Finite Math Examples

$2 x + 3 y - z = 1$ , $3 x + 5 y + z = 5$

Step 1

Solve the equation for $y$ .

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Step 1.1

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

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Step 1.1.1

Subtract $2 x$ from both sides of the equation.

$3 y - z = 1 - 2 x$

$3 x + 5 y + z = 5$

Step 1.1.2

Add $z$ to both sides of the equation.

$3 y = 1 - 2 x + z$

$3 x + 5 y + z = 5$

$3 y = 1 - 2 x + z$

$3 x + 5 y + z = 5$

Step 1.2

Divide each term in $3 y = 1 - 2 x + z$ by $3$ and simplify.

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Step 1.2.1

Divide each term in $3 y = 1 - 2 x + z$ by $3$ .

$\frac{3 y}{3} = \frac{1}{3} + \frac{- 2 x}{3} + \frac{z}{3}$

$3 x + 5 y + z = 5$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $3$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{1}{3} + \frac{- 2 x}{3} + \frac{z}{3}$

$3 x + 5 y + z = 5$

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{1}{3} + \frac{- 2 x}{3} + \frac{z}{3}$

$3 x + 5 y + z = 5$

$y = \frac{1}{3} + \frac{- 2 x}{3} + \frac{z}{3}$

$3 x + 5 y + z = 5$

$y = \frac{1}{3} + \frac{- 2 x}{3} + \frac{z}{3}$

$3 x + 5 y + z = 5$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Move the negative in front of the fraction.

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

$3 x + 5 y + z = 5$

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

$3 x + 5 y + z = 5$

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

$3 x + 5 y + z = 5$

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

$3 x + 5 y + z = 5$

Step 2

Solve the equation for $z$ .

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Step 2.1

Simplify $3 x + 5 (\frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}) + z$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Apply the distributive property.

$3 x + 5 (\frac{1}{3}) + 5 (- \frac{2 x}{3}) + 5 (\frac{z}{3}) + z = 5$

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

Step 2.1.1.2

Simplify.

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Step 2.1.1.2.1

Combine $5$ and $\frac{1}{3}$ .

$3 x + \frac{5}{3} + 5 (- \frac{2 x}{3}) + 5 (\frac{z}{3}) + z = 5$

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

Step 2.1.1.2.2

Multiply $5 (- \frac{2 x}{3})$ .

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Step 2.1.1.2.2.1

Multiply $- 1$ by $5$ .

$3 x + \frac{5}{3} - 5 \frac{2 x}{3} + 5 (\frac{z}{3}) + z = 5$

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

Step 2.1.1.2.2.2

Combine $- 5$ and $\frac{2 x}{3}$ .

$3 x + \frac{5}{3} + \frac{- 5 (2 x)}{3} + 5 (\frac{z}{3}) + z = 5$

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

Step 2.1.1.2.2.3

Multiply $2$ by $- 5$ .

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

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

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

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

Step 2.1.1.2.3

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

$3 x + \frac{5}{3} + \frac{- 10 x}{3} + \frac{5 z}{3} + z = 5$

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

$3 x + \frac{5}{3} + \frac{- 10 x}{3} + \frac{5 z}{3} + z = 5$

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

Step 2.1.1.3

Move the negative in front of the fraction.

$3 x + \frac{5}{3} - \frac{10 x}{3} + \frac{5 z}{3} + z = 5$

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

$3 x + \frac{5}{3} - \frac{10 x}{3} + \frac{5 z}{3} + z = 5$

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

Step 2.1.2

To write $3 x$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

Step 2.1.3

Combine $3 x$ and $\frac{3}{3}$ .

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

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

Step 2.1.4

Combine the numerators over the common denominator.

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

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

Step 2.1.5

Combine the numerators over the common denominator.

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

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

Step 2.1.6

Multiply $3$ by $3$ .

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

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

Step 2.1.7

Subtract $10 x$ from $9 x$ .

$z + \frac{- x + 5 + 5 z}{3} = 5$

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

Step 2.1.8

To write $z$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$z \cdot \frac{3}{3} + \frac{- x + 5 + 5 z}{3} = 5$

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

Step 2.1.9

Simplify terms.

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Step 2.1.9.1

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

$\frac{z \cdot 3}{3} + \frac{- x + 5 + 5 z}{3} = 5$

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

Step 2.1.9.2

Combine the numerators over the common denominator.

$\frac{z \cdot 3 - x + 5 + 5 z}{3} = 5$

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

$\frac{z \cdot 3 - x + 5 + 5 z}{3} = 5$

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

Step 2.1.10

Simplify the numerator.

