Finite Math Examples

$x - 8 y + z = 1$ , $3 x - y + 2 z = 1$

Step 1

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

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Step 1.1

Add $8 y$ to both sides of the equation.

$x + z = 1 + 8 y$

$3 x - y + 2 z = 1$

Step 1.2

Subtract $z$ from both sides of the equation.

$x = 1 + 8 y - z$

$3 x - y + 2 z = 1$

$x = 1 + 8 y - z$

$3 x - y + 2 z = 1$

Step 2

Solve the equation for $y$ .

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Step 2.1

Simplify $3 (1 + 8 y - z) - y + 2 z$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Apply the distributive property.

$3 \cdot 1 + 3 (8 y) + 3 (- z) - y + 2 z = 1$

$x = 1 + 8 y - z$

Step 2.1.1.2

Simplify.

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Step 2.1.1.2.1

Multiply $3$ by $1$ .

$3 + 3 (8 y) + 3 (- z) - y + 2 z = 1$

$x = 1 + 8 y - z$

Step 2.1.1.2.2

Multiply $8$ by $3$ .

$3 + 24 y + 3 (- z) - y + 2 z = 1$

$x = 1 + 8 y - z$

Step 2.1.1.2.3

Multiply $- 1$ by $3$ .

$3 + 24 y - 3 z - y + 2 z = 1$

$x = 1 + 8 y - z$

$3 + 24 y - 3 z - y + 2 z = 1$

$x = 1 + 8 y - z$

$3 + 24 y - 3 z - y + 2 z = 1$

$x = 1 + 8 y - z$

Step 2.1.2

Simplify by adding terms.

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Step 2.1.2.1

Subtract $y$ from $24 y$ .

$3 + 23 y - 3 z + 2 z = 1$

$x = 1 + 8 y - z$

Step 2.1.2.2

Add $- 3 z$ and $2 z$ .

$3 + 23 y - z = 1$

$x = 1 + 8 y - z$

$3 + 23 y - z = 1$

$x = 1 + 8 y - z$

$3 + 23 y - z = 1$

$x = 1 + 8 y - z$

Step 2.2

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

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Step 2.2.1

Subtract $3$ from both sides of the equation.

$23 y - z = 1 - 3$

$x = 1 + 8 y - z$

Step 2.2.2

Add $z$ to both sides of the equation.

$23 y = 1 - 3 + z$

$x = 1 + 8 y - z$

Step 2.2.3

Subtract $3$ from $1$ .

$23 y = - 2 + z$

$x = 1 + 8 y - z$

$23 y = - 2 + z$

$x = 1 + 8 y - z$

Step 2.3

Divide each term in $23 y = - 2 + z$ by $23$ and simplify.

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Step 2.3.1

Divide each term in $23 y = - 2 + z$ by $23$ .

$\frac{23 y}{23} = \frac{- 2}{23} + \frac{z}{23}$

$x = 1 + 8 y - z$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $23$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{23 y}{23} = \frac{- 2}{23} + \frac{z}{23}$

$x = 1 + 8 y - z$

Step 2.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 2}{23} + \frac{z}{23}$

$x = 1 + 8 y - z$

$y = \frac{- 2}{23} + \frac{z}{23}$

$x = 1 + 8 y - z$

$y = \frac{- 2}{23} + \frac{z}{23}$

$x = 1 + 8 y - z$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Move the negative in front of the fraction.

$y = - \frac{2}{23} + \frac{z}{23}$

$x = 1 + 8 y - z$

$y = - \frac{2}{23} + \frac{z}{23}$

$x = 1 + 8 y - z$

$y = - \frac{2}{23} + \frac{z}{23}$

$x = 1 + 8 y - z$

$y = - \frac{2}{23} + \frac{z}{23}$

$x = 1 + 8 y - z$

Step 3

Simplify the right side.

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Step 3.1

Simplify $1 + 8 (- \frac{2}{23} + \frac{z}{23}) - z$ .

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Step 3.1.1

Simplify each term.

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Step 3.1.1.1

Apply the distributive property.

$x = 1 + 8 (- \frac{2}{23}) + 8 (\frac{z}{23}) - z$

$y = - \frac{2}{23} + \frac{z}{23}$

Step 3.1.1.2

Multiply $8 (- \frac{2}{23})$ .

