Finite Math Examples

$x + y + z = 6$ , $x - 2 y - z = 6$ , $2 x - y - 2 z = 6$

Step 1

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

Tap for more steps...

Step 1.1

Subtract $y$ from both sides of the equation.

$x + z = 6 - y$

$x - 2 y - z = 6$

$2 x - y - 2 z = 6$

Step 1.2

Subtract $z$ from both sides of the equation.

$x = 6 - y - z$

$x - 2 y - z = 6$

$2 x - y - 2 z = 6$

$x = 6 - y - z$

$x - 2 y - z = 6$

$2 x - y - 2 z = 6$

Step 2

Replace all occurrences of $x$ with $6 - y - z$ in each equation.

Tap for more steps...

Step 2.1

Replace all occurrences of $x$ in $x - 2 y - z = 6$ with $6 - y - z$ .

$(6 - y - z) - 2 y - z = 6$

$x = 6 - y - z$

$2 x - y - 2 z = 6$

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Simplify $(6 - y - z) - 2 y - z$ .

Tap for more steps...

Step 2.2.1.1

Subtract $2 y$ from $- y$ .

$6 - 3 y - z - z = 6$

$x = 6 - y - z$

$2 x - y - 2 z = 6$

Step 2.2.1.2

Subtract $z$ from $- z$ .

$6 - 3 y - 2 z = 6$

$x = 6 - y - z$

$2 x - y - 2 z = 6$

$6 - 3 y - 2 z = 6$

$x = 6 - y - z$

$2 x - y - 2 z = 6$

$6 - 3 y - 2 z = 6$

$x = 6 - y - z$

$2 x - y - 2 z = 6$

Step 2.3

Replace all occurrences of $x$ in $2 x - y - 2 z = 6$ with $6 - y - z$ .

$2 (6 - y - z) - y - 2 z = 6$

$6 - 3 y - 2 z = 6$

$x = 6 - y - z$

Step 2.4

Simplify the left side.

Tap for more steps...

Step 2.4.1

Simplify $2 (6 - y - z) - y - 2 z$ .

Tap for more steps...

Step 2.4.1.1

Simplify each term.

Tap for more steps...

Step 2.4.1.1.1

Apply the distributive property.

$2 \cdot 6 + 2 (- y) + 2 (- z) - y - 2 z = 6$

$6 - 3 y - 2 z = 6$

$x = 6 - y - z$

Step 2.4.1.1.2

Simplify.

Tap for more steps...

Step 2.4.1.1.2.1

Multiply $2$ by $6$ .

$12 + 2 (- y) + 2 (- z) - y - 2 z = 6$

$6 - 3 y - 2 z = 6$

$x = 6 - y - z$

Step 2.4.1.1.2.2

Multiply $- 1$ by $2$ .

$12 - 2 y + 2 (- z) - y - 2 z = 6$

$6 - 3 y - 2 z = 6$

$x = 6 - y - z$

Step 2.4.1.1.2.3

Multiply $- 1$ by $2$ .

$12 - 2 y - 2 z - y - 2 z = 6$

$6 - 3 y - 2 z = 6$

$x = 6 - y - z$

$12 - 2 y - 2 z - y - 2 z = 6$

$6 - 3 y - 2 z = 6$

$x = 6 - y - z$

$12 - 2 y - 2 z - y - 2 z = 6$

$6 - 3 y - 2 z = 6$

$x = 6 - y - z$

Step 2.4.1.2

Simplify by adding terms.

Tap for more steps...

Step 2.4.1.2.1

Subtract $y$ from $- 2 y$ .

$12 - 3 y - 2 z - 2 z = 6$

$6 - 3 y - 2 z = 6$

$x = 6 - y - z$

Step 2.4.1.2.2

Subtract $2 z$ from $- 2 z$ .

$12 - 3 y - 4 z = 6$

$6 - 3 y - 2 z = 6$

$x = 6 - y - z$

$12 - 3 y - 4 z = 6$

$6 - 3 y - 2 z = 6$

$x = 6 - y - z$

$12 - 3 y - 4 z = 6$

$6 - 3 y - 2 z = 6$

$x = 6 - y - z$

$12 - 3 y - 4 z = 6$

$6 - 3 y - 2 z = 6$

$x = 6 - y - z$

$12 - 3 y - 4 z = 6$

$6 - 3 y - 2 z = 6$

$x = 6 - y - z$

Step 3

Solve for $y$ in $12 - 3 y - 4 z = 6$ .

Tap for more steps...

Step 3.1

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

Tap for more steps...

Step 3.1.1

Subtract $12$ from both sides of the equation.

$- 3 y - 4 z = 6 - 12$

$6 - 3 y - 2 z = 6$

$x = 6 - y - z$

Step 3.1.2

Add $4 z$ to both sides of the equation.

$- 3 y = 6 - 12 + 4 z$

$6 - 3 y - 2 z = 6$

$x = 6 - y - z$

Step 3.1.3

Subtract $12$ from $6$ .

