Finite Math Examples

$2 x - 3 y - z = 0$ , $- x + 2 y + z = 5$

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.

$- 3 y - z = - 2 x$

$- x + 2 y + z = 5$

Step 1.1.2

Add $z$ to both sides of the equation.

$- 3 y = - 2 x + z$

$- x + 2 y + z = 5$

$- 3 y = - 2 x + z$

$- x + 2 y + z = 5$

Step 1.2

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

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Step 1.2.1

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

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

$- x + 2 y + z = 5$

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.

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

$- x + 2 y + z = 5$

Step 1.2.2.1.2

Divide $y$ by $1$ .

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

$- x + 2 y + z = 5$

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

$- x + 2 y + z = 5$

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

$- x + 2 y + z = 5$

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

Dividing two negative values results in a positive value.

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

$- x + 2 y + z = 5$

Step 1.2.3.1.2

Move the negative in front of the fraction.

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

$- x + 2 y + z = 5$

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

$- x + 2 y + z = 5$

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

$- x + 2 y + z = 5$

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

$- x + 2 y + z = 5$

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

$- x + 2 y + z = 5$

Step 2

Solve the equation for $z$ .

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Step 2.1

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

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Apply the distributive property.

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

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

Step 2.1.1.2

Multiply $2 \frac{2 x}{3}$ .

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Step 2.1.1.2.1

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

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

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

Step 2.1.1.2.2

Multiply $2$ by $2$ .

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

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

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

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

Step 2.1.1.3

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

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Step 2.1.1.3.1

Multiply $- 1$ by $2$ .

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

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

Step 2.1.1.3.2

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

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

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

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

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

Step 2.1.1.4

Move the negative in front of the fraction.

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

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

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

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

Step 2.1.2

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

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

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

Step 2.1.3

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

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

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

Step 2.1.4

Combine the numerators over the common denominator.

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

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

Step 2.1.5

Combine the numerators over the common denominator.

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

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

Step 2.1.6

Multiply $3$ by $- 1$ .

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

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

Step 2.1.7

Add $- 3 x$ and $4 x$ .

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

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

Step 2.1.8

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

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

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

Step 2.1.9

Simplify terms.

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Step 2.1.9.1

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

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

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

Step 2.1.9.2

Combine the numerators over the common denominator.

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

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

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

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

Step 2.1.10

Simplify the numerator.

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Step 2.1.10.1

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

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

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

Step 2.1.10.2

Subtract $2 z$ from $3 z$ .

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

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

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

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

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

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

Step 2.2

Multiply both sides of the equation by $3$ .

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

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

Step 2.3

Simplify both sides of the equation.

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Step 2.3.1

Simplify the left side.

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Step 2.3.1.1

Cancel the common factor of $3$ .

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Step 2.3.1.1.1

Cancel the common factor.

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

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

Step 2.3.1.1.2

Rewrite the expression.

$x + z = 3 \cdot 5$

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

$x + z = 3 \cdot 5$

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

$x + z = 3 \cdot 5$

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

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Multiply $3$ by $5$ .

$x + z = 15$

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

$x + z = 15$

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

$x + z = 15$

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

Step 2.4

Subtract $x$ from both sides of the equation.

$z = 15 - x$

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

$z = 15 - x$

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

Step 3

Simplify the right side.

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Step 3.1

Simplify $\frac{2 x}{3} - \frac{15 - x}{3}$ .

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Step 3.1.1

Combine the numerators over the common denominator.

$y = \frac{2 x - (15 - x)}{3}$

$z = 15 - x$

Step 3.1.2

Simplify each term.

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Step 3.1.2.1

Apply the distributive property.

$y = \frac{2 x - 1 \cdot 15 + x}{3}$

$z = 15 - x$

Step 3.1.2.2

Multiply $- 1$ by $15$ .

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

$z = 15 - x$

Step 3.1.2.3

Multiply $- - x$ .

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Step 3.1.2.3.1

Multiply $- 1$ by $- 1$ .

$y = \frac{2 x - 15 + 1 x}{3}$

$z = 15 - x$

Step 3.1.2.3.2

Multiply $x$ by $1$ .

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

$z = 15 - x$

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

$z = 15 - x$

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

$z = 15 - x$

Step 3.1.3

Simplify terms.

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Step 3.1.3.1

Add $2 x$ and $x$ .

$y = \frac{3 x - 15}{3}$

$z = 15 - x$

Step 3.1.3.2

Cancel the common factor of $3 x - 15$ and $3$ .

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Step 3.1.3.2.1

Factor $3$ out of $3 x$ .

$y = \frac{3 (x) - 15}{3}$

$z = 15 - x$

Step 3.1.3.2.2

Factor $3$ out of $- 15$ .

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

$z = 15 - x$

Step 3.1.3.2.3

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

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

$z = 15 - x$

Step 3.1.3.2.4

Cancel the common factors.

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Step 3.1.3.2.4.1

Factor $3$ out of $3$ .

$y = \frac{3 (x - 5)}{3 (1)}$

$z = 15 - x$

Step 3.1.3.2.4.2

Cancel the common factor.

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

$z = 15 - x$

Step 3.1.3.2.4.3

Rewrite the expression.

$y = \frac{x - 5}{1}$

$z = 15 - x$

Step 3.1.3.2.4.4

Divide $x - 5$ by $1$ .

$y = x - 5$

$z = 15 - x$

$y = x - 5$

$z = 15 - x$

$y = x - 5$

$z = 15 - x$

$y = x - 5$

$z = 15 - x$

$y = x - 5$

$z = 15 - x$

$y = x - 5$

$z = 15 - x$

Step 4

Simplify the right side.

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Step 4.1

Reorder $15$ and $- x$ .

$z = - x + 15$

$y = x - 5$

$z = - x + 15$

$y = x - 5$