Finite Math Examples

$2 x - y + 3 z = 15$ , $4 x + y - z = - 7$ , $- 3 x + 4 y + 5 z = 8$

Step 1

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

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Step 1.1

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

$y - z = - 7 - 4 x$

$2 x - y + 3 z = 15$

$- 3 x + 4 y + 5 z = 8$

Step 1.2

Add $z$ to both sides of the equation.

$y = - 7 - 4 x + z$

$2 x - y + 3 z = 15$

$- 3 x + 4 y + 5 z = 8$

$y = - 7 - 4 x + z$

$2 x - y + 3 z = 15$

$- 3 x + 4 y + 5 z = 8$

Step 2

Replace all occurrences of $y$ with $- 7 - 4 x + z$ in each equation.

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Step 2.1

Replace all occurrences of $y$ in $2 x - y + 3 z = 15$ with $- 7 - 4 x + z$ .

$2 x - (- 7 - 4 x + z) + 3 z = 15$

$y = - 7 - 4 x + z$

$- 3 x + 4 y + 5 z = 8$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $2 x - (- 7 - 4 x + z) + 3 z$ .

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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.

$2 x + 7 - (- 4 x) - z + 3 z = 15$

$y = - 7 - 4 x + z$

$- 3 x + 4 y + 5 z = 8$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $- 1$ by $- 7$ .

$2 x + 7 - (- 4 x) - z + 3 z = 15$

$y = - 7 - 4 x + z$

$- 3 x + 4 y + 5 z = 8$

Step 2.2.1.1.2.2

Multiply $- 4$ by $- 1$ .

$2 x + 7 + 4 x - z + 3 z = 15$

$y = - 7 - 4 x + z$

$- 3 x + 4 y + 5 z = 8$

$2 x + 7 + 4 x - z + 3 z = 15$

$y = - 7 - 4 x + z$

$- 3 x + 4 y + 5 z = 8$

$2 x + 7 + 4 x - z + 3 z = 15$

$y = - 7 - 4 x + z$

$- 3 x + 4 y + 5 z = 8$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Add $2 x$ and $4 x$ .

$6 x + 7 - z + 3 z = 15$

$y = - 7 - 4 x + z$

$- 3 x + 4 y + 5 z = 8$

Step 2.2.1.2.2

Add $- z$ and $3 z$ .

$6 x + 7 + 2 z = 15$

$y = - 7 - 4 x + z$

$- 3 x + 4 y + 5 z = 8$

$6 x + 7 + 2 z = 15$

$y = - 7 - 4 x + z$

$- 3 x + 4 y + 5 z = 8$

$6 x + 7 + 2 z = 15$

$y = - 7 - 4 x + z$

$- 3 x + 4 y + 5 z = 8$

$6 x + 7 + 2 z = 15$

$y = - 7 - 4 x + z$

$- 3 x + 4 y + 5 z = 8$

Step 2.3

Replace all occurrences of $y$ in $- 3 x + 4 y + 5 z = 8$ with $- 7 - 4 x + z$ .

$- 3 x + 4 (- 7 - 4 x + z) + 5 z = 8$

$6 x + 7 + 2 z = 15$

$y = - 7 - 4 x + z$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $- 3 x + 4 (- 7 - 4 x + z) + 5 z$ .

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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.

$- 3 x + 4 \cdot - 7 + 4 (- 4 x) + 4 z + 5 z = 8$

$6 x + 7 + 2 z = 15$

$y = - 7 - 4 x + z$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $4$ by $- 7$ .

$- 3 x - 28 + 4 (- 4 x) + 4 z + 5 z = 8$

$6 x + 7 + 2 z = 15$

$y = - 7 - 4 x + z$

Step 2.4.1.1.2.2

Multiply $- 4$ by $4$ .

$- 3 x - 28 - 16 x + 4 z + 5 z = 8$

$6 x + 7 + 2 z = 15$

$y = - 7 - 4 x + z$

$- 3 x - 28 - 16 x + 4 z + 5 z = 8$

$6 x + 7 + 2 z = 15$

$y = - 7 - 4 x + z$

$- 3 x - 28 - 16 x + 4 z + 5 z = 8$

$6 x + 7 + 2 z = 15$

$y = - 7 - 4 x + z$

Step 2.4.1.2

Simplify by adding terms.

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Step 2.4.1.2.1

Subtract $16 x$ from $- 3 x$ .

