Finite Math Examples

$2 x + y - z = 4$ , $- x + y + 3 z = 1$

Step 1

Solve the equation for $x$ .

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Step 1.1

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

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Step 1.1.1

Subtract $y$ from both sides of the equation.

$2 x - z = 4 - y$

$- x + y + 3 z = 1$

Step 1.1.2

Add $z$ to both sides of the equation.

$2 x = 4 - y + z$

$- x + y + 3 z = 1$

$2 x = 4 - y + z$

$- x + y + 3 z = 1$

Step 1.2

Divide each term in $2 x = 4 - y + z$ by $2$ and simplify.

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Step 1.2.1

Divide each term in $2 x = 4 - y + z$ by $2$ .

$\frac{2 x}{2} = \frac{4}{2} + \frac{- y}{2} + \frac{z}{2}$

$- x + y + 3 z = 1$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{4}{2} + \frac{- y}{2} + \frac{z}{2}$

$- x + y + 3 z = 1$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{4}{2} + \frac{- y}{2} + \frac{z}{2}$

$- x + y + 3 z = 1$

$x = \frac{4}{2} + \frac{- y}{2} + \frac{z}{2}$

$- x + y + 3 z = 1$

$x = \frac{4}{2} + \frac{- y}{2} + \frac{z}{2}$

$- x + y + 3 z = 1$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Divide $4$ by $2$ .

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

$- x + y + 3 z = 1$

Step 1.2.3.1.2

Move the negative in front of the fraction.

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

$- x + y + 3 z = 1$

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

$- x + y + 3 z = 1$

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

$- x + y + 3 z = 1$

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

$- x + y + 3 z = 1$

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

$- x + y + 3 z = 1$

Step 2

Solve the equation for $y$ .

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Step 2.1

Simplify $- (2 - \frac{y}{2} + \frac{z}{2}) + y + 3 z$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Apply the distributive property.

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

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

Step 2.1.1.2

Simplify.

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Step 2.1.1.2.1

Multiply $- 1$ by $2$ .

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

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

Step 2.1.1.2.2

Multiply $- - \frac{y}{2}$ .

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Step 2.1.1.2.2.1

Multiply $- 1$ by $- 1$ .

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

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

Step 2.1.1.2.2.2

Multiply $\frac{y}{2}$ by $1$ .

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

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

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

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

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

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

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

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

Step 2.1.2

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

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

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

Step 2.1.3

Simplify terms.

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Step 2.1.3.1

Combine $y$ and $\frac{2}{2}$ .

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

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

Step 2.1.3.2

Combine the numerators over the common denominator.

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

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

Step 2.1.3.3

Combine the numerators over the common denominator.

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

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

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

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

Step 2.1.4

Move $2$ to the left of $y$ .

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

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

Step 2.1.5

Add $y$ and $2 y$ .

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

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

Step 2.1.6

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

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

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

Step 2.1.7

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

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

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

Step 2.1.8

Combine the numerators over the common denominator.

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

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

Step 2.1.9

Multiply $- 2$ by $2$ .

$3 z + \frac{- 4 - z + 3 y}{2} = 1$

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

Step 2.1.10

To write $3 z$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$3 z \cdot \frac{2}{2} + \frac{- 4 - z + 3 y}{2} = 1$

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

Step 2.1.11

Simplify terms.

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Step 2.1.11.1

Combine $3 z$ and $\frac{2}{2}$ .

$\frac{3 z \cdot 2}{2} + \frac{- 4 - z + 3 y}{2} = 1$

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

Step 2.1.11.2

Combine the numerators over the common denominator.

$\frac{3 z \cdot 2 - 4 - z + 3 y}{2} = 1$

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

$\frac{3 z \cdot 2 - 4 - z + 3 y}{2} = 1$

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

Step 2.1.12

Simplify the numerator.

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Step 2.1.12.1

Multiply $2$ by $3$ .

$\frac{6 z - 4 - z + 3 y}{2} = 1$

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

Step 2.1.12.2

Subtract $z$ from $6 z$ .

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

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

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

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

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

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

Step 2.2

Multiply both sides by $2$ .

$\frac{5 z - 4 + 3 y}{2} \cdot 2 = 1 \cdot 2$

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

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

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Step 2.3.1.1.1

Cancel the common factor of $2$ .

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Step 2.3.1.1.1.1

Cancel the common factor.

$\frac{5 z - 4 + 3 y}{2} \cdot 2 = 1 \cdot 2$

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

Step 2.3.1.1.1.2

Rewrite the expression.

$5 z - 4 + 3 y = 1 \cdot 2$

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

$5 z - 4 + 3 y = 1 \cdot 2$

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

Step 2.3.1.1.2

Simplify the expression.

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Step 2.3.1.1.2.1

Move $- 4$ .

$5 z + 3 y - 4 = 1 \cdot 2$

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

Step 2.3.1.1.2.2

Reorder $5 z$ and $3 y$ .