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Step 3.1.1.2.1

Multiply $- 1$ by $8$ .

$x = 1 - 8 (\frac{2}{23}) + 8 (\frac{z}{23}) - z$

$y = - \frac{2}{23} + \frac{z}{23}$

Step 3.1.1.2.2

Combine $- 8$ and $\frac{2}{23}$ .

$x = 1 + \frac{- 8 \cdot 2}{23} + 8 (\frac{z}{23}) - z$

$y = - \frac{2}{23} + \frac{z}{23}$

Step 3.1.1.2.3

Multiply $- 8$ by $2$ .

$x = 1 + \frac{- 16}{23} + 8 (\frac{z}{23}) - z$

$y = - \frac{2}{23} + \frac{z}{23}$

$x = 1 + \frac{- 16}{23} + 8 (\frac{z}{23}) - z$

$y = - \frac{2}{23} + \frac{z}{23}$

Step 3.1.1.3

Combine $8$ and $\frac{z}{23}$ .

$x = 1 + \frac{- 16}{23} + \frac{8 z}{23} - z$

$y = - \frac{2}{23} + \frac{z}{23}$

Step 3.1.1.4

Move the negative in front of the fraction.

$x = 1 - \frac{16}{23} + \frac{8 z}{23} - z$

$y = - \frac{2}{23} + \frac{z}{23}$

$x = 1 - \frac{16}{23} + \frac{8 z}{23} - z$

$y = - \frac{2}{23} + \frac{z}{23}$

Step 3.1.2

Simplify the expression.

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Step 3.1.2.1

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

$x = \frac{23}{23} - \frac{16}{23} + \frac{8 z}{23} - z$

$y = - \frac{2}{23} + \frac{z}{23}$

Step 3.1.2.2

Combine the numerators over the common denominator.

$x = \frac{23 - 16}{23} + \frac{8 z}{23} - z$

$y = - \frac{2}{23} + \frac{z}{23}$

Step 3.1.2.3

Subtract $16$ from $23$ .

$x = \frac{7}{23} + \frac{8 z}{23} - z$

$y = - \frac{2}{23} + \frac{z}{23}$

$x = \frac{7}{23} + \frac{8 z}{23} - z$

$y = - \frac{2}{23} + \frac{z}{23}$

Step 3.1.3

To write $- z$ as a fraction with a common denominator, multiply by $\frac{23}{23}$ .

$x = \frac{7}{23} + \frac{8 z}{23} - z \cdot \frac{23}{23}$

$y = - \frac{2}{23} + \frac{z}{23}$

Step 3.1.4

Combine $- z$ and $\frac{23}{23}$ .

$x = \frac{7}{23} + \frac{8 z}{23} + \frac{- z \cdot 23}{23}$

$y = - \frac{2}{23} + \frac{z}{23}$

Step 3.1.5

Combine the numerators over the common denominator.

$x = \frac{7}{23} + \frac{8 z - z \cdot 23}{23}$

$y = - \frac{2}{23} + \frac{z}{23}$

Step 3.1.6

Combine the numerators over the common denominator.

$x = \frac{7 + 8 z - z \cdot 23}{23}$

$y = - \frac{2}{23} + \frac{z}{23}$

Step 3.1.7

Multiply $23$ by $- 1$ .

$x = \frac{7 + 8 z - 23 z}{23}$

$y = - \frac{2}{23} + \frac{z}{23}$

Step 3.1.8

Subtract $23 z$ from $8 z$ .

$x = \frac{7 - 15 z}{23}$

$y = - \frac{2}{23} + \frac{z}{23}$

$x = \frac{7 - 15 z}{23}$

$y = - \frac{2}{23} + \frac{z}{23}$

$x = \frac{7 - 15 z}{23}$

$y = - \frac{2}{23} + \frac{z}{23}$

Step 4

Simplify the right side.

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Step 4.1

Reorder $- \frac{2}{23}$ and $\frac{z}{23}$ .

$y = \frac{z}{23} - \frac{2}{23}$

$x = \frac{7 - 15 z}{23}$

$y = \frac{z}{23} - \frac{2}{23}$

$x = \frac{7 - 15 z}{23}$