$- 19 x - 28 + 4 z + 5 z = 8$

$6 x + 7 + 2 z = 15$

$y = - 7 - 4 x + z$

Step 2.4.1.2.2

Add $4 z$ and $5 z$ .

$- 19 x - 28 + 9 z = 8$

$6 x + 7 + 2 z = 15$

$y = - 7 - 4 x + z$

$- 19 x - 28 + 9 z = 8$

$6 x + 7 + 2 z = 15$

$y = - 7 - 4 x + z$

$- 19 x - 28 + 9 z = 8$

$6 x + 7 + 2 z = 15$

$y = - 7 - 4 x + z$

$- 19 x - 28 + 9 z = 8$

$6 x + 7 + 2 z = 15$

$y = - 7 - 4 x + z$

$- 19 x - 28 + 9 z = 8$

$6 x + 7 + 2 z = 15$

$y = - 7 - 4 x + z$

Step 3

Solve for $x$ in $- 19 x - 28 + 9 z = 8$ .

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Step 3.1

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

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Step 3.1.1

Add $28$ to both sides of the equation.

$- 19 x + 9 z = 8 + 28$

$6 x + 7 + 2 z = 15$

$y = - 7 - 4 x + z$

Step 3.1.2

Subtract $9 z$ from both sides of the equation.

$- 19 x = 8 + 28 - 9 z$

$6 x + 7 + 2 z = 15$

$y = - 7 - 4 x + z$

Step 3.1.3

Add $8$ and $28$ .

$- 19 x = 36 - 9 z$

$6 x + 7 + 2 z = 15$

$y = - 7 - 4 x + z$

$- 19 x = 36 - 9 z$

$6 x + 7 + 2 z = 15$

$y = - 7 - 4 x + z$

Step 3.2

Divide each term in $- 19 x = 36 - 9 z$ by $- 19$ and simplify.

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Step 3.2.1

Divide each term in $- 19 x = 36 - 9 z$ by $- 19$ .

$\frac{- 19 x}{- 19} = \frac{36}{- 19} + \frac{- 9 z}{- 19}$

$6 x + 7 + 2 z = 15$

$y = - 7 - 4 x + z$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $- 19$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{- 19 x}{- 19} = \frac{36}{- 19} + \frac{- 9 z}{- 19}$

$6 x + 7 + 2 z = 15$

$y = - 7 - 4 x + z$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{36}{- 19} + \frac{- 9 z}{- 19}$

$6 x + 7 + 2 z = 15$

$y = - 7 - 4 x + z$

$x = \frac{36}{- 19} + \frac{- 9 z}{- 19}$

$6 x + 7 + 2 z = 15$

$y = - 7 - 4 x + z$

$x = \frac{36}{- 19} + \frac{- 9 z}{- 19}$

$6 x + 7 + 2 z = 15$

$y = - 7 - 4 x + z$

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

Move the negative in front of the fraction.

$x = - \frac{36}{19} + \frac{- 9 z}{- 19}$

$6 x + 7 + 2 z = 15$

$y = - 7 - 4 x + z$

Step 3.2.3.1.2

Dividing two negative values results in a positive value.

$x = - \frac{36}{19} + \frac{9 z}{19}$

$6 x + 7 + 2 z = 15$

$y = - 7 - 4 x + z$

$x = - \frac{36}{19} + \frac{9 z}{19}$

$6 x + 7 + 2 z = 15$

$y = - 7 - 4 x + z$

$x = - \frac{36}{19} + \frac{9 z}{19}$

$6 x + 7 + 2 z = 15$

$y = - 7 - 4 x + z$

$x = - \frac{36}{19} + \frac{9 z}{19}$

$6 x + 7 + 2 z = 15$

$y = - 7 - 4 x + z$

$x = - \frac{36}{19} + \frac{9 z}{19}$

$6 x + 7 + 2 z = 15$

$y = - 7 - 4 x + z$

Step 4

Replace all occurrences of $x$ with $- \frac{36}{19} + \frac{9 z}{19}$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $6 x + 7 + 2 z = 15$ with $- \frac{36}{19} + \frac{9 z}{19}$ .

$6 (- \frac{36}{19} + \frac{9 z}{19}) + 7 + 2 z = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

$y = - 7 - 4 x + z$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $6 (- \frac{36}{19} + \frac{9 z}{19}) + 7 + 2 z$ .

