Finite Math Examples

$x + y + z = 73$ , $- 2 x + 5 y = 80$ , $x - z = - 9$

Step 1

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

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Step 1.1

Subtract $y$ from both sides of the equation.

$x + z = 73 - y$

$- 2 x + 5 y = 80$

$x - z = - 9$

Step 1.2

Subtract $z$ from both sides of the equation.

$x = 73 - y - z$

$- 2 x + 5 y = 80$

$x - z = - 9$

$x = 73 - y - z$

$- 2 x + 5 y = 80$

$x - z = - 9$

Step 2

Replace all occurrences of $x$ with $73 - y - z$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $- 2 x + 5 y = 80$ with $73 - y - z$ .

$- 2 (73 - y - z) + 5 y = 80$

$x = 73 - y - z$

$x - z = - 9$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $- 2 (73 - y - z) + 5 y$ .

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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.

$- 2 \cdot 73 - 2 (- y) - 2 (- z) + 5 y = 80$

$x = 73 - y - z$

$x - z = - 9$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $- 2$ by $73$ .

$- 146 - 2 (- y) - 2 (- z) + 5 y = 80$

$x = 73 - y - z$

$x - z = - 9$

Step 2.2.1.1.2.2

Multiply $- 1$ by $- 2$ .

$- 146 + 2 y - 2 (- z) + 5 y = 80$

$x = 73 - y - z$

$x - z = - 9$

Step 2.2.1.1.2.3

Multiply $- 1$ by $- 2$ .

$- 146 + 2 y + 2 z + 5 y = 80$

$x = 73 - y - z$

$x - z = - 9$

$- 146 + 2 y + 2 z + 5 y = 80$

$x = 73 - y - z$

$x - z = - 9$

$- 146 + 2 y + 2 z + 5 y = 80$

$x = 73 - y - z$

$x - z = - 9$

Step 2.2.1.2

Add $2 y$ and $5 y$ .

$- 146 + 7 y + 2 z = 80$

$x = 73 - y - z$

$x - z = - 9$

$- 146 + 7 y + 2 z = 80$

$x = 73 - y - z$

$x - z = - 9$

$- 146 + 7 y + 2 z = 80$

$x = 73 - y - z$

$x - z = - 9$

Step 2.3

Replace all occurrences of $x$ in $x - z = - 9$ with $73 - y - z$ .

$(73 - y - z) - z = - 9$

$- 146 + 7 y + 2 z = 80$

$x = 73 - y - z$

Step 2.4

Simplify the left side.

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Step 2.4.1

Subtract $z$ from $- z$ .

$73 - y - 2 z = - 9$

$- 146 + 7 y + 2 z = 80$

$x = 73 - y - z$

$73 - y - 2 z = - 9$

$- 146 + 7 y + 2 z = 80$

$x = 73 - y - z$

$73 - y - 2 z = - 9$

$- 146 + 7 y + 2 z = 80$

$x = 73 - y - z$

Step 3

Solve for $y$ in $73 - y - 2 z = - 9$ .

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Step 3.1

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

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Step 3.1.1

Subtract $73$ from both sides of the equation.

$- y - 2 z = - 9 - 73$

$- 146 + 7 y + 2 z = 80$

$x = 73 - y - z$

Step 3.1.2

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

$- y = - 9 - 73 + 2 z$

$- 146 + 7 y + 2 z = 80$

$x = 73 - y - z$

Step 3.1.3

Subtract $73$ from $- 9$ .

$- y = - 82 + 2 z$

$- 146 + 7 y + 2 z = 80$

$x = 73 - y - z$

$- y = - 82 + 2 z$

$- 146 + 7 y + 2 z = 80$

$x = 73 - y - z$

Step 3.2

Divide each term in $- y = - 82 + 2 z$ by $- 1$ and simplify.

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Step 3.2.1

Divide each term in $- y = - 82 + 2 z$ by $- 1$ .

$\frac{- y}{- 1} = \frac{- 82}{- 1} + \frac{2 z}{- 1}$

$- 146 + 7 y + 2 z = 80$

$x = 73 - y - z$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{- 82}{- 1} + \frac{2 z}{- 1}$

$- 146 + 7 y + 2 z = 80$

$x = 73 - y - z$

Step 3.2.2.2

Divide $y$ by $1$ .

