}{4}) - 40 (\frac{141}{4}) - 120 z + 2840 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

$x = - 2 y - 5 z + 67$

Step 4.2.1.1.2.4

Rewrite the expression.

$- 10 (- 11 z) - 40 (\frac{141}{4}) - 120 z + 2840 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

$x = - 2 y - 5 z + 67$

$- 10 (- 11 z) - 40 (\frac{141}{4}) - 120 z + 2840 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

$x = - 2 y - 5 z + 67$

Step 4.2.1.1.3

Multiply $- 11$ by $- 10$ .

$110 z - 40 (\frac{141}{4}) - 120 z + 2840 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

$x = - 2 y - 5 z + 67$

Step 4.2.1.1.4

Cancel the common factor of $4$ .

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Step 4.2.1.1.4.1

Factor $4$ out of $- 40$ .

$110 z + 4 (- 10) (\frac{141}{4}) - 120 z + 2840 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

$x = - 2 y - 5 z + 67$

Step 4.2.1.1.4.2

Cancel the common factor.

$110 z + 4 \cdot (- 10 (\frac{141}{4})) - 120 z + 2840 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

$x = - 2 y - 5 z + 67$

Step 4.2.1.1.4.3

Rewrite the expression.

$110 z - 10 \cdot 141 - 120 z + 2840 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

$x = - 2 y - 5 z + 67$

$110 z - 10 \cdot 141 - 120 z + 2840 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

$x = - 2 y - 5 z + 67$

Step 4.2.1.1.5

Multiply $- 10$ by $141$ .

$110 z - 1410 - 120 z + 2840 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

$x = - 2 y - 5 z + 67$

$110 z - 1410 - 120 z + 2840 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

$x = - 2 y - 5 z + 67$

Step 4.2.1.2

Simplify by adding terms.

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Step 4.2.1.2.1

Subtract $120 z$ from $110 z$ .

$- 10 z - 1410 + 2840 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

$x = - 2 y - 5 z + 67$

Step 4.2.1.2.2

Add $- 1410$ and $2840$ .

$- 10 z + 1430 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

$x = - 2 y - 5 z + 67$

$- 10 z + 1430 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

$x = - 2 y - 5 z + 67$

$- 10 z + 1430 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

$x = - 2 y - 5 z + 67$

$- 10 z + 1430 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

$x = - 2 y - 5 z + 67$

Step 4.3

Replace all occurrences of $y$ in $x = - 2 y - 5 z + 67$ with $- \frac{11 z}{4} + \frac{141}{4}$ .

$x = - 2 (- \frac{11 z}{4} + \frac{141}{4}) - 5 z + 67$

$- 10 z + 1430 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $- 2 (- \frac{11 z}{4} + \frac{141}{4}) - 5 z + 67$ .

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Step 4.4.1.1

Simplify each term.

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Step 4.4.1.1.1

Apply the distributive property.

$x = - 2 (- \frac{11 z}{4}) - 2 (\frac{141}{4}) - 5 z + 67$

$- 10 z + 1430 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

Step 4.4.1.1.2

Cancel the common factor of $2$ .

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Step 4.4.1.1.2.1

Move the leading negative in $- \frac{11 z}{4}$ into the numerator.

$x = - 2 \frac{- 11 z}{4} - 2 (\frac{141}{4}) - 5 z + 67$

$- 10 z + 1430 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

Step 4.4.1.1.2.2

Factor $2$ out of $- 2$ .

$x = 2 (- 1) (\frac{- 11 z}{4}) - 2 (\frac{141}{4}) - 5 z + 67$

$- 10 z + 1430 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

Step 4.4.1.1.2.3

Factor $2$ out of $4$ .

$x = 2 \cdot (- 1 \frac{- 11 z}{2 \cdot 2}) - 2 (\frac{141}{4}) - 5 z + 67$

$- 10 z + 1430 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

Step 4.4.1.1.2.4

Cancel the common factor.

$x = 2 \cdot (- 1 \frac{- 11 z}{2 \cdot 2}) - 2 (\frac{141}{4}) - 5 z + 67$

$- 10 z + 1430 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

Step 4.4.1.1.2.5

Rewrite the expression.

$x = - 1 \frac{- 11 z}{2} - 2 (\frac{141}{4}) - 5 z + 67$

$- 10 z + 1430 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

$x = - 1 \frac{- 11 z}{2} - 2 (\frac{141}{4}) - 5 z + 67$

$- 10 z + 1430 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

Step 4.4.1.1.3

Cancel the common factor of $2$ .

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Step 4.4.1.1.3.1

Factor $2$ out of $- 2$ .

$x = - 1 \frac{- 11 z}{2} + 2 (- 1) (\frac{141}{4}) - 5 z + 67$

$- 10 z + 1430 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

Step 4.4.1.1.3.2

Factor $2$ out of $4$ .

