Finite Math Examples

$x - y + 3 z = 13$ , $4 x + y + 2 z = 17$ , $3 x + 2 y + 2 z = 1$

Step 1

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

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Step 1.1

Add $y$ to both sides of the equation.

$x + 3 z = 13 + y$

$4 x + y + 2 z = 17$

$3 x + 2 y + 2 z = 1$

Step 1.2

Subtract $3 z$ from both sides of the equation.

$x = 13 + y - 3 z$

$4 x + y + 2 z = 17$

$3 x + 2 y + 2 z = 1$

$x = 13 + y - 3 z$

$4 x + y + 2 z = 17$

$3 x + 2 y + 2 z = 1$

Step 2

Replace all occurrences of $x$ with $13 + y - 3 z$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $4 x + y + 2 z = 17$ with $13 + y - 3 z$ .

$4 (13 + y - 3 z) + y + 2 z = 17$

$x = 13 + y - 3 z$

$3 x + 2 y + 2 z = 1$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $4 (13 + y - 3 z) + y + 2 z$ .

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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.

$4 \cdot 13 + 4 y + 4 (- 3 z) + y + 2 z = 17$

$x = 13 + y - 3 z$

$3 x + 2 y + 2 z = 1$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $4$ by $13$ .

$52 + 4 y + 4 (- 3 z) + y + 2 z = 17$

$x = 13 + y - 3 z$

$3 x + 2 y + 2 z = 1$

Step 2.2.1.1.2.2

Multiply $- 3$ by $4$ .

$52 + 4 y - 12 z + y + 2 z = 17$

$x = 13 + y - 3 z$

$3 x + 2 y + 2 z = 1$

$52 + 4 y - 12 z + y + 2 z = 17$

$x = 13 + y - 3 z$

$3 x + 2 y + 2 z = 1$

$52 + 4 y - 12 z + y + 2 z = 17$

$x = 13 + y - 3 z$

$3 x + 2 y + 2 z = 1$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Add $4 y$ and $y$ .

$52 + 5 y - 12 z + 2 z = 17$

$x = 13 + y - 3 z$

$3 x + 2 y + 2 z = 1$

Step 2.2.1.2.2

Add $- 12 z$ and $2 z$ .

$52 + 5 y - 10 z = 17$

$x = 13 + y - 3 z$

$3 x + 2 y + 2 z = 1$

$52 + 5 y - 10 z = 17$

$x = 13 + y - 3 z$

$3 x + 2 y + 2 z = 1$

$52 + 5 y - 10 z = 17$

$x = 13 + y - 3 z$

$3 x + 2 y + 2 z = 1$

$52 + 5 y - 10 z = 17$

$x = 13 + y - 3 z$

$3 x + 2 y + 2 z = 1$

Step 2.3

Replace all occurrences of $x$ in $3 x + 2 y + 2 z = 1$ with $13 + y - 3 z$ .

$3 (13 + y - 3 z) + 2 y + 2 z = 1$

$52 + 5 y - 10 z = 17$

$x = 13 + y - 3 z$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $3 (13 + y - 3 z) + 2 y + 2 z$ .

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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.

$3 \cdot 13 + 3 y + 3 (- 3 z) + 2 y + 2 z = 1$

$52 + 5 y - 10 z = 17$

$x = 13 + y - 3 z$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $3$ by $13$ .

$39 + 3 y + 3 (- 3 z) + 2 y + 2 z = 1$

$52 + 5 y - 10 z = 17$

$x = 13 + y - 3 z$

Step 2.4.1.1.2.2

Multiply $- 3$ by $3$ .

$39 + 3 y - 9 z + 2 y + 2 z = 1$

$52 + 5 y - 10 z = 17$

$x = 13 + y - 3 z$

$39 + 3 y - 9 z + 2 y + 2 z = 1$

$52 + 5 y - 10 z = 17$

$x = 13 + y - 3 z$

$39 + 3 y - 9 z + 2 y + 2 z = 1$

$52 + 5 y - 10 z = 17$

$x = 13 + y - 3 z$

Step 2.4.1.2

Simplify by adding terms.

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Step 2.4.1.2.1

Add $3 y$ and $2 y$ .