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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.

$6 (- \frac{36}{19}) + 6 (\frac{9 z}{19}) + 7 + 2 z = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

$y = - 7 - 4 x + z$

Step 4.2.1.1.2

Multiply $6 (- \frac{36}{19})$ .

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Step 4.2.1.1.2.1

Multiply $- 1$ by $6$ .

$- 6 (\frac{36}{19}) + 6 (\frac{9 z}{19}) + 7 + 2 z = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

$y = - 7 - 4 x + z$

Step 4.2.1.1.2.2

Combine $- 6$ and $\frac{36}{19}$ .

$\frac{- 6 \cdot 36}{19} + 6 (\frac{9 z}{19}) + 7 + 2 z = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

$y = - 7 - 4 x + z$

Step 4.2.1.1.2.3

Multiply $- 6$ by $36$ .

$\frac{- 216}{19} + 6 (\frac{9 z}{19}) + 7 + 2 z = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

$y = - 7 - 4 x + z$

$\frac{- 216}{19} + 6 (\frac{9 z}{19}) + 7 + 2 z = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

$y = - 7 - 4 x + z$

Step 4.2.1.1.3

Multiply $6 \frac{9 z}{19}$ .

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Step 4.2.1.1.3.1

Combine $6$ and $\frac{9 z}{19}$ .

$\frac{- 216}{19} + \frac{6 (9 z)}{19} + 7 + 2 z = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

$y = - 7 - 4 x + z$

Step 4.2.1.1.3.2

Multiply $9$ by $6$ .

$\frac{- 216}{19} + \frac{54 z}{19} + 7 + 2 z = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

$y = - 7 - 4 x + z$

$\frac{- 216}{19} + \frac{54 z}{19} + 7 + 2 z = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

$y = - 7 - 4 x + z$

Step 4.2.1.1.4

Move the negative in front of the fraction.

$- \frac{216}{19} + \frac{54 z}{19} + 7 + 2 z = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

$y = - 7 - 4 x + z$

$- \frac{216}{19} + \frac{54 z}{19} + 7 + 2 z = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

$y = - 7 - 4 x + z$

Step 4.2.1.2

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

$\frac{54 z}{19} - \frac{216}{19} + 7 \cdot \frac{19}{19} + 2 z = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

$y = - 7 - 4 x + z$

Step 4.2.1.3

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

$\frac{54 z}{19} - \frac{216}{19} + \frac{7 \cdot 19}{19} + 2 z = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

$y = - 7 - 4 x + z$

Step 4.2.1.4

Combine the numerators over the common denominator.

$\frac{54 z}{19} + \frac{- 216 + 7 \cdot 19}{19} + 2 z = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

$y = - 7 - 4 x + z$

Step 4.2.1.5

Simplify the numerator.

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Step 4.2.1.5.1

Multiply $7$ by $19$ .

$\frac{54 z}{19} + \frac{- 216 + 133}{19} + 2 z = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

$y = - 7 - 4 x + z$

Step 4.2.1.5.2

Add $- 216$ and $133$ .

$\frac{54 z}{19} + \frac{- 83}{19} + 2 z = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

$y = - 7 - 4 x + z$

$\frac{54 z}{19} + \frac{- 83}{19} + 2 z = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

$y = - 7 - 4 x + z$

Step 4.2.1.6

Move the negative in front of the fraction.

$\frac{54 z}{19} - \frac{83}{19} + 2 z = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

$y = - 7 - 4 x + z$

Step 4.2.1.7

To write $2 z$ as a fraction with a common denominator, multiply by $\frac{19}{19}$ .

$\frac{54 z}{19} + 2 z \cdot \frac{19}{19} - \frac{83}{19} = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

$y = - 7 - 4 x + z$

Step 4.2.1.8

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

$\frac{54 z}{19} + \frac{2 z \cdot 19}{19} - \frac{83}{19} = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

$y = - 7 - 4 x + z$

Step 4.2.1.9

Combine the numerators over the common denominator.

$\frac{54 z + 2 z \cdot 19}{19} - \frac{83}{19} = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

$y = - 7 - 4 x + z$

Step 4.2.1.10

Combine the numerators over the common denominator.

$\frac{54 z + 2 z \cdot 19 - 83}{19} = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

$y = - 7 - 4 x + z$

Step 4.2.1.11

Multiply $19$ by $2$ .

