Finite Math Examples

$3 x + 7 y - 2 z = 0$ , $7 x - 5 y + 3 z = 1$ , $10 x + 2 y + z = 1$

Step 1

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

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Step 1.1

Subtract $10 x$ from both sides of the equation.

$2 y + z = 1 - 10 x$

$3 x + 7 y - 2 z = 0$

$7 x - 5 y + 3 z = 1$

Step 1.2

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

$z = 1 - 10 x - 2 y$

$3 x + 7 y - 2 z = 0$

$7 x - 5 y + 3 z = 1$

$z = 1 - 10 x - 2 y$

$3 x + 7 y - 2 z = 0$

$7 x - 5 y + 3 z = 1$

Step 2

Replace all occurrences of $z$ with $1 - 10 x - 2 y$ in each equation.

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Step 2.1

Replace all occurrences of $z$ in $3 x + 7 y - 2 z = 0$ with $1 - 10 x - 2 y$ .

$3 x + 7 y - 2 (1 - 10 x - 2 y) = 0$

$z = 1 - 10 x - 2 y$

$7 x - 5 y + 3 z = 1$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $3 x + 7 y - 2 (1 - 10 x - 2 y)$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Apply the distributive property.

$3 x + 7 y - 2 \cdot 1 - 2 (- 10 x) - 2 (- 2 y) = 0$

$z = 1 - 10 x - 2 y$

$7 x - 5 y + 3 z = 1$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $- 2$ by $1$ .

$3 x + 7 y - 2 - 2 (- 10 x) - 2 (- 2 y) = 0$

$z = 1 - 10 x - 2 y$

$7 x - 5 y + 3 z = 1$

Step 2.2.1.1.2.2

Multiply $- 10$ by $- 2$ .

$3 x + 7 y - 2 + 20 x - 2 (- 2 y) = 0$

$z = 1 - 10 x - 2 y$

$7 x - 5 y + 3 z = 1$

Step 2.2.1.1.2.3

Multiply $- 2$ by $- 2$ .

$3 x + 7 y - 2 + 20 x + 4 y = 0$

$z = 1 - 10 x - 2 y$

$7 x - 5 y + 3 z = 1$

$3 x + 7 y - 2 + 20 x + 4 y = 0$

$z = 1 - 10 x - 2 y$

$7 x - 5 y + 3 z = 1$

$3 x + 7 y - 2 + 20 x + 4 y = 0$

$z = 1 - 10 x - 2 y$

$7 x - 5 y + 3 z = 1$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Add $3 x$ and $20 x$ .

$23 x + 7 y - 2 + 4 y = 0$

$z = 1 - 10 x - 2 y$

$7 x - 5 y + 3 z = 1$

Step 2.2.1.2.2

Add $7 y$ and $4 y$ .

$23 x + 11 y - 2 = 0$

$z = 1 - 10 x - 2 y$

$7 x - 5 y + 3 z = 1$

$23 x + 11 y - 2 = 0$

$z = 1 - 10 x - 2 y$

$7 x - 5 y + 3 z = 1$

$23 x + 11 y - 2 = 0$

$z = 1 - 10 x - 2 y$

$7 x - 5 y + 3 z = 1$

$23 x + 11 y - 2 = 0$

$z = 1 - 10 x - 2 y$

$7 x - 5 y + 3 z = 1$

Step 2.3

Replace all occurrences of $z$ in $7 x - 5 y + 3 z = 1$ with $1 - 10 x - 2 y$ .

$7 x - 5 y + 3 (1 - 10 x - 2 y) = 1$

$23 x + 11 y - 2 = 0$

$z = 1 - 10 x - 2 y$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $7 x - 5 y + 3 (1 - 10 x - 2 y)$ .

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Step 2.4.1.1

Simplify each term.

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Step 2.4.1.1.1

Apply the distributive property.