$3 y + 5 z - 4 = 1 \cdot 2$

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

$3 y + 5 z - 4 = 1 \cdot 2$

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

$3 y + 5 z - 4 = 1 \cdot 2$

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

$3 y + 5 z - 4 = 1 \cdot 2$

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

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Multiply $2$ by $1$ .

$3 y + 5 z - 4 = 2$

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

$3 y + 5 z - 4 = 2$

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

$3 y + 5 z - 4 = 2$

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

Step 2.4

Solve for $y$ .

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Step 2.4.1

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

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Step 2.4.1.1

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

$3 y - 4 = 2 - 5 z$

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

Step 2.4.1.2

Add $4$ to both sides of the equation.

$3 y = 2 - 5 z + 4$

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

Step 2.4.1.3

Add $2$ and $4$ .

$3 y = - 5 z + 6$

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

$3 y = - 5 z + 6$

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

Step 2.4.2

Divide each term in $3 y = - 5 z + 6$ by $3$ and simplify.

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Step 2.4.2.1

Divide each term in $3 y = - 5 z + 6$ by $3$ .

$\frac{3 y}{3} = \frac{- 5 z}{3} + \frac{6}{3}$

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

Step 2.4.2.2

Simplify the left side.

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Step 2.4.2.2.1

Cancel the common factor of $3$ .

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Step 2.4.2.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{- 5 z}{3} + \frac{6}{3}$

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

Step 2.4.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 5 z}{3} + \frac{6}{3}$

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

$y = \frac{- 5 z}{3} + \frac{6}{3}$

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

$y = \frac{- 5 z}{3} + \frac{6}{3}$

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

Step 2.4.2.3

Simplify the right side.

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Step 2.4.2.3.1

Simplify each term.

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Step 2.4.2.3.1.1

Move the negative in front of the fraction.

$y = - \frac{5 z}{3} + \frac{6}{3}$

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

Step 2.4.2.3.1.2

Divide $6$ by $3$ .

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

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

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

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

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

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

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

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

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

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

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

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

Step 3

Simplify the right side.

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Step 3.1

Simplify $2 - \frac{- \frac{5 z}{3} + 2}{2} + \frac{z}{2}$ .

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Step 3.1.1

Combine the numerators over the common denominator.

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

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

Step 3.1.2

Simplify each term.

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Step 3.1.2.1

Apply the distributive property.

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

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

Step 3.1.2.2

Multiply $- - \frac{5 z}{3}$ .

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Step 3.1.2.2.1

Multiply $- 1$ by $- 1$ .

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

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

Step 3.1.2.2.2

Multiply $\frac{5 z}{3}$ by $1$ .

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

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

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

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

Step 3.1.2.3

Multiply $- 1$ by $2$ .

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

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

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

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

Step 3.1.3

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

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

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

Step 3.1.4

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

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

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

Step 3.1.5

Combine the numerators over the common denominator.

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

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

Step 3.1.6

Add $5 z$ and $z \cdot 3$ .

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Step 3.1.6.1

Reorder $z$ and $3$ .

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

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

Step 3.1.6.2

Add $5 z$ and $3 \cdot z$ .

$x = 2 + \frac{\frac{8 z}{3} - 2}{2}$

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

$x = 2 + \frac{\frac{8 z}{3} - 2}{2}$

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

Step 3.1.7

Cancel the common factor of $\frac{8 z}{3} - 2$ and $2$ .

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Step 3.1.7.1

Factor $2$ out of $\frac{8 z}{3}$ .

$x = 2 + \frac{2 (\frac{4 z}{3}) - 2}{2}$

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

Step 3.1.7.2

Factor $2$ out of $- 2$ .

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

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

Step 3.1.7.3

Factor $2$ out of $2 \frac{4 z}{3} + 2 (- 1)$ .

$x = 2 + \frac{2 (\frac{4 z}{3} - 1)}{2}$

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

Step 3.1.7.4

Cancel the common factors.

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Step 3.1.7.4.1

Factor $2$ out of $2$ .

$x = 2 + \frac{2 (\frac{4 z}{3} - 1)}{2 (1)}$

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

Step 3.1.7.4.2

Cancel the common factor.

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

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

Step 3.1.7.4.3

Rewrite the expression.

$x = 2 + \frac{\frac{4 z}{3} - 1}{1}$

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

Step 3.1.7.4.4

Divide $\frac{4 z}{3} - 1$ by $1$ .

$x = 2 + \frac{4 z}{3} - 1$

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

$x = 2 + \frac{4 z}{3} - 1$

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

$x = 2 + \frac{4 z}{3} - 1$

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

Step 3.1.8

Subtract $1$ from $2$ .

$x = \frac{4 z}{3} + 1$

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

$x = \frac{4 z}{3} + 1$

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

$x = \frac{4 z}{3} + 1$

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