$- 3 y = - 6 + 4 z$

$6 - 3 y - 2 z = 6$

$x = 6 - y - z$

$- 3 y = - 6 + 4 z$

$6 - 3 y - 2 z = 6$

$x = 6 - y - z$

Step 3.2

Divide each term in $- 3 y = - 6 + 4 z$ by $- 3$ and simplify.

Tap for more steps...

Step 3.2.1

Divide each term in $- 3 y = - 6 + 4 z$ by $- 3$ .

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

$6 - 3 y - 2 z = 6$

$x = 6 - y - z$

Step 3.2.2

Simplify the left side.

Tap for more steps...

Step 3.2.2.1

Cancel the common factor of $- 3$ .

Tap for more steps...

Step 3.2.2.1.1

Cancel the common factor.

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

$6 - 3 y - 2 z = 6$

$x = 6 - y - z$

Step 3.2.2.1.2

Divide $y$ by $1$ .

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

$6 - 3 y - 2 z = 6$

$x = 6 - y - z$

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

$6 - 3 y - 2 z = 6$

$x = 6 - y - z$

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

$6 - 3 y - 2 z = 6$

$x = 6 - y - z$

Step 3.2.3

Simplify the right side.

Tap for more steps...

Step 3.2.3.1

Simplify each term.

Tap for more steps...

Step 3.2.3.1.1

Divide $- 6$ by $- 3$ .

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

$6 - 3 y - 2 z = 6$

$x = 6 - y - z$

Step 3.2.3.1.2

Move the negative in front of the fraction.

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

$6 - 3 y - 2 z = 6$

$x = 6 - y - z$

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

$6 - 3 y - 2 z = 6$

$x = 6 - y - z$

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

$6 - 3 y - 2 z = 6$

$x = 6 - y - z$

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

$6 - 3 y - 2 z = 6$

$x = 6 - y - z$

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

$6 - 3 y - 2 z = 6$

$x = 6 - y - z$

Step 4

Replace all occurrences of $y$ with $2 - \frac{4 z}{3}$ in each equation.

Tap for more steps...

Step 4.1

Replace all occurrences of $y$ in $6 - 3 y - 2 z = 6$ with $2 - \frac{4 z}{3}$ .

$6 - 3 (2 - \frac{4 z}{3}) - 2 z = 6$

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

$x = 6 - y - z$

Step 4.2

Simplify the left side.

Tap for more steps...

Step 4.2.1

Simplify $6 - 3 (2 - \frac{4 z}{3}) - 2 z$ .

Tap for more steps...

Step 4.2.1.1

Simplify each term.

Tap for more steps...

Step 4.2.1.1.1

Apply the distributive property.

$6 - 3 \cdot 2 - 3 (- \frac{4 z}{3}) - 2 z = 6$

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

$x = 6 - y - z$

Step 4.2.1.1.2

Multiply $- 3$ by $2$ .

$6 - 6 - 3 (- \frac{4 z}{3}) - 2 z = 6$

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

$x = 6 - y - z$

Step 4.2.1.1.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.2.1.1.3.1

Move the leading negative in $- \frac{4 z}{3}$ into the numerator.

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

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

$x = 6 - y - z$

Step 4.2.1.1.3.2

Factor $3$ out of $- 3$ .

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

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

$x = 6 - y - z$

Step 4.2.1.1.3.3

Cancel the common factor.

$6 - 6 + 3 \cdot (- 1 \frac{- 4 z}{3}) - 2 z = 6$

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

$x = 6 - y - z$

Step 4.2.1.1.3.4

Rewrite the expression.

$6 - 6 - 1 (- 4 z) - 2 z = 6$

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

$x = 6 - y - z$

$6 - 6 - 1 (- 4 z) - 2 z = 6$

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

$x = 6 - y - z$

Step 4.2.1.1.4

Multiply $- 4$ by $- 1$ .

$6 - 6 + 4 z - 2 z = 6$

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

$x = 6 - y - z$

$6 - 6 + 4 z - 2 z = 6$

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

$x = 6 - y - z$

Step 4.2.1.2

Simplify by adding terms.

Tap for more steps...

Step 4.2.1.2.1

Subtract $6$ from $6$ .

$0 + 4 z - 2 z = 6$

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

$x = 6 - y - z$

Step 4.2.1.2.2

Add $0$ and $4 z$ .

$4 z - 2 z = 6$

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

$x = 6 - y - z$

Step 4.2.1.2.3

Subtract $2 z$ from $4 z$ .

$2 z = 6$

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

$x = 6 - y - z$

$2 z = 6$

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

$x = 6 - y - z$

$2 z = 6$

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

$x = 6 - y - z$

$2 z = 6$

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

$x = 6 - y - z$

Step 4.3

Replace all occurrences of $y$ in $x = 6 - y - z$ with $2 - \frac{4 z}{3}$ .

$x = 6 - (2 - \frac{4 z}{3}) - z$

$2 z = 6$

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

Step 4.4

Simplify the right side.

