Finite Math Examples

$x + 4 y + 5 z = 19$ , $2 x - y + z = 12$

Step 1

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

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Step 1.1

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

$x + 5 z = 19 - 4 y$

$2 x - y + z = 12$

Step 1.2

Subtract $5 z$ from both sides of the equation.

$x = 19 - 4 y - 5 z$

$2 x - y + z = 12$

$x = 19 - 4 y - 5 z$

$2 x - y + z = 12$

Step 2

Solve the equation for $y$ .

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Step 2.1

Simplify $2 (19 - 4 y - 5 z) - y + z$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Apply the distributive property.

$2 \cdot 19 + 2 (- 4 y) + 2 (- 5 z) - y + z = 12$

$x = 19 - 4 y - 5 z$

Step 2.1.1.2

Simplify.

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Step 2.1.1.2.1

Multiply $2$ by $19$ .

$38 + 2 (- 4 y) + 2 (- 5 z) - y + z = 12$

$x = 19 - 4 y - 5 z$

Step 2.1.1.2.2

Multiply $- 4$ by $2$ .

$38 - 8 y + 2 (- 5 z) - y + z = 12$

$x = 19 - 4 y - 5 z$

Step 2.1.1.2.3

Multiply $- 5$ by $2$ .

$38 - 8 y - 10 z - y + z = 12$

$x = 19 - 4 y - 5 z$

$38 - 8 y - 10 z - y + z = 12$

$x = 19 - 4 y - 5 z$

$38 - 8 y - 10 z - y + z = 12$

$x = 19 - 4 y - 5 z$

Step 2.1.2

Simplify by adding terms.

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Step 2.1.2.1

Subtract $y$ from $- 8 y$ .

$38 - 9 y - 10 z + z = 12$

$x = 19 - 4 y - 5 z$

Step 2.1.2.2

Add $- 10 z$ and $z$ .

$38 - 9 y - 9 z = 12$

$x = 19 - 4 y - 5 z$

$38 - 9 y - 9 z = 12$

$x = 19 - 4 y - 5 z$

$38 - 9 y - 9 z = 12$

$x = 19 - 4 y - 5 z$

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 $38$ from both sides of the equation.

$- 9 y - 9 z = 12 - 38$

$x = 19 - 4 y - 5 z$

Step 2.2.2

Add $9 z$ to both sides of the equation.

$- 9 y = 12 - 38 + 9 z$

$x = 19 - 4 y - 5 z$

Step 2.2.3

Subtract $38$ from $12$ .

$- 9 y = - 26 + 9 z$

$x = 19 - 4 y - 5 z$

$- 9 y = - 26 + 9 z$

$x = 19 - 4 y - 5 z$

Step 2.3

Divide each term in $- 9 y = - 26 + 9 z$ by $- 9$ and simplify.

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Step 2.3.1

Divide each term in $- 9 y = - 26 + 9 z$ by $- 9$ .

$\frac{- 9 y}{- 9} = \frac{- 26}{- 9} + \frac{9 z}{- 9}$

$x = 19 - 4 y - 5 z$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $- 9$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{- 9 y}{- 9} = \frac{- 26}{- 9} + \frac{9 z}{- 9}$

$x = 19 - 4 y - 5 z$

Step 2.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 26}{- 9} + \frac{9 z}{- 9}$

$x = 19 - 4 y - 5 z$

$y = \frac{- 26}{- 9} + \frac{9 z}{- 9}$

$x = 19 - 4 y - 5 z$

$y = \frac{- 26}{- 9} + \frac{9 z}{- 9}$

$x = 19 - 4 y - 5 z$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Simplify each term.

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Step 2.3.3.1.1

Dividing two negative values results in a positive value.

$y = \frac{26}{9} + \frac{9 z}{- 9}$

$x = 19 - 4 y - 5 z$

Step 2.3.3.1.2

Cancel the common factor of $9$ and $- 9$ .

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Step 2.3.3.1.2.1

Factor $9$ out of $9 z$ .