$7 x - 5 y + 3 \cdot 1 + 3 (- 10 x) + 3 (- 2 y) = 1$

$23 x + 11 y - 2 = 0$

$z = 1 - 10 x - 2 y$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $3$ by $1$ .

$7 x - 5 y + 3 + 3 (- 10 x) + 3 (- 2 y) = 1$

$23 x + 11 y - 2 = 0$

$z = 1 - 10 x - 2 y$

Step 2.4.1.1.2.2

Multiply $- 10$ by $3$ .

$7 x - 5 y + 3 - 30 x + 3 (- 2 y) = 1$

$23 x + 11 y - 2 = 0$

$z = 1 - 10 x - 2 y$

Step 2.4.1.1.2.3

Multiply $- 2$ by $3$ .

$7 x - 5 y + 3 - 30 x - 6 y = 1$

$23 x + 11 y - 2 = 0$

$z = 1 - 10 x - 2 y$

$7 x - 5 y + 3 - 30 x - 6 y = 1$

$23 x + 11 y - 2 = 0$

$z = 1 - 10 x - 2 y$

$7 x - 5 y + 3 - 30 x - 6 y = 1$

$23 x + 11 y - 2 = 0$

$z = 1 - 10 x - 2 y$

Step 2.4.1.2

Simplify by adding terms.

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Step 2.4.1.2.1

Subtract $30 x$ from $7 x$ .

$- 23 x - 5 y + 3 - 6 y = 1$

$23 x + 11 y - 2 = 0$

$z = 1 - 10 x - 2 y$

Step 2.4.1.2.2

Subtract $6 y$ from $- 5 y$ .

$- 23 x - 11 y + 3 = 1$

$23 x + 11 y - 2 = 0$

$z = 1 - 10 x - 2 y$

$- 23 x - 11 y + 3 = 1$

$23 x + 11 y - 2 = 0$

$z = 1 - 10 x - 2 y$

$- 23 x - 11 y + 3 = 1$

$23 x + 11 y - 2 = 0$

$z = 1 - 10 x - 2 y$

$- 23 x - 11 y + 3 = 1$

$23 x + 11 y - 2 = 0$

$z = 1 - 10 x - 2 y$

$- 23 x - 11 y + 3 = 1$

$23 x + 11 y - 2 = 0$

$z = 1 - 10 x - 2 y$

Step 3

Solve for $x$ in $- 23 x - 11 y + 3 = 1$ .

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Step 3.1

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

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Step 3.1.1

Add $11 y$ to both sides of the equation.

$- 23 x + 3 = 1 + 11 y$

$23 x + 11 y - 2 = 0$

$z = 1 - 10 x - 2 y$

Step 3.1.2

Subtract $3$ from both sides of the equation.

$- 23 x = 1 + 11 y - 3$

$23 x + 11 y - 2 = 0$

$z = 1 - 10 x - 2 y$

Step 3.1.3

Subtract $3$ from $1$ .

$- 23 x = 11 y - 2$

$23 x + 11 y - 2 = 0$

$z = 1 - 10 x - 2 y$

$- 23 x = 11 y - 2$

$23 x + 11 y - 2 = 0$

$z = 1 - 10 x - 2 y$

Step 3.2

Divide each term in $- 23 x = 11 y - 2$ by $- 23$ and simplify.

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Step 3.2.1

Divide each term in $- 23 x = 11 y - 2$ by $- 23$ .

$\frac{- 23 x}{- 23} = \frac{11 y}{- 23} + \frac{- 2}{- 23}$

$23 x + 11 y - 2 = 0$

$z = 1 - 10 x - 2 y$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $- 23$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{- 23 x}{- 23} = \frac{11 y}{- 23} + \frac{- 2}{- 23}$

$23 x + 11 y - 2 = 0$

$z = 1 - 10 x - 2 y$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{11 y}{- 23} + \frac{- 2}{- 23}$

$23 x + 11 y - 2 = 0$

$z = 1 - 10 x - 2 y$

$x = \frac{11 y}{- 23} + \frac{- 2}{- 23}$

$23 x + 11 y - 2 = 0$

$z = 1 - 10 x - 2 y$

$x = \frac{11 y}{- 23} + \frac{- 2}{- 23}$

$23 x + 11 y - 2 = 0$

$z = 1 - 10 x - 2 y$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

Move the negative in front of the fraction.