$y = \frac{- 82}{- 1} + \frac{2 z}{- 1}$

$- 146 + 7 y + 2 z = 80$

$x = 73 - y - z$

$y = \frac{- 82}{- 1} + \frac{2 z}{- 1}$

$- 146 + 7 y + 2 z = 80$

$x = 73 - y - z$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

Divide $- 82$ by $- 1$ .

$y = 82 + \frac{2 z}{- 1}$

$- 146 + 7 y + 2 z = 80$

$x = 73 - y - z$

Step 3.2.3.1.2

Move the negative one from the denominator of $\frac{2 z}{- 1}$ .

$y = 82 - 1 \cdot (2 z)$

$- 146 + 7 y + 2 z = 80$

$x = 73 - y - z$

Step 3.2.3.1.3

Rewrite $- 1 \cdot (2 z)$ as $- (2 z)$ .

$y = 82 - (2 z)$

$- 146 + 7 y + 2 z = 80$

$x = 73 - y - z$

Step 3.2.3.1.4

Multiply $2$ by $- 1$ .

$y = 82 - 2 z$

$- 146 + 7 y + 2 z = 80$

$x = 73 - y - z$

$y = 82 - 2 z$

$- 146 + 7 y + 2 z = 80$

$x = 73 - y - z$

$y = 82 - 2 z$

$- 146 + 7 y + 2 z = 80$

$x = 73 - y - z$

$y = 82 - 2 z$

$- 146 + 7 y + 2 z = 80$

$x = 73 - y - z$

$y = 82 - 2 z$

$- 146 + 7 y + 2 z = 80$

$x = 73 - y - z$

Step 4

Replace all occurrences of $y$ with $82 - 2 z$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $- 146 + 7 y + 2 z = 80$ with $82 - 2 z$ .

$- 146 + 7 (82 - 2 z) + 2 z = 80$

$y = 82 - 2 z$

$x = 73 - y - z$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $- 146 + 7 (82 - 2 z) + 2 z$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Apply the distributive property.

$- 146 + 7 \cdot 82 + 7 (- 2 z) + 2 z = 80$

$y = 82 - 2 z$

$x = 73 - y - z$

Step 4.2.1.1.2

Multiply $7$ by $82$ .

$- 146 + 574 + 7 (- 2 z) + 2 z = 80$

$y = 82 - 2 z$

$x = 73 - y - z$

Step 4.2.1.1.3

Multiply $- 2$ by $7$ .

$- 146 + 574 - 14 z + 2 z = 80$

$y = 82 - 2 z$

$x = 73 - y - z$

$- 146 + 574 - 14 z + 2 z = 80$

$y = 82 - 2 z$

$x = 73 - y - z$

Step 4.2.1.2

Simplify by adding terms.

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Step 4.2.1.2.1

Add $- 146$ and $574$ .

$428 - 14 z + 2 z = 80$

$y = 82 - 2 z$

$x = 73 - y - z$

Step 4.2.1.2.2

Add $- 14 z$ and $2 z$ .

$428 - 12 z = 80$

$y = 82 - 2 z$

$x = 73 - y - z$

$428 - 12 z = 80$

$y = 82 - 2 z$

$x = 73 - y - z$

$428 - 12 z = 80$

$y = 82 - 2 z$

$x = 73 - y - z$

$428 - 12 z = 80$

$y = 82 - 2 z$

$x = 73 - y - z$

Step 4.3

Replace all occurrences of $y$ in $x = 73 - y - z$ with $82 - 2 z$ .

$x = 73 - (82 - 2 z) - z$

$428 - 12 z = 80$

$y = 82 - 2 z$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $73 - (82 - 2 z) - z$ .

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Step 4.4.1.1

Simplify each term.

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Step 4.4.1.1.1

Apply the distributive property.

$x = 73 - 1 \cdot 82 - (- 2 z) - z$

$428 - 12 z = 80$

$y = 82 - 2 z$

Step 4.4.1.1.2

Multiply $- 1$ by $82$ .

$x = 73 - 82 - (- 2 z) - z$

$428 - 12 z = 80$

$y = 82 - 2 z$

Step 4.4.1.1.3

Multiply $- 2$ by $- 1$ .

