Finite Math Examples

$x - y = 1$ , $2 x - 8 z = 8$ , $8 y + z = 6$

Step 1

Add $y$ to both sides of the equation.

$x = 1 + y$

$2 x - 8 z = 8$

$8 y + z = 6$

Step 2

Replace all occurrences of $x$ with $1 + y$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $2 x - 8 z = 8$ with $1 + y$ .

$2 (1 + y) - 8 z = 8$

$x = 1 + y$

$8 y + z = 6$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Apply the distributive property.

$2 \cdot 1 + 2 y - 8 z = 8$

$x = 1 + y$

$8 y + z = 6$

Step 2.2.1.2

Multiply $2$ by $1$ .

$2 + 2 y - 8 z = 8$

$x = 1 + y$

$8 y + z = 6$

$2 + 2 y - 8 z = 8$

$x = 1 + y$

$8 y + z = 6$

$2 + 2 y - 8 z = 8$

$x = 1 + y$

$8 y + z = 6$

$2 + 2 y - 8 z = 8$

$x = 1 + y$

$8 y + z = 6$

Step 3

Reorder $1$ and $y$ .

$x = y + 1$

$2 + 2 y - 8 z = 8$

$8 y + z = 6$

Step 4

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

$z = 6 - 8 y$

$x = y + 1$

$2 + 2 y - 8 z = 8$

Step 5

Replace all occurrences of $z$ with $6 - 8 y$ in each equation.

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Step 5.1

Replace all occurrences of $z$ in $2 + 2 y - 8 z = 8$ with $6 - 8 y$ .

$2 + 2 y - 8 (6 - 8 y) = 8$

$z = 6 - 8 y$

$x = y + 1$

Step 5.2

Simplify the left side.

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Step 5.2.1

Simplify $2 + 2 y - 8 (6 - 8 y)$ .

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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.

$2 + 2 y - 8 \cdot 6 - 8 (- 8 y) = 8$

$z = 6 - 8 y$

$x = y + 1$

Step 5.2.1.1.2

Multiply $- 8$ by $6$ .

$2 + 2 y - 48 - 8 (- 8 y) = 8$

$z = 6 - 8 y$

$x = y + 1$

Step 5.2.1.1.3

Multiply $- 8$ by $- 8$ .

$2 + 2 y - 48 + 64 y = 8$

$z = 6 - 8 y$

$x = y + 1$

$2 + 2 y - 48 + 64 y = 8$

$z = 6 - 8 y$

$x = y + 1$

Step 5.2.1.2

Simplify by adding terms.

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Step 5.2.1.2.1

Subtract $48$ from $2$ .

$2 y - 46 + 64 y = 8$

$z = 6 - 8 y$

$x = y + 1$

Step 5.2.1.2.2

Add $2 y$ and $64 y$ .

$66 y - 46 = 8$

$z = 6 - 8 y$

$x = y + 1$

$66 y - 46 = 8$

$z = 6 - 8 y$

$x = y + 1$

$66 y - 46 = 8$

$z = 6 - 8 y$

$x = y + 1$

$66 y - 46 = 8$

$z = 6 - 8 y$

$x = y + 1$

$66 y - 46 = 8$

$z = 6 - 8 y$

$x = y + 1$

Step 6

Solve for $y$ in $66 y - 46 = 8$ .

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Step 6.1

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

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Step 6.1.1

Add $46$ to both sides of the equation.

$66 y = 8 + 46$

$z = 6 - 8 y$

$x = y + 1$

Step 6.1.2

Add $8$ and $46$ .

$66 y = 54$

$z = 6 - 8 y$

$x = y + 1$

$66 y = 54$

$z = 6 - 8 y$

$x = y + 1$

Step 6.2

Divide each term in $66 y = 54$ by $66$ and simplify.

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Step 6.2.1

Divide each term in $66 y = 54$ by $66$ .

$\frac{66 y}{66} = \frac{54}{66}$

$z = 6 - 8 y$

$x = y + 1$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $66$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{66 y}{66} = \frac{54}{66}$

$z = 6 - 8 y$

$x = y + 1$

Step 6.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{54}{66}$

$z = 6 - 8 y$

$x = y + 1$

$y = \frac{54}{66}$

$z = 6 - 8 y$

$x = y + 1$

$y = \frac{54}{66}$

$z = 6 - 8 y$

$x = y + 1$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Cancel the common factor of $54$ and $66$ .

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Step 6.2.3.1.1

Factor $6$ out of $54$ .

$y = \frac{6 (9)}{66}$

$z = 6 - 8 y$

$x = y + 1$

Step 6.2.3.1.2

Cancel the common factors.

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Step 6.2.3.1.2.1

Factor $6$ out of $66$ .

$y = \frac{6 \cdot 9}{6 \cdot 11}$

$z = 6 - 8 y$

$x = y + 1$

Step 6.2.3.1.2.2

Cancel the common factor.

$y = \frac{6 \cdot 9}{6 \cdot 11}$

$z = 6 - 8 y$

$x = y + 1$

Step 6.2.3.1.2.3

Rewrite the expression.