$\frac{54 z + 38 z - 83}{19} = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

$y = - 7 - 4 x + z$

Step 4.2.1.12

Add $54 z$ and $38 z$ .

$\frac{92 z - 83}{19} = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

$y = - 7 - 4 x + z$

$\frac{92 z - 83}{19} = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

$y = - 7 - 4 x + z$

$\frac{92 z - 83}{19} = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

$y = - 7 - 4 x + z$

Step 4.3

Replace all occurrences of $x$ in $y = - 7 - 4 x + z$ with $- \frac{36}{19} + \frac{9 z}{19}$ .

$y = - 7 - 4 (- \frac{36}{19} + \frac{9 z}{19}) + z$

$\frac{92 z - 83}{19} = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $- 7 - 4 (- \frac{36}{19} + \frac{9 z}{19}) + z$ .

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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.

$y = - 7 - 4 (- \frac{36}{19}) - 4 \frac{9 z}{19} + z$

$\frac{92 z - 83}{19} = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

Step 4.4.1.1.2

Multiply $- 4 (- \frac{36}{19})$ .

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Step 4.4.1.1.2.1

Multiply $- 1$ by $- 4$ .

$y = - 7 + 4 (\frac{36}{19}) - 4 \frac{9 z}{19} + z$

$\frac{92 z - 83}{19} = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

Step 4.4.1.1.2.2

Combine $4$ and $\frac{36}{19}$ .

$y = - 7 + \frac{4 \cdot 36}{19} - 4 \frac{9 z}{19} + z$

$\frac{92 z - 83}{19} = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

Step 4.4.1.1.2.3

Multiply $4$ by $36$ .

$y = - 7 + \frac{144}{19} - 4 \frac{9 z}{19} + z$

$\frac{92 z - 83}{19} = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

$y = - 7 + \frac{144}{19} - 4 \frac{9 z}{19} + z$

$\frac{92 z - 83}{19} = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

Step 4.4.1.1.3

Multiply $- 4 \frac{9 z}{19}$ .

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Step 4.4.1.1.3.1

Combine $- 4$ and $\frac{9 z}{19}$ .

$y = - 7 + \frac{144}{19} + \frac{- 4 (9 z)}{19} + z$

$\frac{92 z - 83}{19} = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

Step 4.4.1.1.3.2

Multiply $9$ by $- 4$ .

$y = - 7 + \frac{144}{19} + \frac{- 36 z}{19} + z$

$\frac{92 z - 83}{19} = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

$y = - 7 + \frac{144}{19} + \frac{- 36 z}{19} + z$

$\frac{92 z - 83}{19} = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

Step 4.4.1.1.4

Move the negative in front of the fraction.

$y = - 7 + \frac{144}{19} - \frac{36 z}{19} + z$

$\frac{92 z - 83}{19} = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

$y = - 7 + \frac{144}{19} - \frac{36 z}{19} + z$

$\frac{92 z - 83}{19} = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

Step 4.4.1.2

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

$y = - 7 \cdot \frac{19}{19} + \frac{144}{19} - \frac{36 z}{19} + z$

$\frac{92 z - 83}{19} = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

Step 4.4.1.3

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

$y = \frac{- 7 \cdot 19}{19} + \frac{144}{19} - \frac{36 z}{19} + z$

$\frac{92 z - 83}{19} = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

Step 4.4.1.4

Combine the numerators over the common denominator.

$y = \frac{- 7 \cdot 19 + 144}{19} - \frac{36 z}{19} + z$

$\frac{92 z - 83}{19} = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

Step 4.4.1.5

Simplify the numerator.

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Step 4.4.1.5.1

Multiply $- 7$ by $19$ .

$y = \frac{- 133 + 144}{19} - \frac{36 z}{19} + z$

$\frac{92 z - 83}{19} = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

Step 4.4.1.5.2

Add $- 133$ and $144$ .

$y = \frac{11}{19} - \frac{36 z}{19} + z$

$\frac{92 z - 83}{19} = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

$y = \frac{11}{19} - \frac{36 z}{19} + z$

$\frac{92 z - 83}{19} = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

Step 4.4.1.6

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

$y = \frac{11}{19} - \frac{36 z}{19} + z \cdot \frac{19}{19}$

$\frac{92 z - 83}{19} = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

Step 4.4.1.7

Simplify terms.