$x = 73 - 82 + 2 z - z$

$428 - 12 z = 80$

$y = 82 - 2 z$

$x = 73 - 82 + 2 z - z$

$428 - 12 z = 80$

$y = 82 - 2 z$

Step 4.4.1.2

Simplify by adding terms.

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Step 4.4.1.2.1

Subtract $82$ from $73$ .

$x = - 9 + 2 z - z$

$428 - 12 z = 80$

$y = 82 - 2 z$

Step 4.4.1.2.2

Subtract $z$ from $2 z$ .

$x = - 9 + z$

$428 - 12 z = 80$

$y = 82 - 2 z$

$x = - 9 + z$

$428 - 12 z = 80$

$y = 82 - 2 z$

$x = - 9 + z$

$428 - 12 z = 80$

$y = 82 - 2 z$

$x = - 9 + z$

$428 - 12 z = 80$

$y = 82 - 2 z$

$x = - 9 + z$

$428 - 12 z = 80$

$y = 82 - 2 z$

Step 5

Solve for $z$ in $428 - 12 z = 80$ .

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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 $428$ from both sides of the equation.

$- 12 z = 80 - 428$

$x = - 9 + z$

$y = 82 - 2 z$

Step 5.1.2

Subtract $428$ from $80$ .

$- 12 z = - 348$

$x = - 9 + z$

$y = 82 - 2 z$

$- 12 z = - 348$

$x = - 9 + z$

$y = 82 - 2 z$

Step 5.2

Divide each term in $- 12 z = - 348$ by $- 12$ and simplify.

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Step 5.2.1

Divide each term in $- 12 z = - 348$ by $- 12$ .

$\frac{- 12 z}{- 12} = \frac{- 348}{- 12}$

$x = - 9 + z$

$y = 82 - 2 z$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $- 12$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{- 12 z}{- 12} = \frac{- 348}{- 12}$

$x = - 9 + z$

$y = 82 - 2 z$

Step 5.2.2.1.2

Divide $z$ by $1$ .

$z = \frac{- 348}{- 12}$

$x = - 9 + z$

$y = 82 - 2 z$

$z = \frac{- 348}{- 12}$

$x = - 9 + z$

$y = 82 - 2 z$

$z = \frac{- 348}{- 12}$

$x = - 9 + z$

$y = 82 - 2 z$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $- 348$ by $- 12$ .

$z = 29$

$x = - 9 + z$

$y = 82 - 2 z$

$z = 29$

$x = - 9 + z$

$y = 82 - 2 z$

$z = 29$

$x = - 9 + z$

$y = 82 - 2 z$

$z = 29$

$x = - 9 + z$

$y = 82 - 2 z$

Step 6

Replace all occurrences of $z$ with $29$ in each equation.

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Step 6.1

Replace all occurrences of $z$ in $x = - 9 + z$ with $29$ .

$x = - 9 + 29$

$z = 29$

$y = 82 - 2 z$

Step 6.2

Simplify $x = - 9 + 29$ .

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Step 6.2.1

Simplify the left side.

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Step 6.2.1.1

Remove parentheses.

$x = - 9 + 29$

$z = 29$

$y = 82 - 2 z$

$x = - 9 + 29$

$z = 29$

$y = 82 - 2 z$

Step 6.2.2

Simplify the right side.

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Step 6.2.2.1

Add $- 9$ and $29$ .

$x = 20$

$z = 29$

$y = 82 - 2 z$

$x = 20$

$z = 29$

$y = 82 - 2 z$

$x = 20$

$z = 29$

$y = 82 - 2 z$

Step 6.3

Replace all occurrences of $z$ in $y = 82 - 2 z$ with $29$ .

$y = 82 - 2 \cdot 29$

$x = 20$

$z = 29$

Step 6.4

Simplify the right side.

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Step 6.4.1

Simplify $82 - 2 \cdot 29$ .

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Step 6.4.1.1

Multiply $- 2$ by $29$ .

$y = 82 - 58$

$x = 20$

$z = 29$

Step 6.4.1.2

Subtract $58$ from $82$ .

$y = 24$

$x = 20$

$z = 29$

$y = 24$

$x = 20$

$z = 29$

$y = 24$

$x = 20$

$z = 29$

$y = 24$

$x = 20$

$z = 29$

Step 7

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

$(20, 24, 29)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(20, 24, 29)$

Equation Form:

$x = 20, y = 24, z = 29$