$39 + 5 y - 9 z + 2 z = 1$

$52 + 5 y - 10 z = 17$

$x = 13 + y - 3 z$

Step 2.4.1.2.2

Add $- 9 z$ and $2 z$ .

$39 + 5 y - 7 z = 1$

$52 + 5 y - 10 z = 17$

$x = 13 + y - 3 z$

$39 + 5 y - 7 z = 1$

$52 + 5 y - 10 z = 17$

$x = 13 + y - 3 z$

$39 + 5 y - 7 z = 1$

$52 + 5 y - 10 z = 17$

$x = 13 + y - 3 z$

$39 + 5 y - 7 z = 1$

$52 + 5 y - 10 z = 17$

$x = 13 + y - 3 z$

$39 + 5 y - 7 z = 1$

$52 + 5 y - 10 z = 17$

$x = 13 + y - 3 z$

Step 3

Solve for $y$ in $39 + 5 y - 7 z = 1$ .

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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 $39$ from both sides of the equation.

$5 y - 7 z = 1 - 39$

$52 + 5 y - 10 z = 17$

$x = 13 + y - 3 z$

Step 3.1.2

Add $7 z$ to both sides of the equation.

$5 y = 1 - 39 + 7 z$

$52 + 5 y - 10 z = 17$

$x = 13 + y - 3 z$

Step 3.1.3

Subtract $39$ from $1$ .

$5 y = - 38 + 7 z$

$52 + 5 y - 10 z = 17$

$x = 13 + y - 3 z$

$5 y = - 38 + 7 z$

$52 + 5 y - 10 z = 17$

$x = 13 + y - 3 z$

Step 3.2

Divide each term in $5 y = - 38 + 7 z$ by $5$ and simplify.

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Step 3.2.1

Divide each term in $5 y = - 38 + 7 z$ by $5$ .

$\frac{5 y}{5} = \frac{- 38}{5} + \frac{7 z}{5}$

$52 + 5 y - 10 z = 17$

$x = 13 + y - 3 z$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $5$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{5 y}{5} = \frac{- 38}{5} + \frac{7 z}{5}$

$52 + 5 y - 10 z = 17$

$x = 13 + y - 3 z$

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 38}{5} + \frac{7 z}{5}$

$52 + 5 y - 10 z = 17$

$x = 13 + y - 3 z$

$y = \frac{- 38}{5} + \frac{7 z}{5}$

$52 + 5 y - 10 z = 17$

$x = 13 + y - 3 z$

$y = \frac{- 38}{5} + \frac{7 z}{5}$

$52 + 5 y - 10 z = 17$

$x = 13 + y - 3 z$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Move the negative in front of the fraction.

$y = - \frac{38}{5} + \frac{7 z}{5}$

$52 + 5 y - 10 z = 17$

$x = 13 + y - 3 z$

$y = - \frac{38}{5} + \frac{7 z}{5}$

$52 + 5 y - 10 z = 17$

$x = 13 + y - 3 z$

$y = - \frac{38}{5} + \frac{7 z}{5}$

$52 + 5 y - 10 z = 17$

$x = 13 + y - 3 z$

$y = - \frac{38}{5} + \frac{7 z}{5}$

$52 + 5 y - 10 z = 17$

$x = 13 + y - 3 z$

Step 4

Replace all occurrences of $y$ with $- \frac{38}{5} + \frac{7 z}{5}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $52 + 5 y - 10 z = 17$ with $- \frac{38}{5} + \frac{7 z}{5}$ .

$52 + 5 (- \frac{38}{5} + \frac{7 z}{5}) - 10 z = 17$

$y = - \frac{38}{5} + \frac{7 z}{5}$

$x = 13 + y - 3 z$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $52 + 5 (- \frac{38}{5} + \frac{7 z}{5}) - 10 z$ .

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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.

$52 + 5 (- \frac{38}{5}) + 5 (\frac{7 z}{5}) - 10 z = 17$

$y = - \frac{38}{5} + \frac{7 z}{5}$

$x = 13 + y - 3 z$

Step 4.2.1.1.2

Cancel the common factor of $5$ .

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Step 4.2.1.1.2.1

Move the leading negative in $- \frac{38}{5}$ into the numerator.

