Finite Math Examples

$x + 3 y + 4 z = 10$ , $- 3 x + 7 y + 42 = 2$

Step 1

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

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Step 1.1

Subtract $3 y$ from both sides of the equation.

$x + 4 z = 10 - 3 y$

$- 3 x + 7 y + 42 = 2$

Step 1.2

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

$x = 10 - 3 y - 4 z$

$- 3 x + 7 y + 42 = 2$

$x = 10 - 3 y - 4 z$

$- 3 x + 7 y + 42 = 2$

Step 2

Solve the equation for $y$ .

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Step 2.1

Simplify $- 3 (10 - 3 y - 4 z) + 7 y + 42$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Apply the distributive property.

$- 3 \cdot 10 - 3 (- 3 y) - 3 (- 4 z) + 7 y + 42 = 2$

$x = 10 - 3 y - 4 z$

Step 2.1.1.2

Simplify.

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Step 2.1.1.2.1

Multiply $- 3$ by $10$ .

$- 30 - 3 (- 3 y) - 3 (- 4 z) + 7 y + 42 = 2$

$x = 10 - 3 y - 4 z$

Step 2.1.1.2.2

Multiply $- 3$ by $- 3$ .

$- 30 + 9 y - 3 (- 4 z) + 7 y + 42 = 2$

$x = 10 - 3 y - 4 z$

Step 2.1.1.2.3

Multiply $- 4$ by $- 3$ .

$- 30 + 9 y + 12 z + 7 y + 42 = 2$

$x = 10 - 3 y - 4 z$

$- 30 + 9 y + 12 z + 7 y + 42 = 2$

$x = 10 - 3 y - 4 z$

$- 30 + 9 y + 12 z + 7 y + 42 = 2$

$x = 10 - 3 y - 4 z$

Step 2.1.2

Simplify by adding terms.

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Step 2.1.2.1

Add $- 30$ and $42$ .

$9 y + 12 z + 7 y + 12 = 2$

$x = 10 - 3 y - 4 z$

Step 2.1.2.2

Add $9 y$ and $7 y$ .

$16 y + 12 z + 12 = 2$

$x = 10 - 3 y - 4 z$

$16 y + 12 z + 12 = 2$

$x = 10 - 3 y - 4 z$

$16 y + 12 z + 12 = 2$

$x = 10 - 3 y - 4 z$

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 $12 z$ from both sides of the equation.

$16 y + 12 = 2 - 12 z$

$x = 10 - 3 y - 4 z$

Step 2.2.2

Subtract $12$ from both sides of the equation.

$16 y = 2 - 12 z - 12$

$x = 10 - 3 y - 4 z$

Step 2.2.3

Subtract $12$ from $2$ .

$16 y = - 12 z - 10$

$x = 10 - 3 y - 4 z$

$16 y = - 12 z - 10$

$x = 10 - 3 y - 4 z$

Step 2.3

Divide each term in $16 y = - 12 z - 10$ by $16$ and simplify.

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Step 2.3.1

Divide each term in $16 y = - 12 z - 10$ by $16$ .

$\frac{16 y}{16} = \frac{- 12 z}{16} + \frac{- 10}{16}$

$x = 10 - 3 y - 4 z$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $16$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{16 y}{16} = \frac{- 12 z}{16} + \frac{- 10}{16}$

$x = 10 - 3 y - 4 z$

Step 2.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 12 z}{16} + \frac{- 10}{16}$

$x = 10 - 3 y - 4 z$

$y = \frac{- 12 z}{16} + \frac{- 10}{16}$

$x = 10 - 3 y - 4 z$

$y = \frac{- 12 z}{16} + \frac{- 10}{16}$

$x = 10 - 3 y - 4 z$

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

Cancel the common factor of $- 12$ and $16$ .

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Step 2.3.3.1.1.1

Factor $4$ out of $- 12 z$ .

$y = \frac{4 (- 3 z)}{16} + \frac{- 10}{16}$

$x = 10 - 3 y - 4 z$

Step 2.3.3.1.1.2

Cancel the common factors.

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Step 2.3.3.1.1.2.1

Factor $4$ out of $16$ .

$y = \frac{4 (- 3 z)}{4 (4)} + \frac{- 10}{16}$

$x = 10 - 3 y - 4 z$

Step 2.3.3.1.1.2.2

Cancel the common factor.

$y = \frac{4 (- 3 z)}{4 \cdot 4} + \frac{- 10}{16}$

$x = 10 - 3 y - 4 z$

Step 2.3.3.1.1.2.3

Rewrite the expression.

$y = \frac{- 3 z}{4} + \frac{- 10}{16}$

$x = 10 - 3 y - 4 z$

$y = \frac{- 3 z}{4} + \frac{- 10}{16}$

$x = 10 - 3 y - 4 z$

$y = \frac{- 3 z}{4} + \frac{- 10}{16}$

$x = 10 - 3 y - 4 z$

Step 2.3.3.1.2

Move the negative in front of the fraction.

