Finite Math Examples

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

Step 1

Replace all occurrences of $z$ with $- 2 x$ in each equation.

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Step 1.1

Replace all occurrences of $z$ in $y = 3 + z$ with $- 2 x$ .

$y = 3 - 2 x$

$y = - 1 + x$

$z = - 2 x$

Step 1.2

Simplify the left side.

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Step 1.2.1

Remove parentheses.

$y = 3 - 2 x$

$y = - 1 + x$

$z = - 2 x$

$y = 3 - 2 x$

$y = - 1 + x$

$z = - 2 x$

$y = 3 - 2 x$

$y = - 1 + x$

$z = - 2 x$

Step 2

Reorder $3$ and $- 2 x$ .

$y = - 2 x + 3$

$y = - 1 + x$

$z = - 2 x$

Step 3

Replace all occurrences of $y$ with $- 2 x + 3$ in each equation.

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Step 3.1

Replace all occurrences of $y$ in $y = - 1 + x$ with $- 2 x + 3$ .

$- 2 x + 3 = - 1 + x$

$y = - 2 x + 3$

$z = - 2 x$

Step 3.2

Simplify the left side.

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Step 3.2.1

Remove parentheses.

$- 2 x + 3 = - 1 + x$

$y = - 2 x + 3$

$z = - 2 x$

$- 2 x + 3 = - 1 + x$

$y = - 2 x + 3$

$z = - 2 x$

$- 2 x + 3 = - 1 + x$

$y = - 2 x + 3$

$z = - 2 x$

Step 4

Solve for $x$ in $- 2 x + 3 = - 1 + x$ .

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Step 4.1

Move all terms containing $x$ to the left side of the equation.

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Step 4.1.1

Subtract $x$ from both sides of the equation.

$- 2 x + 3 - x = - 1$

$y = - 2 x + 3$

$z = - 2 x$

Step 4.1.2

Subtract $x$ from $- 2 x$ .

$- 3 x + 3 = - 1$

$y = - 2 x + 3$

$z = - 2 x$

$- 3 x + 3 = - 1$

$y = - 2 x + 3$

$z = - 2 x$

Step 4.2

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

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Step 4.2.1

Subtract $3$ from both sides of the equation.

$- 3 x = - 1 - 3$

$y = - 2 x + 3$

$z = - 2 x$

Step 4.2.2

Subtract $3$ from $- 1$ .

$- 3 x = - 4$

$y = - 2 x + 3$

$z = - 2 x$

$- 3 x = - 4$

$y = - 2 x + 3$

$z = - 2 x$

Step 4.3

Divide each term in $- 3 x = - 4$ by $- 3$ and simplify.

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Step 4.3.1

Divide each term in $- 3 x = - 4$ by $- 3$ .

$\frac{- 3 x}{- 3} = \frac{- 4}{- 3}$

$y = - 2 x + 3$

$z = - 2 x$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $- 3$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{- 3 x}{- 3} = \frac{- 4}{- 3}$

$y = - 2 x + 3$

$z = - 2 x$

Step 4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 4}{- 3}$

$y = - 2 x + 3$

$z = - 2 x$

$x = \frac{- 4}{- 3}$

$y = - 2 x + 3$

$z = - 2 x$

$x = \frac{- 4}{- 3}$

$y = - 2 x + 3$

$z = - 2 x$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Dividing two negative values results in a positive value.

$x = \frac{4}{3}$

$y = - 2 x + 3$

$z = - 2 x$

$x = \frac{4}{3}$

$y = - 2 x + 3$

$z = - 2 x$

$x = \frac{4}{3}$

$y = - 2 x + 3$

$z = - 2 x$

$x = \frac{4}{3}$

$y = - 2 x + 3$

$z = - 2 x$

Step 5

Replace all occurrences of $x$ with $\frac{4}{3}$ in each equation.

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Step 5.1

Replace all occurrences of $x$ in $y = - 2 x + 3$ with $\frac{4}{3}$ .