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Step 4.4.1.7.1

Combine $z$ and $\frac{19}{19}$ .

$y = \frac{11}{19} - \frac{36 z}{19} + \frac{z \cdot 19}{19}$

$\frac{92 z - 83}{19} = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

Step 4.4.1.7.2

Combine the numerators over the common denominator.

$y = \frac{11}{19} + \frac{- 36 z + z \cdot 19}{19}$

$\frac{92 z - 83}{19} = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

Step 4.4.1.7.3

Combine the numerators over the common denominator.

$y = \frac{11 - 36 z + z \cdot 19}{19}$

$\frac{92 z - 83}{19} = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

$y = \frac{11 - 36 z + z \cdot 19}{19}$

$\frac{92 z - 83}{19} = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

Step 4.4.1.8

Move $19$ to the left of $z$ .

$y = \frac{11 - 36 z + 19 z}{19}$

$\frac{92 z - 83}{19} = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

Step 4.4.1.9

Add $- 36 z$ and $19 z$ .

$y = \frac{11 - 17 z}{19}$

$\frac{92 z - 83}{19} = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

$y = \frac{11 - 17 z}{19}$

$\frac{92 z - 83}{19} = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

$y = \frac{11 - 17 z}{19}$

$\frac{92 z - 83}{19} = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

$y = \frac{11 - 17 z}{19}$

$\frac{92 z - 83}{19} = 15$

$x = - \frac{36}{19} + \frac{9 z}{19}$

Step 5

Solve for $z$ in $\frac{92 z - 83}{19} = 15$ .

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Step 5.1

Multiply both sides by $19$ .

$\frac{92 z - 83}{19} \cdot 19 = 15 \cdot 19$

$y = \frac{11 - 17 z}{19}$

$x = - \frac{36}{19} + \frac{9 z}{19}$

Step 5.2

Simplify.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

Cancel the common factor of $19$ .

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Step 5.2.1.1.1

Cancel the common factor.

$\frac{92 z - 83}{19} \cdot 19 = 15 \cdot 19$

$y = \frac{11 - 17 z}{19}$

$x = - \frac{36}{19} + \frac{9 z}{19}$

Step 5.2.1.1.2

Rewrite the expression.

$92 z - 83 = 15 \cdot 19$

$y = \frac{11 - 17 z}{19}$

$x = - \frac{36}{19} + \frac{9 z}{19}$

$92 z - 83 = 15 \cdot 19$

$y = \frac{11 - 17 z}{19}$

$x = - \frac{36}{19} + \frac{9 z}{19}$

$92 z - 83 = 15 \cdot 19$

$y = \frac{11 - 17 z}{19}$

$x = - \frac{36}{19} + \frac{9 z}{19}$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Multiply $15$ by $19$ .

$92 z - 83 = 285$

$y = \frac{11 - 17 z}{19}$

$x = - \frac{36}{19} + \frac{9 z}{19}$

$92 z - 83 = 285$

$y = \frac{11 - 17 z}{19}$

$x = - \frac{36}{19} + \frac{9 z}{19}$

$92 z - 83 = 285$

$y = \frac{11 - 17 z}{19}$

$x = - \frac{36}{19} + \frac{9 z}{19}$

Step 5.3

Solve for $z$ .

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Step 5.3.1

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

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Step 5.3.1.1

Add $83$ to both sides of the equation.

$92 z = 285 + 83$

$y = \frac{11 - 17 z}{19}$

$x = - \frac{36}{19} + \frac{9 z}{19}$

Step 5.3.1.2

Add $285$ and $83$ .

$92 z = 368$

$y = \frac{11 - 17 z}{19}$

$x = - \frac{36}{19} + \frac{9 z}{19}$

$92 z = 368$

$y = \frac{11 - 17 z}{19}$

$x = - \frac{36}{19} + \frac{9 z}{19}$

Step 5.3.2

Divide each term in $92 z = 368$ by $92$ and simplify.

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Step 5.3.2.1

Divide each term in $92 z = 368$ by $92$ .

$\frac{92 z}{92} = \frac{368}{92}$

$y = \frac{11 - 17 z}{19}$

$x = - \frac{36}{19} + \frac{9 z}{19}$

Step 5.3.2.2

Simplify the left side.

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Step 5.3.2.2.1

Cancel the common factor of $92$ .

