Finite Math Examples

$x - 3 z = - 5$ , $2 x - y + 2 z = 16$ , $7 z - 3 y - 5 z = 19$

Step 1

Add $3 z$ to both sides of the equation.

$x = - 5 + 3 z$

$2 x - y + 2 z = 16$

$- 3 y + 2 z = 19$

Step 2

Replace all occurrences of $x$ with $- 5 + 3 z$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $2 x - y + 2 z = 16$ with $- 5 + 3 z$ .

$2 (- 5 + 3 z) - y + 2 z = 16$

$x = - 5 + 3 z$

$- 3 y + 2 z = 19$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $2 (- 5 + 3 z) - y + 2 z$ .

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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.

$2 \cdot - 5 + 2 (3 z) - y + 2 z = 16$

$x = - 5 + 3 z$

$- 3 y + 2 z = 19$

Step 2.2.1.1.2

Multiply $2$ by $- 5$ .

$- 10 + 2 (3 z) - y + 2 z = 16$

$x = - 5 + 3 z$

$- 3 y + 2 z = 19$

Step 2.2.1.1.3

Multiply $3$ by $2$ .

$- 10 + 6 z - y + 2 z = 16$

$x = - 5 + 3 z$

$- 3 y + 2 z = 19$

$- 10 + 6 z - y + 2 z = 16$

$x = - 5 + 3 z$

$- 3 y + 2 z = 19$

Step 2.2.1.2

Add $6 z$ and $2 z$ .

$- 10 - y + 8 z = 16$

$x = - 5 + 3 z$

$- 3 y + 2 z = 19$

$- 10 - y + 8 z = 16$

$x = - 5 + 3 z$

$- 3 y + 2 z = 19$

$- 10 - y + 8 z = 16$

$x = - 5 + 3 z$

$- 3 y + 2 z = 19$

$- 10 - y + 8 z = 16$

$x = - 5 + 3 z$

$- 3 y + 2 z = 19$

Step 3

Reorder $- 5$ and $3 z$ .

$x = 3 z - 5$

$- 10 - y + 8 z = 16$

$- 3 y + 2 z = 19$

Step 4

Solve for $y$ in $- 10 - y + 8 z = 16$ .

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Step 4.1

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

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Step 4.1.1

Add $10$ to both sides of the equation.

$- y + 8 z = 16 + 10$

$x = 3 z - 5$

$- 3 y + 2 z = 19$

Step 4.1.2

Subtract $8 z$ from both sides of the equation.

$- y = 16 + 10 - 8 z$

$x = 3 z - 5$

$- 3 y + 2 z = 19$

Step 4.1.3

Add $16$ and $10$ .

$- y = 26 - 8 z$

$x = 3 z - 5$

$- 3 y + 2 z = 19$

$- y = 26 - 8 z$

$x = 3 z - 5$

$- 3 y + 2 z = 19$

Step 4.2

Divide each term in $- y = 26 - 8 z$ by $- 1$ and simplify.

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Step 4.2.1

Divide each term in $- y = 26 - 8 z$ by $- 1$ .

$\frac{- y}{- 1} = \frac{26}{- 1} + \frac{- 8 z}{- 1}$

$x = 3 z - 5$

$- 3 y + 2 z = 19$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{26}{- 1} + \frac{- 8 z}{- 1}$

$x = 3 z - 5$

$- 3 y + 2 z = 19$

Step 4.2.2.2

Divide $y$ by $1$ .

$y = \frac{26}{- 1} + \frac{- 8 z}{- 1}$

$x = 3 z - 5$

$- 3 y + 2 z = 19$

$y = \frac{26}{- 1} + \frac{- 8 z}{- 1}$

$x = 3 z - 5$

$- 3 y + 2 z = 19$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Simplify each term.

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Step 4.2.3.1.1

Divide $26$ by $- 1$ .

$y = - 26 + \frac{- 8 z}{- 1}$

$x = 3 z - 5$

$- 3 y + 2 z = 19$

Step 4.2.3.1.2

Move the negative one from the denominator of $\frac{- 8 z}{- 1}$ .

$y = - 26 - 1 \cdot (- 8 z)$

$x = 3 z - 5$

$- 3 y + 2 z = 19$

Step 4.2.3.1.3

Rewrite $- 1 \cdot (- 8 z)$ as $- (- 8 z)$ .

$y = - 26 - (- 8 z)$

$x = 3 z - 5$

$- 3 y + 2 z = 19$

Step 4.2.3.1.4

Multiply $- 8$ by $- 1$ .

$y = - 26 + 8 z$

$x = 3 z - 5$

$- 3 y + 2 z = 19$

$y = - 26 + 8 z$

$x = 3 z - 5$

$- 3 y + 2 z = 19$

$y = - 26 + 8 z$

$x = 3 z - 5$

$- 3 y + 2 z = 19$

$y = - 26 + 8 z$

$x = 3 z - 5$

$- 3 y + 2 z = 19$

$y = - 26 + 8 z$

$x = 3 z - 5$

$- 3 y + 2 z = 19$

Step 5

Replace all occurrences of $y$ with $- 26 + 8 z$ in each equation.