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

$x = \frac{4}{3}$

$z = - 2 x$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $- 2 (\frac{4}{3}) + 3$ .

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Step 5.2.1.1

Simplify each term.

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Step 5.2.1.1.1

Multiply $- 2 (\frac{4}{3})$ .

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Step 5.2.1.1.1.1

Combine $- 2$ and $\frac{4}{3}$ .

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

$x = \frac{4}{3}$

$z = - 2 x$

Step 5.2.1.1.1.2

Multiply $- 2$ by $4$ .

$y = \frac{- 8}{3} + 3$

$x = \frac{4}{3}$

$z = - 2 x$

$y = \frac{- 8}{3} + 3$

$x = \frac{4}{3}$

$z = - 2 x$

Step 5.2.1.1.2

Move the negative in front of the fraction.

$y = - \frac{8}{3} + 3$

$x = \frac{4}{3}$

$z = - 2 x$

$y = - \frac{8}{3} + 3$

$x = \frac{4}{3}$

$z = - 2 x$

Step 5.2.1.2

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

$y = - \frac{8}{3} + 3 \cdot \frac{3}{3}$

$x = \frac{4}{3}$

$z = - 2 x$

Step 5.2.1.3

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

$y = - \frac{8}{3} + \frac{3 \cdot 3}{3}$

$x = \frac{4}{3}$

$z = - 2 x$

Step 5.2.1.4

Combine the numerators over the common denominator.

$y = \frac{- 8 + 3 \cdot 3}{3}$

$x = \frac{4}{3}$

$z = - 2 x$

Step 5.2.1.5

Simplify the numerator.

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Step 5.2.1.5.1

Multiply $3$ by $3$ .

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

$x = \frac{4}{3}$

$z = - 2 x$

Step 5.2.1.5.2

Add $- 8$ and $9$ .

$y = \frac{1}{3}$

$x = \frac{4}{3}$

$z = - 2 x$

$y = \frac{1}{3}$

$x = \frac{4}{3}$

$z = - 2 x$

$y = \frac{1}{3}$

$x = \frac{4}{3}$

$z = - 2 x$

$y = \frac{1}{3}$

$x = \frac{4}{3}$

$z = - 2 x$

Step 5.3

Replace all occurrences of $x$ in $z = - 2 x$ with $\frac{4}{3}$ .

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

$y = \frac{1}{3}$

$x = \frac{4}{3}$

Step 5.4

Simplify the right side.

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Step 5.4.1

Simplify $- 2 (\frac{4}{3})$ .

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Step 5.4.1.1

Multiply $- 2 (\frac{4}{3})$ .

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Step 5.4.1.1.1

Combine $- 2$ and $\frac{4}{3}$ .

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

$y = \frac{1}{3}$

$x = \frac{4}{3}$

Step 5.4.1.1.2

Multiply $- 2$ by $4$ .

$z = \frac{- 8}{3}$

$y = \frac{1}{3}$

$x = \frac{4}{3}$

$z = \frac{- 8}{3}$

$y = \frac{1}{3}$

$x = \frac{4}{3}$

Step 5.4.1.2

Move the negative in front of the fraction.

$z = - \frac{8}{3}$

$y = \frac{1}{3}$

$x = \frac{4}{3}$

$z = - \frac{8}{3}$

$y = \frac{1}{3}$

$x = \frac{4}{3}$

$z = - \frac{8}{3}$

$y = \frac{1}{3}$

$x = \frac{4}{3}$

$z = - \frac{8}{3}$

$y = \frac{1}{3}$

$x = \frac{4}{3}$

Step 6

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(\frac{4}{3}, \frac{1}{3}, - \frac{8}{3})$

Step 7

The result can be shown in multiple forms.

Point Form:

$(\frac{4}{3}, \frac{1}{3}, - \frac{8}{3})$

Equation Form:

$x = \frac{4}{3}, y = \frac{1}{3}, z = - \frac{8}{3}$