$x = - \frac{11 y}{23} + \frac{- 2}{- 23}$

$23 x + 11 y - 2 = 0$

$z = 1 - 10 x - 2 y$

Step 3.2.3.1.2

Dividing two negative values results in a positive value.

$x = - \frac{11 y}{23} + \frac{2}{23}$

$23 x + 11 y - 2 = 0$

$z = 1 - 10 x - 2 y$

$x = - \frac{11 y}{23} + \frac{2}{23}$

$23 x + 11 y - 2 = 0$

$z = 1 - 10 x - 2 y$

$x = - \frac{11 y}{23} + \frac{2}{23}$

$23 x + 11 y - 2 = 0$

$z = 1 - 10 x - 2 y$

$x = - \frac{11 y}{23} + \frac{2}{23}$

$23 x + 11 y - 2 = 0$

$z = 1 - 10 x - 2 y$

$x = - \frac{11 y}{23} + \frac{2}{23}$

$23 x + 11 y - 2 = 0$

$z = 1 - 10 x - 2 y$

Step 4

Replace all occurrences of $x$ with $- \frac{11 y}{23} + \frac{2}{23}$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $23 x + 11 y - 2 = 0$ with $- \frac{11 y}{23} + \frac{2}{23}$ .

$23 (- \frac{11 y}{23} + \frac{2}{23}) + 11 y - 2 = 0$

$x = - \frac{11 y}{23} + \frac{2}{23}$

$z = 1 - 10 x - 2 y$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $23 (- \frac{11 y}{23} + \frac{2}{23}) + 11 y - 2$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Apply the distributive property.

$23 (- \frac{11 y}{23}) + 23 (\frac{2}{23}) + 11 y - 2 = 0$

$x = - \frac{11 y}{23} + \frac{2}{23}$

$z = 1 - 10 x - 2 y$

Step 4.2.1.1.2

Cancel the common factor of $23$ .

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Step 4.2.1.1.2.1

Move the leading negative in $- \frac{11 y}{23}$ into the numerator.

$23 (\frac{- 11 y}{23}) + 23 (\frac{2}{23}) + 11 y - 2 = 0$

$x = - \frac{11 y}{23} + \frac{2}{23}$

$z = 1 - 10 x - 2 y$

Step 4.2.1.1.2.2

Cancel the common factor.

$23 (\frac{- 11 y}{23}) + 23 (\frac{2}{23}) + 11 y - 2 = 0$

$x = - \frac{11 y}{23} + \frac{2}{23}$

$z = 1 - 10 x - 2 y$

Step 4.2.1.1.2.3

Rewrite the expression.

$- 11 y + 23 (\frac{2}{23}) + 11 y - 2 = 0$

$x = - \frac{11 y}{23} + \frac{2}{23}$

$z = 1 - 10 x - 2 y$

$- 11 y + 23 (\frac{2}{23}) + 11 y - 2 = 0$

$x = - \frac{11 y}{23} + \frac{2}{23}$

$z = 1 - 10 x - 2 y$

Step 4.2.1.1.3

Cancel the common factor of $23$ .

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Step 4.2.1.1.3.1

Cancel the common factor.

$- 11 y + 23 (\frac{2}{23}) + 11 y - 2 = 0$

$x = - \frac{11 y}{23} + \frac{2}{23}$

$z = 1 - 10 x - 2 y$

Step 4.2.1.1.3.2

Rewrite the expression.