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Step 2.1.10.1

Move $3$ to the left of $z$ .

$\frac{3 \cdot z - x + 5 + 5 z}{3} = 5$

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

Step 2.1.10.2

Add $3 z$ and $5 z$ .

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

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

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

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

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

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

Step 2.2

Multiply both sides by $3$ .

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

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

Step 2.3

Simplify.

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Step 2.3.1

Simplify the left side.

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Step 2.3.1.1

Simplify $\frac{8 z - x + 5}{3} \cdot 3$ .

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Step 2.3.1.1.1

Cancel the common factor of $3$ .

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Step 2.3.1.1.1.1

Cancel the common factor.

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

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

Step 2.3.1.1.1.2

Rewrite the expression.

$8 z - x + 5 = 5 \cdot 3$

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

$8 z - x + 5 = 5 \cdot 3$

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

Step 2.3.1.1.2

Reorder $8 z$ and $- x$ .

$- x + 8 z + 5 = 5 \cdot 3$

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

$- x + 8 z + 5 = 5 \cdot 3$

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

$- x + 8 z + 5 = 5 \cdot 3$

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Multiply $5$ by $3$ .

$- x + 8 z + 5 = 15$

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

$- x + 8 z + 5 = 15$

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

$- x + 8 z + 5 = 15$

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

Step 2.4

Solve for $z$ .

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Step 2.4.1

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

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Step 2.4.1.1

Add $x$ to both sides of the equation.

$8 z + 5 = 15 + x$

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

Step 2.4.1.2

Subtract $5$ from both sides of the equation.

$8 z = 15 + x - 5$

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

Step 2.4.1.3

Subtract $5$ from $15$ .

$8 z = x + 10$

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

$8 z = x + 10$

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

Step 2.4.2

Divide each term in $8 z = x + 10$ by $8$ and simplify.

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Step 2.4.2.1

Divide each term in $8 z = x + 10$ by $8$ .

$\frac{8 z}{8} = \frac{x}{8} + \frac{10}{8}$

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

Step 2.4.2.2

Simplify the left side.

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Step 2.4.2.2.1

Cancel the common factor of $8$ .

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Step 2.4.2.2.1.1

Cancel the common factor.

$\frac{8 z}{8} = \frac{x}{8} + \frac{10}{8}$

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

Step 2.4.2.2.1.2

Divide $z$ by $1$ .

$z = \frac{x}{8} + \frac{10}{8}$

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

$z = \frac{x}{8} + \frac{10}{8}$

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

$z = \frac{x}{8} + \frac{10}{8}$

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

Step 2.4.2.3

Simplify the right side.

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Step 2.4.2.3.1

Cancel the common factor of $10$ and $8$ .

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Step 2.4.2.3.1.1

Factor $2$ out of $10$ .

$z = \frac{x}{8} + \frac{2 (5)}{8}$

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

Step 2.4.2.3.1.2

Cancel the common factors.

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Step 2.4.2.3.1.2.1

Factor $2$ out of $8$ .

$z = \frac{x}{8} + \frac{2 \cdot 5}{2 \cdot 4}$

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

Step 2.4.2.3.1.2.2

Cancel the common factor.

$z = \frac{x}{8} + \frac{2 \cdot 5}{2 \cdot 4}$

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

Step 2.4.2.3.1.2.3

Rewrite the expression.

$z = \frac{x}{8} + \frac{5}{4}$

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

$z = \frac{x}{8} + \frac{5}{4}$

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

$z = \frac{x}{8} + \frac{5}{4}$

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

$z = \frac{x}{8} + \frac{5}{4}$

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

$z = \frac{x}{8} + \frac{5}{4}$

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

$z = \frac{x}{8} + \frac{5}{4}$

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

$z = \frac{x}{8} + \frac{5}{4}$

$y = \frac{1}{3} - \frac{2 x}{3} + \frac{z}{3}$

Step 3

Simplify the right side.

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Step 3.1

Simplify $\frac{1}{3} - \frac{2 x}{3} + \frac{\frac{x}{8} + \frac{5}{4}}{3}$ .

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Step 3.1.1

Combine the numerators over the common denominator.

$y = \frac{1 - 2 x + \frac{x}{8} + \frac{5}{4}}{3}$

$z = \frac{x}{8} + \frac{5}{4}$

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.

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

$z = \frac{x}{8} + \frac{5}{4}$

Step 3.1.2.2

Combine the numerators over the common denominator.

