Finite Math Examples

$x - 5 y + z = 3$ , $x + 2 y - 2 z = - 12$ , $2 x + 2 z = 18$

Step 1

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

Tap for more steps...

Step 1.1

Add $5 y$ to both sides of the equation.

$x + z = 3 + 5 y$

$x + 2 y - 2 z = - 12$

$2 x + 2 z = 18$

Step 1.2

Subtract $z$ from both sides of the equation.

$x = 3 + 5 y - z$

$x + 2 y - 2 z = - 12$

$2 x + 2 z = 18$

$x = 3 + 5 y - z$

$x + 2 y - 2 z = - 12$

$2 x + 2 z = 18$

Step 2

Replace all occurrences of $x$ with $3 + 5 y - z$ in each equation.

Tap for more steps...

Step 2.1

Replace all occurrences of $x$ in $x + 2 y - 2 z = - 12$ with $3 + 5 y - z$ .

$(3 + 5 y - z) + 2 y - 2 z = - 12$

$x = 3 + 5 y - z$

$2 x + 2 z = 18$

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Simplify $(3 + 5 y - z) + 2 y - 2 z$ .

Tap for more steps...

Step 2.2.1.1

Add $5 y$ and $2 y$ .

$3 + 7 y - z - 2 z = - 12$

$x = 3 + 5 y - z$

$2 x + 2 z = 18$

Step 2.2.1.2

Subtract $2 z$ from $- z$ .

$3 + 7 y - 3 z = - 12$

$x = 3 + 5 y - z$

$2 x + 2 z = 18$

$3 + 7 y - 3 z = - 12$

$x = 3 + 5 y - z$

$2 x + 2 z = 18$

$3 + 7 y - 3 z = - 12$

$x = 3 + 5 y - z$

$2 x + 2 z = 18$

Step 2.3

Replace all occurrences of $x$ in $2 x + 2 z = 18$ with $3 + 5 y - z$ .

$2 (3 + 5 y - z) + 2 z = 18$

$3 + 7 y - 3 z = - 12$

$x = 3 + 5 y - z$

Step 2.4

Simplify the left side.

Tap for more steps...

Step 2.4.1

Simplify $2 (3 + 5 y - z) + 2 z$ .

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.

$2 \cdot 3 + 2 (5 y) + 2 (- z) + 2 z = 18$

$3 + 7 y - 3 z = - 12$

$x = 3 + 5 y - z$

Step 2.4.1.1.2

Simplify.

Tap for more steps...

Step 2.4.1.1.2.1

Multiply $2$ by $3$ .

$6 + 2 (5 y) + 2 (- z) + 2 z = 18$

$3 + 7 y - 3 z = - 12$

$x = 3 + 5 y - z$

Step 2.4.1.1.2.2

Multiply $5$ by $2$ .

$6 + 10 y + 2 (- z) + 2 z = 18$

$3 + 7 y - 3 z = - 12$

$x = 3 + 5 y - z$

Step 2.4.1.1.2.3

Multiply $- 1$ by $2$ .

$6 + 10 y - 2 z + 2 z = 18$

$3 + 7 y - 3 z = - 12$

$x = 3 + 5 y - z$

$6 + 10 y - 2 z + 2 z = 18$

$3 + 7 y - 3 z = - 12$

$x = 3 + 5 y - z$

$6 + 10 y - 2 z + 2 z = 18$

$3 + 7 y - 3 z = - 12$

$x = 3 + 5 y - z$

Step 2.4.1.2

Combine the opposite terms in $6 + 10 y - 2 z + 2 z$ .

Tap for more steps...

Step 2.4.1.2.1

Add $- 2 z$ and $2 z$ .

$6 + 10 y + 0 = 18$

$3 + 7 y - 3 z = - 12$

$x = 3 + 5 y - z$

Step 2.4.1.2.2

Add $6 + 10 y$ and $0$ .

$6 + 10 y = 18$

$3 + 7 y - 3 z = - 12$

$x = 3 + 5 y - z$

$6 + 10 y = 18$

$3 + 7 y - 3 z = - 12$

$x = 3 + 5 y - z$

$6 + 10 y = 18$

$3 + 7 y - 3 z = - 12$

$x = 3 + 5 y - z$

$6 + 10 y = 18$

$3 + 7 y - 3 z = - 12$

$x = 3 + 5 y - z$

$6 + 10 y = 18$

$3 + 7 y - 3 z = - 12$

$x = 3 + 5 y - z$

Step 3

Solve for $y$ in $6 + 10 y = 18$ .