$- 11 y + 2 + 11 y - 2 = 0$

$x = - \frac{11 y}{23} + \frac{2}{23}$

$z = 1 - 10 x - 2 y$

$- 11 y + 2 + 11 y - 2 = 0$

$x = - \frac{11 y}{23} + \frac{2}{23}$

$z = 1 - 10 x - 2 y$

$- 11 y + 2 + 11 y - 2 = 0$

$x = - \frac{11 y}{23} + \frac{2}{23}$

$z = 1 - 10 x - 2 y$

Step 4.2.1.2

Combine the opposite terms in $- 11 y + 2 + 11 y - 2$ .

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Step 4.2.1.2.1

Add $- 11 y$ and $11 y$ .

$0 + 2 - 2 = 0$

$x = - \frac{11 y}{23} + \frac{2}{23}$

$z = 1 - 10 x - 2 y$

Step 4.2.1.2.2

Add $0$ and $2$ .

$2 - 2 = 0$

$x = - \frac{11 y}{23} + \frac{2}{23}$

$z = 1 - 10 x - 2 y$

Step 4.2.1.2.3

Subtract $2$ from $2$ .

$0 = 0$

$x = - \frac{11 y}{23} + \frac{2}{23}$

$z = 1 - 10 x - 2 y$

$0 = 0$

$x = - \frac{11 y}{23} + \frac{2}{23}$

$z = 1 - 10 x - 2 y$

$0 = 0$

$x = - \frac{11 y}{23} + \frac{2}{23}$

$z = 1 - 10 x - 2 y$

$0 = 0$

$x = - \frac{11 y}{23} + \frac{2}{23}$

$z = 1 - 10 x - 2 y$

Step 4.3

Replace all occurrences of $x$ in $z = 1 - 10 x - 2 y$ with $- \frac{11 y}{23} + \frac{2}{23}$ .

$z = 1 - 10 (- \frac{11 y}{23} + \frac{2}{23}) - 2 y$

$0 = 0$

$x = - \frac{11 y}{23} + \frac{2}{23}$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $1 - 10 (- \frac{11 y}{23} + \frac{2}{23}) - 2 y$ .

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Step 4.4.1.1

Simplify each term.

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Step 4.4.1.1.1

Apply the distributive property.

$z = 1 - 10 (- \frac{11 y}{23}) - 10 (\frac{2}{23}) - 2 y$

$0 = 0$

$x = - \frac{11 y}{23} + \frac{2}{23}$

Step 4.4.1.1.2

Multiply $- 10 (- \frac{11 y}{23})$ .

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Step 4.4.1.1.2.1

Multiply $- 1$ by $- 10$ .

$z = 1 + 10 (\frac{11 y}{23}) - 10 (\frac{2}{23}) - 2 y$

$0 = 0$

$x = - \frac{11 y}{23} + \frac{2}{23}$

Step 4.4.1.1.2.2

Combine $10$ and $\frac{11 y}{23}$ .

$z = 1 + \frac{10 (11 y)}{23} - 10 (\frac{2}{23}) - 2 y$

$0 = 0$

$x = - \frac{11 y}{23} + \frac{2}{23}$

Step 4.4.1.1.2.3

Multiply $11$ by $10$ .

$z = 1 + \frac{110 y}{23} - 10 (\frac{2}{23}) - 2 y$

$0 = 0$

$x = - \frac{11 y}{23} + \frac{2}{23}$

$z = 1 + \frac{110 y}{23} - 10 (\frac{2}{23}) - 2 y$

$0 = 0$

$x = - \frac{11 y}{23} + \frac{2}{23}$

Step 4.4.1.1.3

Multiply $- 10 (\frac{2}{23})$ .

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Step 4.4.1.1.3.1

Combine $- 10$ and $\frac{2}{23}$ .

$z = 1 + \frac{110 y}{23} + \frac{- 10 \cdot 2}{23} - 2 y$

$0 = 0$

$x = - \frac{11 y}{23} + \frac{2}{23}$

Step 4.4.1.1.3.2

Multiply $- 10$ by $2$ .