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Step 5.1

Replace all occurrences of $y$ in $- 3 y + 2 z = 19$ with $- 26 + 8 z$ .

$- 3 (- 26 + 8 z) + 2 z = 19$

$y = - 26 + 8 z$

$x = 3 z - 5$

Step 5.2

Simplify the left side.

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Step 5.2.1

Simplify $- 3 (- 26 + 8 z) + 2 z$ .

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Step 5.2.1.1

Simplify each term.

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Step 5.2.1.1.1

Apply the distributive property.

$- 3 \cdot - 26 - 3 (8 z) + 2 z = 19$

$y = - 26 + 8 z$

$x = 3 z - 5$

Step 5.2.1.1.2

Multiply $- 3$ by $- 26$ .

$78 - 3 (8 z) + 2 z = 19$

$y = - 26 + 8 z$

$x = 3 z - 5$

Step 5.2.1.1.3

Multiply $8$ by $- 3$ .

$78 - 24 z + 2 z = 19$

$y = - 26 + 8 z$

$x = 3 z - 5$

$78 - 24 z + 2 z = 19$

$y = - 26 + 8 z$

$x = 3 z - 5$

Step 5.2.1.2

Add $- 24 z$ and $2 z$ .

$78 - 22 z = 19$

$y = - 26 + 8 z$

$x = 3 z - 5$

$78 - 22 z = 19$

$y = - 26 + 8 z$

$x = 3 z - 5$

$78 - 22 z = 19$

$y = - 26 + 8 z$

$x = 3 z - 5$

$78 - 22 z = 19$

$y = - 26 + 8 z$

$x = 3 z - 5$

Step 6

Solve for $z$ in $78 - 22 z = 19$ .

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Step 6.1

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

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Step 6.1.1

Subtract $78$ from both sides of the equation.

$- 22 z = 19 - 78$

$y = - 26 + 8 z$

$x = 3 z - 5$

Step 6.1.2

Subtract $78$ from $19$ .

$- 22 z = - 59$

$y = - 26 + 8 z$

$x = 3 z - 5$

$- 22 z = - 59$

$y = - 26 + 8 z$

$x = 3 z - 5$

Step 6.2

Divide each term in $- 22 z = - 59$ by $- 22$ and simplify.

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Step 6.2.1

Divide each term in $- 22 z = - 59$ by $- 22$ .

$\frac{- 22 z}{- 22} = \frac{- 59}{- 22}$

$y = - 26 + 8 z$

$x = 3 z - 5$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $- 22$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{- 22 z}{- 22} = \frac{- 59}{- 22}$

$y = - 26 + 8 z$

$x = 3 z - 5$

Step 6.2.2.1.2

Divide $z$ by $1$ .

$z = \frac{- 59}{- 22}$

$y = - 26 + 8 z$

$x = 3 z - 5$

$z = \frac{- 59}{- 22}$

$y = - 26 + 8 z$

$x = 3 z - 5$

$z = \frac{- 59}{- 22}$

$y = - 26 + 8 z$

$x = 3 z - 5$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Dividing two negative values results in a positive value.

$z = \frac{59}{22}$

$y = - 26 + 8 z$

$x = 3 z - 5$

$z = \frac{59}{22}$

$y = - 26 + 8 z$

$x = 3 z - 5$

$z = \frac{59}{22}$

$y = - 26 + 8 z$

$x = 3 z - 5$

$z = \frac{59}{22}$

$y = - 26 + 8 z$

$x = 3 z - 5$

Step 7

Replace all occurrences of $z$ with $\frac{59}{22}$ in each equation.

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Step 7.1

Replace all occurrences of $z$ in $y = - 26 + 8 z$ with $\frac{59}{22}$ .

$y = - 26 + 8 (\frac{59}{22})$

$z = \frac{59}{22}$

$x = 3 z - 5$

Step 7.2

Simplify the right side.

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Step 7.2.1

Simplify $- 26 + 8 (\frac{59}{22})$ .

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Step 7.2.1.1

Simplify each term.

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Step 7.2.1.1.1

Cancel the common factor of $2$ .

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Step 7.2.1.1.1.1

Factor $2$ out of $8$ .

$y = - 26 + 2 (4) (\frac{59}{22})$

$z = \frac{59}{22}$

$x = 3 z - 5$

Step 7.2.1.1.1.2

Factor $2$ out of $22$ .

$y = - 26 + 2 \cdot (4 (\frac{59}{2 \cdot 11}))$

$z = \frac{59}{22}$

$x = 3 z - 5$

Step 7.2.1.1.1.3

Cancel the common factor.

$y = - 26 + 2 \cdot (4 (\frac{59}{2 \cdot 11}))$

$z = \frac{59}{22}$

$x = 3 z - 5$

Step 7.2.1.1.1.4

Rewrite the expression.

$y = - 26 + 4 (\frac{59}{11})$

$z = \frac{59}{22}$

$x = 3 z - 5$

$y = - 26 + 4 (\frac{59}{11})$

$z = \frac{59}{22}$

$x = 3 z - 5$

Step 7.2.1.1.2

Combine $4$ and $\frac{59}{11}$ .

