Finite Math Examples

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

Step 1

Reorder $- 2$ and $x$ .

$y = x - 2$

$y = 3 + z$

$y = 1 - x$

Step 2

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

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Step 2.1

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

$x - 2 = x - 2$

$y = 3 + z$

$y = 1 - x$

Step 2.2

Simplify the left side.

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Step 2.2.1

Remove parentheses.

$x - 2 = x - 2$

$y = 3 + z$

$y = 1 - x$

$x - 2 = x - 2$

$y = 3 + z$

$y = 1 - x$

Step 2.3

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

$x - 2 = 3 + z$

$x - 2 = x - 2$

$y = 1 - x$

Step 2.4

Simplify the left side.

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Step 2.4.1

Remove parentheses.

$x - 2 = 3 + z$

$x - 2 = x - 2$

$y = 1 - x$

$x - 2 = 3 + z$

$x - 2 = x - 2$

$y = 1 - x$

Step 2.5

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

$x - 2 = 1 - x$

$x - 2 = 3 + z$

$x - 2 = x - 2$

Step 2.6

Simplify the left side.

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Step 2.6.1

Remove parentheses.

$x - 2 = 1 - x$

$x - 2 = 3 + z$

$x - 2 = x - 2$

$x - 2 = 1 - x$

$x - 2 = 3 + z$

$x - 2 = x - 2$

$x - 2 = 1 - x$

$x - 2 = 3 + z$

$x - 2 = x - 2$

Step 3

Remove any equations from the system that are always true.

$x - 2 = 1 - x$

$x - 2 = 3 + z$

Step 4

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

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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

Add $x$ to both sides of the equation.

$x - 2 + x = 1$

$x - 2 = 3 + z$

Step 4.1.2

Add $x$ and $x$ .

$2 x - 2 = 1$

$x - 2 = 3 + z$

$2 x - 2 = 1$

$x - 2 = 3 + z$

Step 4.2

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

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Step 4.2.1

Add $2$ to both sides of the equation.

$2 x = 1 + 2$

$x - 2 = 3 + z$

Step 4.2.2

Add $1$ and $2$ .

$2 x = 3$

$x - 2 = 3 + z$

$2 x = 3$

$x - 2 = 3 + z$

Step 4.3

Divide each term in $2 x = 3$ by $2$ and simplify.

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Step 4.3.1

Divide each term in $2 x = 3$ by $2$ .

$\frac{2 x}{2} = \frac{3}{2}$

$x - 2 = 3 + z$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $2$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{3}{2}$

$x - 2 = 3 + z$

Step 4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{3}{2}$

$x - 2 = 3 + z$

$x = \frac{3}{2}$

$x - 2 = 3 + z$

$x = \frac{3}{2}$

$x - 2 = 3 + z$

$x = \frac{3}{2}$

$x - 2 = 3 + z$

$x = \frac{3}{2}$

$x - 2 = 3 + z$

Step 5

Replace all occurrences of $x$ with $\frac{3}{2}$ in each equation.

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Step 5.1

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

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

$x = \frac{3}{2}$

Step 5.2

Simplify the left side.

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Step 5.2.1

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

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Step 5.2.1.1

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

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

$x = \frac{3}{2}$

Step 5.2.1.2

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

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

$x = \frac{3}{2}$

Step 5.2.1.3

Combine the numerators over the common denominator.

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

$x = \frac{3}{2}$

Step 5.2.1.4

Simplify the numerator.

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Step 5.2.1.4.1

Multiply $- 2$ by $2$ .

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

$x = \frac{3}{2}$

Step 5.2.1.4.2

Subtract $4$ from $3$ .

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

$x = \frac{3}{2}$

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

$x = \frac{3}{2}$

Step 5.2.1.5

Move the negative in front of the fraction.

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

$x = \frac{3}{2}$

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

$x = \frac{3}{2}$

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

$x = \frac{3}{2}$

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

$x = \frac{3}{2}$

Step 6

Solve for $z$ in $- \frac{1}{2} = 3 + z$ .

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Step 6.1

Rewrite the equation as $3 + z = - \frac{1}{2}$ .

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

$x = \frac{3}{2}$

Step 6.2

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

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Step 6.2.1

Subtract $3$ from both sides of the equation.

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

$x = \frac{3}{2}$

Step 6.2.2

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

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

$x = \frac{3}{2}$

Step 6.2.3

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

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

$x = \frac{3}{2}$

Step 6.2.4

Combine the numerators over the common denominator.

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

$x = \frac{3}{2}$

Step 6.2.5

Simplify the numerator.

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Step 6.2.5.1

Multiply $- 3$ by $2$ .

$z = \frac{- 1 - 6}{2}$

$x = \frac{3}{2}$

Step 6.2.5.2

Subtract $6$ from $- 1$ .

$z = \frac{- 7}{2}$

$x = \frac{3}{2}$

$z = \frac{- 7}{2}$

$x = \frac{3}{2}$

Step 6.2.6

Move the negative in front of the fraction.

$z = - \frac{7}{2}$

$x = \frac{3}{2}$

$z = - \frac{7}{2}$

$x = \frac{3}{2}$

$z = - \frac{7}{2}$

$x = \frac{3}{2}$

Step 7

Solve the system of equations.

$z = - \frac{7}{2} x = \frac{3}{2}$

Step 8

List all of the solutions.

$z = - \frac{7}{2}, x = \frac{3}{2}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- \frac{7}{2}, \frac{3}{2}$

Decimal Form:

$- 3.5, 1.5$

Mixed Number Form:

$- 3 \frac{1}{2}, 1 \frac{1}{2}$