$z = 1 + \frac{110 y}{23} + \frac{- 20}{23} - 2 y$

$0 = 0$

$x = - \frac{11 y}{23} + \frac{2}{23}$

$z = 1 + \frac{110 y}{23} + \frac{- 20}{23} - 2 y$

$0 = 0$

$x = - \frac{11 y}{23} + \frac{2}{23}$

Step 4.4.1.1.4

Move the negative in front of the fraction.

$z = 1 + \frac{110 y}{23} - \frac{20}{23} - 2 y$

$0 = 0$

$x = - \frac{11 y}{23} + \frac{2}{23}$

$z = 1 + \frac{110 y}{23} - \frac{20}{23} - 2 y$

$0 = 0$

$x = - \frac{11 y}{23} + \frac{2}{23}$

Step 4.4.1.2

Simplify the expression.

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Step 4.4.1.2.1

Write $1$ as a fraction with a common denominator.

$z = \frac{110 y}{23} + \frac{23}{23} - \frac{20}{23} - 2 y$

$0 = 0$

$x = - \frac{11 y}{23} + \frac{2}{23}$

Step 4.4.1.2.2

Combine the numerators over the common denominator.

$z = \frac{110 y}{23} + \frac{23 - 20}{23} - 2 y$

$0 = 0$

$x = - \frac{11 y}{23} + \frac{2}{23}$

Step 4.4.1.2.3

Subtract $20$ from $23$ .

$z = \frac{110 y}{23} + \frac{3}{23} - 2 y$

$0 = 0$

$x = - \frac{11 y}{23} + \frac{2}{23}$

$z = \frac{110 y}{23} + \frac{3}{23} - 2 y$

$0 = 0$

$x = - \frac{11 y}{23} + \frac{2}{23}$

Step 4.4.1.3

To write $- 2 y$ as a fraction with a common denominator, multiply by $\frac{23}{23}$ .

$z = \frac{110 y}{23} - 2 y \cdot \frac{23}{23} + \frac{3}{23}$

$0 = 0$

$x = - \frac{11 y}{23} + \frac{2}{23}$

Step 4.4.1.4

Combine $- 2 y$ and $\frac{23}{23}$ .

$z = \frac{110 y}{23} + \frac{- 2 y \cdot 23}{23} + \frac{3}{23}$

$0 = 0$

$x = - \frac{11 y}{23} + \frac{2}{23}$

Step 4.4.1.5

Combine the numerators over the common denominator.

$z = \frac{110 y - 2 y \cdot 23}{23} + \frac{3}{23}$

$0 = 0$

$x = - \frac{11 y}{23} + \frac{2}{23}$

Step 4.4.1.6

Combine the numerators over the common denominator.

$z = \frac{110 y - 2 y \cdot 23 + 3}{23}$

$0 = 0$

$x = - \frac{11 y}{23} + \frac{2}{23}$

Step 4.4.1.7

Multiply $23$ by $- 2$ .

$z = \frac{110 y - 46 y + 3}{23}$

$0 = 0$

$x = - \frac{11 y}{23} + \frac{2}{23}$

Step 4.4.1.8

Subtract $46 y$ from $110 y$ .

$z = \frac{64 y + 3}{23}$

$0 = 0$

$x = - \frac{11 y}{23} + \frac{2}{23}$

$z = \frac{64 y + 3}{23}$

$0 = 0$

$x = - \frac{11 y}{23} + \frac{2}{23}$

$z = \frac{64 y + 3}{23}$

$0 = 0$

$x = - \frac{11 y}{23} + \frac{2}{23}$

$z = \frac{64 y + 3}{23}$

$0 = 0$

$x = - \frac{11 y}{23} + \frac{2}{23}$

Step 5

Remove any equations from the system that are always true.

$z = \frac{64 y + 3}{23}$

$x = - \frac{11 y}{23} + \frac{2}{23}$