$y = - \frac{3 z}{4} + \frac{- 10}{16}$

$x = 10 - 3 y - 4 z$

Step 2.3.3.1.3

Cancel the common factor of $- 10$ and $16$ .

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Step 2.3.3.1.3.1

Factor $2$ out of $- 10$ .

$y = - \frac{3 z}{4} + \frac{2 (- 5)}{16}$

$x = 10 - 3 y - 4 z$

Step 2.3.3.1.3.2

Cancel the common factors.

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Step 2.3.3.1.3.2.1

Factor $2$ out of $16$ .

$y = - \frac{3 z}{4} + \frac{2 \cdot - 5}{2 \cdot 8}$

$x = 10 - 3 y - 4 z$

Step 2.3.3.1.3.2.2

Cancel the common factor.

$y = - \frac{3 z}{4} + \frac{2 \cdot - 5}{2 \cdot 8}$

$x = 10 - 3 y - 4 z$

Step 2.3.3.1.3.2.3

Rewrite the expression.

$y = - \frac{3 z}{4} + \frac{- 5}{8}$

$x = 10 - 3 y - 4 z$

$y = - \frac{3 z}{4} + \frac{- 5}{8}$

$x = 10 - 3 y - 4 z$

$y = - \frac{3 z}{4} + \frac{- 5}{8}$

$x = 10 - 3 y - 4 z$

Step 2.3.3.1.4

Move the negative in front of the fraction.

$y = - \frac{3 z}{4} - \frac{5}{8}$

$x = 10 - 3 y - 4 z$

$y = - \frac{3 z}{4} - \frac{5}{8}$

$x = 10 - 3 y - 4 z$

$y = - \frac{3 z}{4} - \frac{5}{8}$

$x = 10 - 3 y - 4 z$

$y = - \frac{3 z}{4} - \frac{5}{8}$

$x = 10 - 3 y - 4 z$

$y = - \frac{3 z}{4} - \frac{5}{8}$

$x = 10 - 3 y - 4 z$

Step 3

Simplify the right side.

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Step 3.1

Simplify $10 - 3 (- \frac{3 z}{4} - \frac{5}{8}) - 4 z$ .

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Step 3.1.1

Simplify each term.

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Step 3.1.1.1

Apply the distributive property.

$x = 10 - 3 (- \frac{3 z}{4}) - 3 (- \frac{5}{8}) - 4 z$

$y = - \frac{3 z}{4} - \frac{5}{8}$

Step 3.1.1.2

Multiply $- 3 (- \frac{3 z}{4})$ .

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Step 3.1.1.2.1

Multiply $- 1$ by $- 3$ .

$x = 10 + 3 (\frac{3 z}{4}) - 3 (- \frac{5}{8}) - 4 z$

$y = - \frac{3 z}{4} - \frac{5}{8}$

Step 3.1.1.2.2

Combine $3$ and $\frac{3 z}{4}$ .

$x = 10 + \frac{3 (3 z)}{4} - 3 (- \frac{5}{8}) - 4 z$

$y = - \frac{3 z}{4} - \frac{5}{8}$

Step 3.1.1.2.3

Multiply $3$ by $3$ .

$x = 10 + \frac{9 z}{4} - 3 (- \frac{5}{8}) - 4 z$

$y = - \frac{3 z}{4} - \frac{5}{8}$

$x = 10 + \frac{9 z}{4} - 3 (- \frac{5}{8}) - 4 z$

$y = - \frac{3 z}{4} - \frac{5}{8}$

Step 3.1.1.3

Multiply $- 3 (- \frac{5}{8})$ .

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Step 3.1.1.3.1

Multiply $- 1$ by $- 3$ .

$x = 10 + \frac{9 z}{4} + 3 (\frac{5}{8}) - 4 z$

$y = - \frac{3 z}{4} - \frac{5}{8}$

Step 3.1.1.3.2

Combine $3$ and $\frac{5}{8}$ .

$x = 10 + \frac{9 z}{4} + \frac{3 \cdot 5}{8} - 4 z$

$y = - \frac{3 z}{4} - \frac{5}{8}$

Step 3.1.1.3.3

Multiply $3$ by $5$ .

$x = 10 + \frac{9 z}{4} + \frac{15}{8} - 4 z$

$y = - \frac{3 z}{4} - \frac{5}{8}$

$x = 10 + \frac{9 z}{4} + \frac{15}{8} - 4 z$

$y = - \frac{3 z}{4} - \frac{5}{8}$

$x = 10 + \frac{9 z}{4} + \frac{15}{8} - 4 z$

$y = - \frac{3 z}{4} - \frac{5}{8}$

Step 3.1.2

To write $10$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$x = \frac{9 z}{4} + 10 \cdot \frac{8}{8} + \frac{15}{8} - 4 z$

$y = - \frac{3 z}{4} - \frac{5}{8}$

Step 3.1.3

Combine $10$ and $\frac{8}{8}$ .