$52 + 5 (\frac{- 38}{5}) + 5 (\frac{7 z}{5}) - 10 z = 17$

$y = - \frac{38}{5} + \frac{7 z}{5}$

$x = 13 + y - 3 z$

Step 4.2.1.1.2.2

Cancel the common factor.

$52 + 5 (\frac{- 38}{5}) + 5 (\frac{7 z}{5}) - 10 z = 17$

$y = - \frac{38}{5} + \frac{7 z}{5}$

$x = 13 + y - 3 z$

Step 4.2.1.1.2.3

Rewrite the expression.

$52 - 38 + 5 (\frac{7 z}{5}) - 10 z = 17$

$y = - \frac{38}{5} + \frac{7 z}{5}$

$x = 13 + y - 3 z$

$52 - 38 + 5 (\frac{7 z}{5}) - 10 z = 17$

$y = - \frac{38}{5} + \frac{7 z}{5}$

$x = 13 + y - 3 z$

Step 4.2.1.1.3

Cancel the common factor of $5$ .

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Step 4.2.1.1.3.1

Cancel the common factor.

$52 - 38 + 5 (\frac{7 z}{5}) - 10 z = 17$

$y = - \frac{38}{5} + \frac{7 z}{5}$

$x = 13 + y - 3 z$

Step 4.2.1.1.3.2

Rewrite the expression.

$52 - 38 + 7 z - 10 z = 17$

$y = - \frac{38}{5} + \frac{7 z}{5}$

$x = 13 + y - 3 z$

$52 - 38 + 7 z - 10 z = 17$

$y = - \frac{38}{5} + \frac{7 z}{5}$

$x = 13 + y - 3 z$

$52 - 38 + 7 z - 10 z = 17$

$y = - \frac{38}{5} + \frac{7 z}{5}$

$x = 13 + y - 3 z$

Step 4.2.1.2

Simplify by adding terms.

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Step 4.2.1.2.1

Subtract $38$ from $52$ .

$14 + 7 z - 10 z = 17$

$y = - \frac{38}{5} + \frac{7 z}{5}$

$x = 13 + y - 3 z$

Step 4.2.1.2.2

Subtract $10 z$ from $7 z$ .

$14 - 3 z = 17$

$y = - \frac{38}{5} + \frac{7 z}{5}$

$x = 13 + y - 3 z$

$14 - 3 z = 17$

$y = - \frac{38}{5} + \frac{7 z}{5}$

$x = 13 + y - 3 z$

$14 - 3 z = 17$

$y = - \frac{38}{5} + \frac{7 z}{5}$

$x = 13 + y - 3 z$

$14 - 3 z = 17$

$y = - \frac{38}{5} + \frac{7 z}{5}$

$x = 13 + y - 3 z$

Step 4.3

Replace all occurrences of $y$ in $x = 13 + y - 3 z$ with $- \frac{38}{5} + \frac{7 z}{5}$ .

$x = 13 - \frac{38}{5} + \frac{7 z}{5} - 3 z$

$14 - 3 z = 17$

$y = - \frac{38}{5} + \frac{7 z}{5}$

Step 4.4

Simplify $x = 13 - \frac{38}{5} + \frac{7 z}{5} - 3 z$ .

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Step 4.4.1

Simplify the left side.

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Step 4.4.1.1

Remove parentheses.

$x = 13 - \frac{38}{5} + \frac{7 z}{5} - 3 z$

$14 - 3 z = 17$

$y = - \frac{38}{5} + \frac{7 z}{5}$

$x = 13 - \frac{38}{5} + \frac{7 z}{5} - 3 z$

$14 - 3 z = 17$

$y = - \frac{38}{5} + \frac{7 z}{5}$

Step 4.4.2

Simplify the right side.

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Step 4.4.2.1

Simplify $13 - \frac{38}{5} + \frac{7 z}{5} - 3 z$ .

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Step 4.4.2.1.1

To write $13$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$x = 13 \cdot \frac{5}{5} - \frac{38}{5} + \frac{7 z}{5} - 3 z$

$14 - 3 z = 17$

$y = - \frac{38}{5} + \frac{7 z}{5}$

Step 4.4.2.1.2

Combine $13$ and $\frac{5}{5}$ .