$x = - 1 \frac{- 11 z}{2} + 2 \cdot (- 1 \frac{141}{2 \cdot 2}) - 5 z + 67$

$- 10 z + 1430 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

Step 4.4.1.1.3.3

Cancel the common factor.

$x = - 1 \frac{- 11 z}{2} + 2 \cdot (- 1 \frac{141}{2 \cdot 2}) - 5 z + 67$

$- 10 z + 1430 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

Step 4.4.1.1.3.4

Rewrite the expression.

$x = - 1 \frac{- 11 z}{2} - 1 (\frac{141}{2}) - 5 z + 67$

$- 10 z + 1430 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

$x = - 1 \frac{- 11 z}{2} - 1 (\frac{141}{2}) - 5 z + 67$

$- 10 z + 1430 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

Step 4.4.1.1.4

Simplify each term.

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Step 4.4.1.1.4.1

Move the negative in front of the fraction.

$x = - 1 (- \frac{11 z}{2}) - 1 (\frac{141}{2}) - 5 z + 67$

$- 10 z + 1430 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

Step 4.4.1.1.4.2

Multiply $- 1 (- \frac{11 z}{2})$ .

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Step 4.4.1.1.4.2.1

Multiply $- 1$ by $- 1$ .

$x = 1 (\frac{11 z}{2}) - 1 (\frac{141}{2}) - 5 z + 67$

$- 10 z + 1430 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

Step 4.4.1.1.4.2.2

Multiply $\frac{11 z}{2}$ by $1$ .

$x = \frac{11 z}{2} - 1 (\frac{141}{2}) - 5 z + 67$

$- 10 z + 1430 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

$x = \frac{11 z}{2} - 1 (\frac{141}{2}) - 5 z + 67$

$- 10 z + 1430 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

Step 4.4.1.1.4.3

Rewrite $- 1 (\frac{141}{2})$ as $- (\frac{141}{2})$ .

$x = \frac{11 z}{2} - \frac{141}{2} - 5 z + 67$

$- 10 z + 1430 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

$x = \frac{11 z}{2} - \frac{141}{2} - 5 z + 67$

$- 10 z + 1430 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

$x = \frac{11 z}{2} - \frac{141}{2} - 5 z + 67$

$- 10 z + 1430 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

Step 4.4.1.2

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

$x = \frac{11 z}{2} - 5 z \cdot \frac{2}{2} - \frac{141}{2} + 67$

$- 10 z + 1430 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

Step 4.4.1.3

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

$x = \frac{11 z}{2} + \frac{- 5 z \cdot 2}{2} - \frac{141}{2} + 67$

$- 10 z + 1430 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

Step 4.4.1.4

Combine the numerators over the common denominator.

$x = \frac{11 z - 5 z \cdot 2}{2} - \frac{141}{2} + 67$

$- 10 z + 1430 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

Step 4.4.1.5

Combine the numerators over the common denominator.

$x = 67 + \frac{11 z - 5 z \cdot 2 - 141}{2}$

$- 10 z + 1430 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

Step 4.4.1.6

Multiply $2$ by $- 5$ .

$x = 67 + \frac{11 z - 10 z - 141}{2}$

$- 10 z + 1430 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

Step 4.4.1.7

Subtract $10 z$ from $11 z$ .

$x = 67 + \frac{z - 141}{2}$

$- 10 z + 1430 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

Step 4.4.1.8

To write $67$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = 67 \cdot \frac{2}{2} + \frac{z - 141}{2}$

$- 10 z + 1430 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

Step 4.4.1.9

Simplify terms.

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Step 4.4.1.9.1

Combine $67$ and $\frac{2}{2}$ .

$x = \frac{67 \cdot 2}{2} + \frac{z - 141}{2}$

$- 10 z + 1430 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

Step 4.4.1.9.2

Combine the numerators over the common denominator.

$x = \frac{67 \cdot 2 + z - 141}{2}$

$- 10 z + 1430 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

$x = \frac{67 \cdot 2 + z - 141}{2}$

$- 10 z + 1430 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

Step 4.4.1.10

Simplify the numerator.

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Step 4.4.1.10.1

Multiply $67$ by $2$ .

$x = \frac{134 + z - 141}{2}$

$- 10 z + 1430 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

Step 4.4.1.10.2

Subtract $141$ from $134$ .

$x = \frac{z - 7}{2}$

$- 10 z + 1430 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

$x = \frac{z - 7}{2}$

$- 10 z + 1430 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

$x = \frac{z - 7}{2}$

$- 10 z + 1430 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

$x = \frac{z - 7}{2}$

$- 10 z + 1430 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

$x = \frac{z - 7}{2}$

$- 10 z + 1430 = 1320$

$y = - \frac{11 z}{4} + \frac{141}{4}$

Step 5

Solve for $z$ in $- 10 z + 1430 = 1320$ .