$x = \frac{9 z}{4} + \frac{10 \cdot 8}{8} + \frac{15}{8} - 4 z$

$y = - \frac{3 z}{4} - \frac{5}{8}$

Step 3.1.4

Combine the numerators over the common denominator.

$x = \frac{9 z}{4} + \frac{10 \cdot 8 + 15}{8} - 4 z$

$y = - \frac{3 z}{4} - \frac{5}{8}$

Step 3.1.5

Simplify the numerator.

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Step 3.1.5.1

Multiply $10$ by $8$ .

$x = \frac{9 z}{4} + \frac{80 + 15}{8} - 4 z$

$y = - \frac{3 z}{4} - \frac{5}{8}$

Step 3.1.5.2

Add $80$ and $15$ .

$x = \frac{9 z}{4} + \frac{95}{8} - 4 z$

$y = - \frac{3 z}{4} - \frac{5}{8}$

$x = \frac{9 z}{4} + \frac{95}{8} - 4 z$

$y = - \frac{3 z}{4} - \frac{5}{8}$

Step 3.1.6

To write $- 4 z$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = \frac{9 z}{4} - 4 z \cdot \frac{4}{4} + \frac{95}{8}$

$y = - \frac{3 z}{4} - \frac{5}{8}$

Step 3.1.7

Simplify terms.

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Step 3.1.7.1

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

$x = \frac{9 z}{4} + \frac{- 4 z \cdot 4}{4} + \frac{95}{8}$

$y = - \frac{3 z}{4} - \frac{5}{8}$

Step 3.1.7.2

Combine the numerators over the common denominator.

$x = \frac{9 z - 4 z \cdot 4}{4} + \frac{95}{8}$

$y = - \frac{3 z}{4} - \frac{5}{8}$

$x = \frac{9 z - 4 z \cdot 4}{4} + \frac{95}{8}$

$y = - \frac{3 z}{4} - \frac{5}{8}$

Step 3.1.8

Simplify each term.

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Step 3.1.8.1

Simplify the numerator.

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Step 3.1.8.1.1

Factor $z$ out of $9 z - 4 z \cdot 4$ .

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Step 3.1.8.1.1.1

Factor $z$ out of $9 z$ .

$x = \frac{z \cdot 9 - 4 z \cdot 4}{4} + \frac{95}{8}$

$y = - \frac{3 z}{4} - \frac{5}{8}$

Step 3.1.8.1.1.2

Factor $z$ out of $- 4 z \cdot 4$ .

$x = \frac{z \cdot 9 + z (- 4 \cdot 4)}{4} + \frac{95}{8}$

$y = - \frac{3 z}{4} - \frac{5}{8}$

Step 3.1.8.1.1.3

Factor $z$ out of $z \cdot 9 + z (- 4 \cdot 4)$ .

$x = \frac{z (9 - 4 \cdot 4)}{4} + \frac{95}{8}$

$y = - \frac{3 z}{4} - \frac{5}{8}$

$x = \frac{z (9 - 4 \cdot 4)}{4} + \frac{95}{8}$

$y = - \frac{3 z}{4} - \frac{5}{8}$

Step 3.1.8.1.2

Multiply $- 4$ by $4$ .

$x = \frac{z (9 - 16)}{4} + \frac{95}{8}$

$y = - \frac{3 z}{4} - \frac{5}{8}$

Step 3.1.8.1.3

Subtract $16$ from $9$ .

$x = \frac{z \cdot - 7}{4} + \frac{95}{8}$

$y = - \frac{3 z}{4} - \frac{5}{8}$

$x = \frac{z \cdot - 7}{4} + \frac{95}{8}$

$y = - \frac{3 z}{4} - \frac{5}{8}$

Step 3.1.8.2

Move $- 7$ to the left of $z$ .

$x = \frac{- 7 \cdot z}{4} + \frac{95}{8}$

$y = - \frac{3 z}{4} - \frac{5}{8}$

Step 3.1.8.3

Move the negative in front of the fraction.

$x = - \frac{7 z}{4} + \frac{95}{8}$

$y = - \frac{3 z}{4} - \frac{5}{8}$

$x = - \frac{7 z}{4} + \frac{95}{8}$

$y = - \frac{3 z}{4} - \frac{5}{8}$

$x = - \frac{7 z}{4} + \frac{95}{8}$

$y = - \frac{3 z}{4} - \frac{5}{8}$

$x = - \frac{7 z}{4} + \frac{95}{8}$

$y = - \frac{3 z}{4} - \frac{5}{8}$