$x = \frac{13 \cdot 5}{5} - \frac{38}{5} + \frac{7 z}{5} - 3 z$

$14 - 3 z = 17$

$y = - \frac{38}{5} + \frac{7 z}{5}$

Step 4.4.2.1.3

Combine the numerators over the common denominator.

$x = \frac{13 \cdot 5 - 38}{5} + \frac{7 z}{5} - 3 z$

$14 - 3 z = 17$

$y = - \frac{38}{5} + \frac{7 z}{5}$

Step 4.4.2.1.4

Simplify the numerator.

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Step 4.4.2.1.4.1

Multiply $13$ by $5$ .

$x = \frac{65 - 38}{5} + \frac{7 z}{5} - 3 z$

$14 - 3 z = 17$

$y = - \frac{38}{5} + \frac{7 z}{5}$

Step 4.4.2.1.4.2

Subtract $38$ from $65$ .

$x = \frac{27}{5} + \frac{7 z}{5} - 3 z$

$14 - 3 z = 17$

$y = - \frac{38}{5} + \frac{7 z}{5}$

$x = \frac{27}{5} + \frac{7 z}{5} - 3 z$

$14 - 3 z = 17$

$y = - \frac{38}{5} + \frac{7 z}{5}$

Step 4.4.2.1.5

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

$x = \frac{27}{5} + \frac{7 z}{5} - 3 z \cdot \frac{5}{5}$

$14 - 3 z = 17$

$y = - \frac{38}{5} + \frac{7 z}{5}$

Step 4.4.2.1.6

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

$x = \frac{27}{5} + \frac{7 z}{5} + \frac{- 3 z \cdot 5}{5}$

$14 - 3 z = 17$

$y = - \frac{38}{5} + \frac{7 z}{5}$

Step 4.4.2.1.7

Combine the numerators over the common denominator.

$x = \frac{27}{5} + \frac{7 z - 3 z \cdot 5}{5}$

$14 - 3 z = 17$

$y = - \frac{38}{5} + \frac{7 z}{5}$

Step 4.4.2.1.8

Combine the numerators over the common denominator.

$x = \frac{27 + 7 z - 3 z \cdot 5}{5}$

$14 - 3 z = 17$

$y = - \frac{38}{5} + \frac{7 z}{5}$

Step 4.4.2.1.9

Multiply $5$ by $- 3$ .

$x = \frac{27 + 7 z - 15 z}{5}$

$14 - 3 z = 17$

$y = - \frac{38}{5} + \frac{7 z}{5}$

Step 4.4.2.1.10

Subtract $15 z$ from $7 z$ .

$x = \frac{27 - 8 z}{5}$

$14 - 3 z = 17$

$y = - \frac{38}{5} + \frac{7 z}{5}$

$x = \frac{27 - 8 z}{5}$

$14 - 3 z = 17$

$y = - \frac{38}{5} + \frac{7 z}{5}$

$x = \frac{27 - 8 z}{5}$

$14 - 3 z = 17$

$y = - \frac{38}{5} + \frac{7 z}{5}$

$x = \frac{27 - 8 z}{5}$

$14 - 3 z = 17$

$y = - \frac{38}{5} + \frac{7 z}{5}$

$x = \frac{27 - 8 z}{5}$

$14 - 3 z = 17$

$y = - \frac{38}{5} + \frac{7 z}{5}$

Step 5

Solve for $z$ in $14 - 3 z = 17$ .

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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 $14$ from both sides of the equation.

$- 3 z = 17 - 14$

$x = \frac{27 - 8 z}{5}$

$y = - \frac{38}{5} + \frac{7 z}{5}$

Step 5.1.2

Subtract $14$ from $17$ .

$- 3 z = 3$

$x = \frac{27 - 8 z}{5}$

$y = - \frac{38}{5} + \frac{7 z}{5}$

$- 3 z = 3$

$x = \frac{27 - 8 z}{5}$

$y = - \frac{38}{5} + \frac{7 z}{5}$

Step 5.2

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

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Step 5.2.1

Divide each term in $- 3 z = 3$ by $- 3$ .