$y = \frac{26}{9} + \frac{9 (z)}{- 9}$

$x = 19 - 4 y - 5 z$

Step 2.3.3.1.2.2

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

$y = \frac{26}{9} - 1 \cdot z$

$x = 19 - 4 y - 5 z$

$y = \frac{26}{9} - 1 \cdot z$

$x = 19 - 4 y - 5 z$

Step 2.3.3.1.3

Rewrite $- 1 \cdot z$ as $- z$ .

$y = \frac{26}{9} - z$

$x = 19 - 4 y - 5 z$

$y = \frac{26}{9} - z$

$x = 19 - 4 y - 5 z$

$y = \frac{26}{9} - z$

$x = 19 - 4 y - 5 z$

$y = \frac{26}{9} - z$

$x = 19 - 4 y - 5 z$

$y = \frac{26}{9} - z$

$x = 19 - 4 y - 5 z$

Step 3

Simplify the right side.

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Step 3.1

Simplify $19 - 4 (\frac{26}{9} - z) - 5 z$ .

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Step 3.1.1

Simplify each term.

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Step 3.1.1.1

Apply the distributive property.

$x = 19 - 4 (\frac{26}{9}) - 4 (- z) - 5 z$

$y = \frac{26}{9} - z$

Step 3.1.1.2

Multiply $- 4 (\frac{26}{9})$ .

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Step 3.1.1.2.1

Combine $- 4$ and $\frac{26}{9}$ .

$x = 19 + \frac{- 4 \cdot 26}{9} - 4 (- z) - 5 z$

$y = \frac{26}{9} - z$

Step 3.1.1.2.2

Multiply $- 4$ by $26$ .

$x = 19 + \frac{- 104}{9} - 4 (- z) - 5 z$

$y = \frac{26}{9} - z$

$x = 19 + \frac{- 104}{9} - 4 (- z) - 5 z$

$y = \frac{26}{9} - z$

Step 3.1.1.3

Multiply $- 1$ by $- 4$ .

$x = 19 + \frac{- 104}{9} + 4 z - 5 z$

$y = \frac{26}{9} - z$

Step 3.1.1.4

Move the negative in front of the fraction.

$x = 19 - \frac{104}{9} + 4 z - 5 z$

$y = \frac{26}{9} - z$

$x = 19 - \frac{104}{9} + 4 z - 5 z$

$y = \frac{26}{9} - z$

Step 3.1.2

To write $19$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$x = 19 \cdot \frac{9}{9} - \frac{104}{9} + 4 z - 5 z$

$y = \frac{26}{9} - z$

Step 3.1.3

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

$x = \frac{19 \cdot 9}{9} - \frac{104}{9} + 4 z - 5 z$

$y = \frac{26}{9} - z$

Step 3.1.4

Combine the numerators over the common denominator.

$x = \frac{19 \cdot 9 - 104}{9} + 4 z - 5 z$

$y = \frac{26}{9} - z$

Step 3.1.5

Simplify the numerator.

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Step 3.1.5.1

Multiply $19$ by $9$ .

$x = \frac{171 - 104}{9} + 4 z - 5 z$

$y = \frac{26}{9} - z$

Step 3.1.5.2

Subtract $104$ from $171$ .

$x = \frac{67}{9} + 4 z - 5 z$

$y = \frac{26}{9} - z$

$x = \frac{67}{9} + 4 z - 5 z$

$y = \frac{26}{9} - z$

Step 3.1.6

Subtract $5 z$ from $4 z$ .

$x = \frac{67}{9} - z$

$y = \frac{26}{9} - z$

$x = \frac{67}{9} - z$

$y = \frac{26}{9} - z$

$x = \frac{67}{9} - z$

$y = \frac{26}{9} - z$

Step 4

Simplify the right side.

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Step 4.1

Reorder $\frac{67}{9}$ and $- z$ .

$x = - z + \frac{67}{9}$

$y = \frac{26}{9} - z$

$x = - z + \frac{67}{9}$

$y = \frac{26}{9} - z$

Step 5

Simplify the right side.

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Step 5.1

Reorder $\frac{26}{9}$ and $- z$ .

$y = - z + \frac{26}{9}$

$x = - z + \frac{67}{9}$

$y = - z + \frac{26}{9}$

$x = - z + \frac{67}{9}$