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

$z = \frac{x}{8} + \frac{5}{4}$

Step 3.1.2.3

Add $4$ and $5$ .

$y = \frac{- 2 x + \frac{x}{8} + \frac{9}{4}}{3}$

$z = \frac{x}{8} + \frac{5}{4}$

$y = \frac{- 2 x + \frac{x}{8} + \frac{9}{4}}{3}$

$z = \frac{x}{8} + \frac{5}{4}$

Step 3.1.3

To write $- 2 x$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$y = \frac{- 2 x \cdot \frac{8}{8} + \frac{x}{8} + \frac{9}{4}}{3}$

$z = \frac{x}{8} + \frac{5}{4}$

Step 3.1.4

Simplify terms.

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Step 3.1.4.1

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

$y = \frac{\frac{- 2 x \cdot 8}{8} + \frac{x}{8} + \frac{9}{4}}{3}$

$z = \frac{x}{8} + \frac{5}{4}$

Step 3.1.4.2

Combine the numerators over the common denominator.

$y = \frac{\frac{- 2 x \cdot 8 + x}{8} + \frac{9}{4}}{3}$

$z = \frac{x}{8} + \frac{5}{4}$

Step 3.1.4.3

Simplify each term.

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Step 3.1.4.3.1

Simplify the numerator.

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Step 3.1.4.3.1.1

Factor $x$ out of $- 2 x \cdot 8 + x$ .

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Step 3.1.4.3.1.1.1

Factor $x$ out of $- 2 x \cdot 8$ .

$y = \frac{\frac{x (- 2 \cdot 8) + x}{8} + \frac{9}{4}}{3}$

$z = \frac{x}{8} + \frac{5}{4}$

Step 3.1.4.3.1.1.2

Raise $x$ to the power of $1$ .

$y = \frac{\frac{x (- 2 \cdot 8) + x}{8} + \frac{9}{4}}{3}$

$z = \frac{x}{8} + \frac{5}{4}$

Step 3.1.4.3.1.1.3

Factor $x$ out of $x^{1}$ .

$y = \frac{\frac{x (- 2 \cdot 8) + x \cdot 1}{8} + \frac{9}{4}}{3}$

$z = \frac{x}{8} + \frac{5}{4}$

Step 3.1.4.3.1.1.4

Factor $x$ out of $x (- 2 \cdot 8) + x \cdot 1$ .

$y = \frac{\frac{x (- 2 \cdot 8 + 1)}{8} + \frac{9}{4}}{3}$

$z = \frac{x}{8} + \frac{5}{4}$

$y = \frac{\frac{x (- 2 \cdot 8 + 1)}{8} + \frac{9}{4}}{3}$

$z = \frac{x}{8} + \frac{5}{4}$

Step 3.1.4.3.1.2

Multiply $- 2$ by $8$ .

$y = \frac{\frac{x (- 16 + 1)}{8} + \frac{9}{4}}{3}$

$z = \frac{x}{8} + \frac{5}{4}$

Step 3.1.4.3.1.3

Add $- 16$ and $1$ .

$y = \frac{\frac{x \cdot - 15}{8} + \frac{9}{4}}{3}$

$z = \frac{x}{8} + \frac{5}{4}$

$y = \frac{\frac{x \cdot - 15}{8} + \frac{9}{4}}{3}$

$z = \frac{x}{8} + \frac{5}{4}$

Step 3.1.4.3.2

Move $- 15$ to the left of $x$ .

$y = \frac{\frac{- 15 \cdot x}{8} + \frac{9}{4}}{3}$

$z = \frac{x}{8} + \frac{5}{4}$

Step 3.1.4.3.3

Move the negative in front of the fraction.

$y = \frac{- \frac{15 x}{8} + \frac{9}{4}}{3}$

$z = \frac{x}{8} + \frac{5}{4}$

$y = \frac{- \frac{15 x}{8} + \frac{9}{4}}{3}$

$z = \frac{x}{8} + \frac{5}{4}$

$y = \frac{- \frac{15 x}{8} + \frac{9}{4}}{3}$

$z = \frac{x}{8} + \frac{5}{4}$

Step 3.1.5

Simplify the numerator.

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Step 3.1.5.1

To write $\frac{9}{4}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{- \frac{15 x}{8} + \frac{9}{4} \cdot \frac{2}{2}}{3}$

$z = \frac{x}{8} + \frac{5}{4}$

Step 3.1.5.2

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1.5.2.1

Multiply $\frac{9}{4}$ by $\frac{2}{2}$ .