$y = - 26 + \frac{4 \cdot 59}{11}$

$z = \frac{59}{22}$

$x = 3 z - 5$

Step 7.2.1.1.3

Multiply $4$ by $59$ .

$y = - 26 + \frac{236}{11}$

$z = \frac{59}{22}$

$x = 3 z - 5$

$y = - 26 + \frac{236}{11}$

$z = \frac{59}{22}$

$x = 3 z - 5$

Step 7.2.1.2

To write $- 26$ as a fraction with a common denominator, multiply by $\frac{11}{11}$ .

$y = - 26 \cdot \frac{11}{11} + \frac{236}{11}$

$z = \frac{59}{22}$

$x = 3 z - 5$

Step 7.2.1.3

Combine $- 26$ and $\frac{11}{11}$ .

$y = \frac{- 26 \cdot 11}{11} + \frac{236}{11}$

$z = \frac{59}{22}$

$x = 3 z - 5$

Step 7.2.1.4

Combine the numerators over the common denominator.

$y = \frac{- 26 \cdot 11 + 236}{11}$

$z = \frac{59}{22}$

$x = 3 z - 5$

Step 7.2.1.5

Simplify the numerator.

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Step 7.2.1.5.1

Multiply $- 26$ by $11$ .

$y = \frac{- 286 + 236}{11}$

$z = \frac{59}{22}$

$x = 3 z - 5$

Step 7.2.1.5.2

Add $- 286$ and $236$ .

$y = \frac{- 50}{11}$

$z = \frac{59}{22}$

$x = 3 z - 5$

$y = \frac{- 50}{11}$

$z = \frac{59}{22}$

$x = 3 z - 5$

Step 7.2.1.6

Move the negative in front of the fraction.

$y = - \frac{50}{11}$

$z = \frac{59}{22}$

$x = 3 z - 5$

$y = - \frac{50}{11}$

$z = \frac{59}{22}$

$x = 3 z - 5$

$y = - \frac{50}{11}$

$z = \frac{59}{22}$

$x = 3 z - 5$

Step 7.3

Replace all occurrences of $z$ in $x = 3 z - 5$ with $\frac{59}{22}$ .

$x = 3 (\frac{59}{22}) - 5$

$y = - \frac{50}{11}$

$z = \frac{59}{22}$

Step 7.4

Simplify the right side.

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Step 7.4.1

Simplify $3 (\frac{59}{22}) - 5$ .

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Step 7.4.1.1

Multiply $3 (\frac{59}{22})$ .

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Step 7.4.1.1.1

Combine $3$ and $\frac{59}{22}$ .

$x = \frac{3 \cdot 59}{22} - 5$

$y = - \frac{50}{11}$

$z = \frac{59}{22}$

Step 7.4.1.1.2

Multiply $3$ by $59$ .

$x = \frac{177}{22} - 5$

$y = - \frac{50}{11}$

$z = \frac{59}{22}$

$x = \frac{177}{22} - 5$

$y = - \frac{50}{11}$

$z = \frac{59}{22}$

Step 7.4.1.2

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{22}{22}$ .

$x = \frac{177}{22} - 5 \cdot \frac{22}{22}$

$y = - \frac{50}{11}$

$z = \frac{59}{22}$

Step 7.4.1.3

Combine $- 5$ and $\frac{22}{22}$ .

$x = \frac{177}{22} + \frac{- 5 \cdot 22}{22}$

$y = - \frac{50}{11}$

$z = \frac{59}{22}$

Step 7.4.1.4

Combine the numerators over the common denominator.

$x = \frac{177 - 5 \cdot 22}{22}$

$y = - \frac{50}{11}$

$z = \frac{59}{22}$

Step 7.4.1.5

Simplify the numerator.

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Step 7.4.1.5.1

Multiply $- 5$ by $22$ .

$x = \frac{177 - 110}{22}$

$y = - \frac{50}{11}$

$z = \frac{59}{22}$

Step 7.4.1.5.2

Subtract $110$ from $177$ .

$x = \frac{67}{22}$

$y = - \frac{50}{11}$

$z = \frac{59}{22}$

$x = \frac{67}{22}$

$y = - \frac{50}{11}$

$z = \frac{59}{22}$

$x = \frac{67}{22}$

$y = - \frac{50}{11}$

$z = \frac{59}{22}$

$x = \frac{67}{22}$

$y = - \frac{50}{11}$

$z = \frac{59}{22}$

$x = \frac{67}{22}$

$y = - \frac{50}{11}$

$z = \frac{59}{22}$

Step 8

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

$(\frac{67}{22}, - \frac{50}{11}, \frac{59}{22})$

Step 9

The result can be shown in multiple forms.

Point Form:

$(\frac{67}{22}, - \frac{50}{11}, \frac{59}{22})$

Equation Form:

$x = \frac{67}{22}, y = - \frac{50}{11}, z = \frac{59}{22}$