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Step 5.1

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

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Step 5.1.1

Subtract $1430$ from both sides of the equation.

$- 10 z = 1320 - 1430$

$x = \frac{z - 7}{2}$

$y = - \frac{11 z}{4} + \frac{141}{4}$

Step 5.1.2

Subtract $1430$ from $1320$ .

$- 10 z = - 110$

$x = \frac{z - 7}{2}$

$y = - \frac{11 z}{4} + \frac{141}{4}$

$- 10 z = - 110$

$x = \frac{z - 7}{2}$

$y = - \frac{11 z}{4} + \frac{141}{4}$

Step 5.2

Divide each term in $- 10 z = - 110$ by $- 10$ and simplify.

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Step 5.2.1

Divide each term in $- 10 z = - 110$ by $- 10$ .

$\frac{- 10 z}{- 10} = \frac{- 110}{- 10}$

$x = \frac{z - 7}{2}$

$y = - \frac{11 z}{4} + \frac{141}{4}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $- 10$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{- 10 z}{- 10} = \frac{- 110}{- 10}$

$x = \frac{z - 7}{2}$

$y = - \frac{11 z}{4} + \frac{141}{4}$

Step 5.2.2.1.2

Divide $z$ by $1$ .

$z = \frac{- 110}{- 10}$

$x = \frac{z - 7}{2}$

$y = - \frac{11 z}{4} + \frac{141}{4}$

$z = \frac{- 110}{- 10}$

$x = \frac{z - 7}{2}$

$y = - \frac{11 z}{4} + \frac{141}{4}$

$z = \frac{- 110}{- 10}$

$x = \frac{z - 7}{2}$

$y = - \frac{11 z}{4} + \frac{141}{4}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $- 110$ by $- 10$ .

$z = 11$

$x = \frac{z - 7}{2}$

$y = - \frac{11 z}{4} + \frac{141}{4}$

$z = 11$

$x = \frac{z - 7}{2}$

$y = - \frac{11 z}{4} + \frac{141}{4}$

$z = 11$

$x = \frac{z - 7}{2}$

$y = - \frac{11 z}{4} + \frac{141}{4}$

$z = 11$

$x = \frac{z - 7}{2}$

$y = - \frac{11 z}{4} + \frac{141}{4}$

Step 6

Replace all occurrences of $z$ with $11$ in each equation.

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Step 6.1

Replace all occurrences of $z$ in $x = \frac{z - 7}{2}$ with $11$ .

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

$z = 11$

$y = - \frac{11 z}{4} + \frac{141}{4}$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $\frac{(11) - 7}{2}$ .

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Step 6.2.1.1

Subtract $7$ from $11$ .

$x = \frac{4}{2}$

$z = 11$

$y = - \frac{11 z}{4} + \frac{141}{4}$

Step 6.2.1.2

Divide $4$ by $2$ .

$x = 2$

$z = 11$

$y = - \frac{11 z}{4} + \frac{141}{4}$

$x = 2$

$z = 11$

$y = - \frac{11 z}{4} + \frac{141}{4}$

$x = 2$

$z = 11$

$y = - \frac{11 z}{4} + \frac{141}{4}$

Step 6.3

Replace all occurrences of $z$ in $y = - \frac{11 z}{4} + \frac{141}{4}$ with $11$ .

$y = - \frac{11 (11)}{4} + \frac{141}{4}$

$x = 2$

$z = 11$

Step 6.4

Simplify the right side.

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Step 6.4.1

Simplify $- \frac{11 (11)}{4} + \frac{141}{4}$ .

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Step 6.4.1.1

Combine the numerators over the common denominator.

$y = \frac{- 11 \cdot 11 + 141}{4}$

$x = 2$

$z = 11$

Step 6.4.1.2

Simplify the expression.

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Step 6.4.1.2.1

Multiply $- 11$ by $11$ .

$y = \frac{- 121 + 141}{4}$

$x = 2$

$z = 11$

Step 6.4.1.2.2

Add $- 121$ and $141$ .

$y = \frac{20}{4}$

$x = 2$

$z = 11$

Step 6.4.1.2.3

Divide $20$ by $4$ .

$y = 5$

$x = 2$

$z = 11$

$y = 5$

$x = 2$

$z = 11$

$y = 5$

$x = 2$

$z = 11$

$y = 5$

$x = 2$

$z = 11$

$y = 5$

$x = 2$

$z = 11$

Step 7

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

$(2, 5, 11)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(2, 5, 11)$

Equation Form:

$x = 2, y = 5, z = 11$