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Step 5.3.2.2.1.1

Cancel the common factor.

$\frac{92 z}{92} = \frac{368}{92}$

$y = \frac{11 - 17 z}{19}$

$x = - \frac{36}{19} + \frac{9 z}{19}$

Step 5.3.2.2.1.2

Divide $z$ by $1$ .

$z = \frac{368}{92}$

$y = \frac{11 - 17 z}{19}$

$x = - \frac{36}{19} + \frac{9 z}{19}$

$z = \frac{368}{92}$

$y = \frac{11 - 17 z}{19}$

$x = - \frac{36}{19} + \frac{9 z}{19}$

$z = \frac{368}{92}$

$y = \frac{11 - 17 z}{19}$

$x = - \frac{36}{19} + \frac{9 z}{19}$

Step 5.3.2.3

Simplify the right side.

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Step 5.3.2.3.1

Divide $368$ by $92$ .

$z = 4$

$y = \frac{11 - 17 z}{19}$

$x = - \frac{36}{19} + \frac{9 z}{19}$

$z = 4$

$y = \frac{11 - 17 z}{19}$

$x = - \frac{36}{19} + \frac{9 z}{19}$

$z = 4$

$y = \frac{11 - 17 z}{19}$

$x = - \frac{36}{19} + \frac{9 z}{19}$

$z = 4$

$y = \frac{11 - 17 z}{19}$

$x = - \frac{36}{19} + \frac{9 z}{19}$

$z = 4$

$y = \frac{11 - 17 z}{19}$

$x = - \frac{36}{19} + \frac{9 z}{19}$

Step 6

Replace all occurrences of $z$ with $4$ in each equation.

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Step 6.1

Replace all occurrences of $z$ in $y = \frac{11 - 17 z}{19}$ with $4$ .

$y = \frac{11 - 17 \cdot 4}{19}$

$z = 4$

$x = - \frac{36}{19} + \frac{9 z}{19}$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $\frac{11 - 17 \cdot 4}{19}$ .

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Step 6.2.1.1

Simplify the numerator.

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Step 6.2.1.1.1

Multiply $- 17$ by $4$ .

$y = \frac{11 - 68}{19}$

$z = 4$

$x = - \frac{36}{19} + \frac{9 z}{19}$

Step 6.2.1.1.2

Subtract $68$ from $11$ .

$y = \frac{- 57}{19}$

$z = 4$

$x = - \frac{36}{19} + \frac{9 z}{19}$

$y = \frac{- 57}{19}$

$z = 4$

$x = - \frac{36}{19} + \frac{9 z}{19}$

Step 6.2.1.2

Divide $- 57$ by $19$ .

$y = - 3$

$z = 4$

$x = - \frac{36}{19} + \frac{9 z}{19}$

$y = - 3$

$z = 4$

$x = - \frac{36}{19} + \frac{9 z}{19}$

$y = - 3$

$z = 4$

$x = - \frac{36}{19} + \frac{9 z}{19}$

Step 6.3

Replace all occurrences of $z$ in $x = - \frac{36}{19} + \frac{9 z}{19}$ with $4$ .

$x = - \frac{36}{19} + \frac{9 (4)}{19}$

$y = - 3$

$z = 4$

Step 6.4

Simplify the right side.

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Step 6.4.1

Simplify $- \frac{36}{19} + \frac{9 (4)}{19}$ .

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Step 6.4.1.1

Combine the numerators over the common denominator.

$x = \frac{- 36 + 9 (4)}{19}$

$y = - 3$

$z = 4$

Step 6.4.1.2

Simplify the expression.

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Step 6.4.1.2.1

Multiply $9$ by $4$ .

$x = \frac{- 36 + 36}{19}$

$y = - 3$

$z = 4$

Step 6.4.1.2.2

Add $- 36$ and $36$ .

$x = \frac{0}{19}$

$y = - 3$

$z = 4$

Step 6.4.1.2.3

Divide $0$ by $19$ .

$x = 0$

$y = - 3$

$z = 4$

$x = 0$

$y = - 3$

$z = 4$

$x = 0$

$y = - 3$

$z = 4$

$x = 0$

$y = - 3$

$z = 4$

$x = 0$

$y = - 3$

$z = 4$

Step 7

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

$(0, - 3, 4)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(0, - 3, 4)$

Equation Form:

$x = 0, y = - 3, z = 4$