Tap for more steps...

Step 4.4.1

Simplify $6 - (2 - \frac{4 z}{3}) - z$ .

Tap for more steps...

Step 4.4.1.1

Simplify each term.

Tap for more steps...

Step 4.4.1.1.1

Apply the distributive property.

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

$2 z = 6$

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

Step 4.4.1.1.2

Multiply $- 1$ by $2$ .

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

$2 z = 6$

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

Step 4.4.1.1.3

Multiply $- - \frac{4 z}{3}$ .

Tap for more steps...

Step 4.4.1.1.3.1

Multiply $- 1$ by $- 1$ .

$x = 6 - 2 + 1 (\frac{4 z}{3}) - z$

$2 z = 6$

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

Step 4.4.1.1.3.2

Multiply $\frac{4 z}{3}$ by $1$ .

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

$2 z = 6$

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

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

$2 z = 6$

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

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

$2 z = 6$

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

Step 4.4.1.2

Subtract $2$ from $6$ .

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

$2 z = 6$

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

Step 4.4.1.3

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

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

$2 z = 6$

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

Step 4.4.1.4

Simplify terms.

Tap for more steps...

Step 4.4.1.4.1

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

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

$2 z = 6$

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

Step 4.4.1.4.2

Combine the numerators over the common denominator.

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

$2 z = 6$

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

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

$2 z = 6$

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

Step 4.4.1.5

Simplify each term.

Tap for more steps...

Step 4.4.1.5.1

Simplify the numerator.

Tap for more steps...

Step 4.4.1.5.1.1

Factor $z$ out of $4 z - z \cdot 3$ .

Tap for more steps...

Step 4.4.1.5.1.1.1

Factor $z$ out of $4 z$ .

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

$2 z = 6$

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

Step 4.4.1.5.1.1.2

Factor $z$ out of $- z \cdot 3$ .

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

$2 z = 6$

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

Step 4.4.1.5.1.1.3

Factor $z$ out of $z \cdot 4 + z (- 1 \cdot 3)$ .

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

$2 z = 6$

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

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

$2 z = 6$

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

Step 4.4.1.5.1.2

Multiply $- 1$ by $3$ .

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

$2 z = 6$

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

Step 4.4.1.5.1.3

Subtract $3$ from $4$ .

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

$2 z = 6$

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

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

$2 z = 6$

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

Step 4.4.1.5.2

Multiply $z$ by $1$ .

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

$2 z = 6$

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

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

$2 z = 6$

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

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

$2 z = 6$

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

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

$2 z = 6$

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

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

$2 z = 6$

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

Step 5

Divide each term in $2 z = 6$ by $2$ and simplify.

Tap for more steps...

Step 5.1

Divide each term in $2 z = 6$ by $2$ .

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

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

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

Step 5.2

Simplify the left side.

Tap for more steps...

Step 5.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.2.1.1

Cancel the common factor.

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

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

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

Step 5.2.1.2

Divide $z$ by $1$ .

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

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

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

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

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

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

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

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

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

Step 5.3

Simplify the right side.

Tap for more steps...

Step 5.3.1

Divide $6$ by $2$ .

$z = 3$

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

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

$z = 3$

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

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

$z = 3$

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

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

Step 6

Replace all occurrences of $z$ with $3$ in each equation.

Tap for more steps...

Step 6.1

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

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

$z = 3$

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

Step 6.2

Simplify the right side.

Tap for more steps...

Step 6.2.1

Simplify $4 + \frac{3}{3}$ .

Tap for more steps...

Step 6.2.1.1

Divide $3$ by $3$ .

$x = 4 + 1$

$z = 3$

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

Step 6.2.1.2

Add $4$ and $1$ .

$x = 5$

$z = 3$

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

$x = 5$

$z = 3$

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

$x = 5$

$z = 3$

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

Step 6.3

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

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

$x = 5$

$z = 3$

Step 6.4

Simplify the right side.

Tap for more steps...

Step 6.4.1

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

Tap for more steps...

Step 6.4.1.1

Simplify each term.

Tap for more steps...

Step 6.4.1.1.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 6.4.1.1.1.1

Cancel the common factor.

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

$x = 5$

$z = 3$

Step 6.4.1.1.1.2

Divide $4$ by $1$ .

$y = 2 - 1 \cdot 4$

$x = 5$

$z = 3$

$y = 2 - 1 \cdot 4$

$x = 5$

$z = 3$

Step 6.4.1.1.2

Multiply $- 1$ by $4$ .

$y = 2 - 4$

$x = 5$

$z = 3$

$y = 2 - 4$

$x = 5$

$z = 3$

Step 6.4.1.2

Subtract $4$ from $2$ .

$y = - 2$

$x = 5$

$z = 3$

$y = - 2$

$x = 5$

$z = 3$

$y = - 2$

$x = 5$

$z = 3$

$y = - 2$

$x = 5$

$z = 3$

Step 7

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

$(5, - 2, 3)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(5, - 2, 3)$

Equation Form:

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