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

$x = \frac{27 - 8 z}{5}$

$y = - \frac{38}{5} + \frac{7 z}{5}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $- 3$ .

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Step 5.2.2.1.1

Cancel the common factor.

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

$x = \frac{27 - 8 z}{5}$

$y = - \frac{38}{5} + \frac{7 z}{5}$

Step 5.2.2.1.2

Divide $z$ by $1$ .

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

$x = \frac{27 - 8 z}{5}$

$y = - \frac{38}{5} + \frac{7 z}{5}$

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

$x = \frac{27 - 8 z}{5}$

$y = - \frac{38}{5} + \frac{7 z}{5}$

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

$x = \frac{27 - 8 z}{5}$

$y = - \frac{38}{5} + \frac{7 z}{5}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $3$ by $- 3$ .

$z = - 1$

$x = \frac{27 - 8 z}{5}$

$y = - \frac{38}{5} + \frac{7 z}{5}$

$z = - 1$

$x = \frac{27 - 8 z}{5}$

$y = - \frac{38}{5} + \frac{7 z}{5}$

$z = - 1$

$x = \frac{27 - 8 z}{5}$

$y = - \frac{38}{5} + \frac{7 z}{5}$

$z = - 1$

$x = \frac{27 - 8 z}{5}$

$y = - \frac{38}{5} + \frac{7 z}{5}$

Step 6

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

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Step 6.1

Replace all occurrences of $z$ in $x = \frac{27 - 8 z}{5}$ with $- 1$ .

$x = \frac{27 - 8 \cdot - 1}{5}$

$z = - 1$

$y = - \frac{38}{5} + \frac{7 z}{5}$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $\frac{27 - 8 \cdot - 1}{5}$ .

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Step 6.2.1.1

Simplify the numerator.

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Step 6.2.1.1.1

Multiply $- 8$ by $- 1$ .

$x = \frac{27 + 8}{5}$

$z = - 1$

$y = - \frac{38}{5} + \frac{7 z}{5}$

Step 6.2.1.1.2

Add $27$ and $8$ .

$x = \frac{35}{5}$

$z = - 1$

$y = - \frac{38}{5} + \frac{7 z}{5}$

$x = \frac{35}{5}$

$z = - 1$

$y = - \frac{38}{5} + \frac{7 z}{5}$

Step 6.2.1.2

Divide $35$ by $5$ .

$x = 7$

$z = - 1$

$y = - \frac{38}{5} + \frac{7 z}{5}$

$x = 7$

$z = - 1$

$y = - \frac{38}{5} + \frac{7 z}{5}$

$x = 7$

$z = - 1$

$y = - \frac{38}{5} + \frac{7 z}{5}$

Step 6.3

Replace all occurrences of $z$ in $y = - \frac{38}{5} + \frac{7 z}{5}$ with $- 1$ .

$y = - \frac{38}{5} + \frac{7 (- 1)}{5}$

$x = 7$

$z = - 1$

Step 6.4

Simplify the right side.

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Step 6.4.1

Simplify $- \frac{38}{5} + \frac{7 (- 1)}{5}$ .

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Step 6.4.1.1

Combine the numerators over the common denominator.

$y = \frac{- 38 + 7 (- 1)}{5}$

$x = 7$

$z = - 1$

Step 6.4.1.2

Simplify the expression.

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Step 6.4.1.2.1

Multiply $7$ by $- 1$ .

$y = \frac{- 38 - 7}{5}$

$x = 7$

$z = - 1$

Step 6.4.1.2.2

Subtract $7$ from $- 38$ .

$y = \frac{- 45}{5}$

$x = 7$

$z = - 1$

Step 6.4.1.2.3

Divide $- 45$ by $5$ .

$y = - 9$

$x = 7$

$z = - 1$

$y = - 9$

$x = 7$

$z = - 1$

$y = - 9$

$x = 7$

$z = - 1$

$y = - 9$

$x = 7$

$z = - 1$

$y = - 9$

$x = 7$

$z = - 1$

Step 7

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

$(7, - 9, - 1)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(7, - 9, - 1)$

Equation Form:

$x = 7, y = - 9, z = - 1$