$y = \frac{9}{11}$

$z = 6 - 8 y$

$x = y + 1$

$y = \frac{9}{11}$

$z = 6 - 8 y$

$x = y + 1$

$y = \frac{9}{11}$

$z = 6 - 8 y$

$x = y + 1$

$y = \frac{9}{11}$

$z = 6 - 8 y$

$x = y + 1$

$y = \frac{9}{11}$

$z = 6 - 8 y$

$x = y + 1$

$y = \frac{9}{11}$

$z = 6 - 8 y$

$x = y + 1$

Step 7

Replace all occurrences of $y$ with $\frac{9}{11}$ in each equation.

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Step 7.1

Replace all occurrences of $y$ in $z = 6 - 8 y$ with $\frac{9}{11}$ .

$z = 6 - 8 (\frac{9}{11})$

$y = \frac{9}{11}$

$x = y + 1$

Step 7.2

Simplify the right side.

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Step 7.2.1

Simplify $6 - 8 (\frac{9}{11})$ .

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Step 7.2.1.1

Simplify each term.

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Step 7.2.1.1.1

Multiply $- 8 (\frac{9}{11})$ .

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Step 7.2.1.1.1.1

Combine $- 8$ and $\frac{9}{11}$ .

$z = 6 + \frac{- 8 \cdot 9}{11}$

$y = \frac{9}{11}$

$x = y + 1$

Step 7.2.1.1.1.2

Multiply $- 8$ by $9$ .

$z = 6 + \frac{- 72}{11}$

$y = \frac{9}{11}$

$x = y + 1$

$z = 6 + \frac{- 72}{11}$

$y = \frac{9}{11}$

$x = y + 1$

Step 7.2.1.1.2

Move the negative in front of the fraction.

$z = 6 - \frac{72}{11}$

$y = \frac{9}{11}$

$x = y + 1$

$z = 6 - \frac{72}{11}$

$y = \frac{9}{11}$

$x = y + 1$

Step 7.2.1.2

To write $6$ as a fraction with a common denominator, multiply by $\frac{11}{11}$ .

$z = 6 \cdot \frac{11}{11} - \frac{72}{11}$

$y = \frac{9}{11}$

$x = y + 1$

Step 7.2.1.3

Combine $6$ and $\frac{11}{11}$ .

$z = \frac{6 \cdot 11}{11} - \frac{72}{11}$

$y = \frac{9}{11}$

$x = y + 1$

Step 7.2.1.4

Combine the numerators over the common denominator.

$z = \frac{6 \cdot 11 - 72}{11}$

$y = \frac{9}{11}$

$x = y + 1$

Step 7.2.1.5

Simplify the numerator.

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Step 7.2.1.5.1

Multiply $6$ by $11$ .

$z = \frac{66 - 72}{11}$

$y = \frac{9}{11}$

$x = y + 1$

Step 7.2.1.5.2

Subtract $72$ from $66$ .

$z = \frac{- 6}{11}$

$y = \frac{9}{11}$

$x = y + 1$

$z = \frac{- 6}{11}$

$y = \frac{9}{11}$

$x = y + 1$

Step 7.2.1.6

Move the negative in front of the fraction.

$z = - \frac{6}{11}$

$y = \frac{9}{11}$

$x = y + 1$

$z = - \frac{6}{11}$

$y = \frac{9}{11}$

$x = y + 1$

$z = - \frac{6}{11}$

$y = \frac{9}{11}$

$x = y + 1$

Step 7.3

Replace all occurrences of $y$ in $x = y + 1$ with $\frac{9}{11}$ .

$x = (\frac{9}{11}) + 1$

$z = - \frac{6}{11}$

$y = \frac{9}{11}$

Step 7.4

Simplify the right side.

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Step 7.4.1

Simplify $(\frac{9}{11}) + 1$ .

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Step 7.4.1.1

Write $1$ as a fraction with a common denominator.

$x = \frac{9}{11} + \frac{11}{11}$

$z = - \frac{6}{11}$

$y = \frac{9}{11}$

Step 7.4.1.2

Combine the numerators over the common denominator.

$x = \frac{9 + 11}{11}$

$z = - \frac{6}{11}$

$y = \frac{9}{11}$

Step 7.4.1.3

Add $9$ and $11$ .

$x = \frac{20}{11}$

$z = - \frac{6}{11}$

$y = \frac{9}{11}$

$x = \frac{20}{11}$

$z = - \frac{6}{11}$

$y = \frac{9}{11}$

$x = \frac{20}{11}$

$z = - \frac{6}{11}$

$y = \frac{9}{11}$

$x = \frac{20}{11}$

$z = - \frac{6}{11}$

$y = \frac{9}{11}$

Step 8

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

$(\frac{20}{11}, \frac{9}{11}, - \frac{6}{11})$

Step 9

The result can be shown in multiple forms.

Point Form:

$(\frac{20}{11}, \frac{9}{11}, - \frac{6}{11})$

Equation Form:

$x = \frac{20}{11}, y = \frac{9}{11}, z = - \frac{6}{11}$