$y = \frac{- \frac{15 x}{8} + \frac{9 \cdot 2}{4 \cdot 2}}{3}$

$z = \frac{x}{8} + \frac{5}{4}$

Step 3.1.5.2.2

Multiply $4$ by $2$ .

$y = \frac{- \frac{15 x}{8} + \frac{9 \cdot 2}{8}}{3}$

$z = \frac{x}{8} + \frac{5}{4}$

$y = \frac{- \frac{15 x}{8} + \frac{9 \cdot 2}{8}}{3}$

$z = \frac{x}{8} + \frac{5}{4}$

Step 3.1.5.3

Combine the numerators over the common denominator.

$y = \frac{\frac{- 15 x + 9 \cdot 2}{8}}{3}$

$z = \frac{x}{8} + \frac{5}{4}$

Step 3.1.5.4

Simplify the numerator.

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Step 3.1.5.4.1

Factor $3$ out of $- 15 x + 9 \cdot 2$ .

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Step 3.1.5.4.1.1

Factor $3$ out of $- 15 x$ .

$y = \frac{\frac{3 (- 5 x) + 9 \cdot 2}{8}}{3}$

$z = \frac{x}{8} + \frac{5}{4}$

Step 3.1.5.4.1.2

Factor $3$ out of $9 \cdot 2$ .

$y = \frac{\frac{3 (- 5 x) + 3 (3 \cdot 2)}{8}}{3}$

$z = \frac{x}{8} + \frac{5}{4}$

Step 3.1.5.4.1.3

Factor $3$ out of $3 (- 5 x) + 3 (3 \cdot 2)$ .

$y = \frac{\frac{3 (- 5 x + 3 \cdot 2)}{8}}{3}$

$z = \frac{x}{8} + \frac{5}{4}$

$y = \frac{\frac{3 (- 5 x + 3 \cdot 2)}{8}}{3}$

$z = \frac{x}{8} + \frac{5}{4}$

Step 3.1.5.4.2

Multiply $3$ by $2$ .

$y = \frac{\frac{3 (- 5 x + 6)}{8}}{3}$

$z = \frac{x}{8} + \frac{5}{4}$

$y = \frac{\frac{3 (- 5 x + 6)}{8}}{3}$

$z = \frac{x}{8} + \frac{5}{4}$

$y = \frac{\frac{3 (- 5 x + 6)}{8}}{3}$

$z = \frac{x}{8} + \frac{5}{4}$

Step 3.1.6

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{3 (- 5 x + 6)}{8} \cdot \frac{1}{3}$

$z = \frac{x}{8} + \frac{5}{4}$

Step 3.1.7

Cancel the common factor of $3$ .

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Step 3.1.7.1

Cancel the common factor.

$y = \frac{3 (- 5 x + 6)}{8} \cdot \frac{1}{3}$

$z = \frac{x}{8} + \frac{5}{4}$

Step 3.1.7.2

Rewrite the expression.

$y = \frac{- 5 x + 6}{8}$

$z = \frac{x}{8} + \frac{5}{4}$

$y = \frac{- 5 x + 6}{8}$

$z = \frac{x}{8} + \frac{5}{4}$

Step 3.1.8

Factor $- 1$ out of $- 5 x$ .

$y = \frac{- (5 x) + 6}{8}$

$z = \frac{x}{8} + \frac{5}{4}$

Step 3.1.9

Rewrite $6$ as $- 1 (- 6)$ .

$y = \frac{- (5 x) - 1 \cdot - 6}{8}$

$z = \frac{x}{8} + \frac{5}{4}$

Step 3.1.10

Factor $- 1$ out of $- (5 x) - 1 (- 6)$ .

$y = \frac{- (5 x - 6)}{8}$

$z = \frac{x}{8} + \frac{5}{4}$

Step 3.1.11

Simplify the expression.

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Step 3.1.11.1

Rewrite $- (5 x - 6)$ as $- 1 (5 x - 6)$ .

$y = \frac{- 1 (5 x - 6)}{8}$

$z = \frac{x}{8} + \frac{5}{4}$

Step 3.1.11.2

Move the negative in front of the fraction.

$y = - \frac{5 x - 6}{8}$

$z = \frac{x}{8} + \frac{5}{4}$

$y = - \frac{5 x - 6}{8}$

$z = \frac{x}{8} + \frac{5}{4}$

$y = - \frac{5 x - 6}{8}$

$z = \frac{x}{8} + \frac{5}{4}$

$y = - \frac{5 x - 6}{8}$

$z = \frac{x}{8} + \frac{5}{4}$