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 $6$ from both sides of the equation.

$10 y = 18 - 6$

$3 + 7 y - 3 z = - 12$

$x = 3 + 5 y - z$

Step 3.1.2

Subtract $6$ from $18$ .

$10 y = 12$

$3 + 7 y - 3 z = - 12$

$x = 3 + 5 y - z$

$10 y = 12$

$3 + 7 y - 3 z = - 12$

$x = 3 + 5 y - z$

Step 3.2

Divide each term in $10 y = 12$ by $10$ and simplify.

Tap for more steps...

Step 3.2.1

Divide each term in $10 y = 12$ by $10$ .

$\frac{10 y}{10} = \frac{12}{10}$

$3 + 7 y - 3 z = - 12$

$x = 3 + 5 y - z$

Step 3.2.2

Simplify the left side.

Tap for more steps...

Step 3.2.2.1

Cancel the common factor of $10$ .

Tap for more steps...

Step 3.2.2.1.1

Cancel the common factor.

$\frac{10 y}{10} = \frac{12}{10}$

$3 + 7 y - 3 z = - 12$

$x = 3 + 5 y - z$

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{12}{10}$

$3 + 7 y - 3 z = - 12$

$x = 3 + 5 y - z$

$y = \frac{12}{10}$

$3 + 7 y - 3 z = - 12$

$x = 3 + 5 y - z$

$y = \frac{12}{10}$

$3 + 7 y - 3 z = - 12$

$x = 3 + 5 y - z$

Step 3.2.3

Simplify the right side.

Tap for more steps...

Step 3.2.3.1

Cancel the common factor of $12$ and $10$ .

Tap for more steps...

Step 3.2.3.1.1

Factor $2$ out of $12$ .

$y = \frac{2 (6)}{10}$

$3 + 7 y - 3 z = - 12$

$x = 3 + 5 y - z$

Step 3.2.3.1.2

Cancel the common factors.

Tap for more steps...

Step 3.2.3.1.2.1

Factor $2$ out of $10$ .

$y = \frac{2 \cdot 6}{2 \cdot 5}$

$3 + 7 y - 3 z = - 12$

$x = 3 + 5 y - z$

Step 3.2.3.1.2.2

Cancel the common factor.

$y = \frac{2 \cdot 6}{2 \cdot 5}$

$3 + 7 y - 3 z = - 12$

$x = 3 + 5 y - z$

Step 3.2.3.1.2.3

Rewrite the expression.

$y = \frac{6}{5}$

$3 + 7 y - 3 z = - 12$

$x = 3 + 5 y - z$

$y = \frac{6}{5}$

$3 + 7 y - 3 z = - 12$

$x = 3 + 5 y - z$

$y = \frac{6}{5}$

$3 + 7 y - 3 z = - 12$

$x = 3 + 5 y - z$

$y = \frac{6}{5}$

$3 + 7 y - 3 z = - 12$

$x = 3 + 5 y - z$

$y = \frac{6}{5}$

$3 + 7 y - 3 z = - 12$

$x = 3 + 5 y - z$

$y = \frac{6}{5}$

$3 + 7 y - 3 z = - 12$

$x = 3 + 5 y - z$

Step 4

Replace all occurrences of $y$ with $\frac{6}{5}$ in each equation.

Tap for more steps...

Step 4.1

Replace all occurrences of $y$ in $3 + 7 y - 3 z = - 12$ with $\frac{6}{5}$ .

$3 + 7 (\frac{6}{5}) - 3 z = - 12$

$y = \frac{6}{5}$

$x = 3 + 5 y - z$

Step 4.2

Simplify the left side.

Tap for more steps...

Step 4.2.1

Simplify $3 + 7 (\frac{6}{5}) - 3 z$ .

Tap for more steps...

Step 4.2.1.1

Multiply $7 (\frac{6}{5})$ .

Tap for more steps...

Step 4.2.1.1.1

Combine $7$ and $\frac{6}{5}$ .

$3 + \frac{7 \cdot 6}{5} - 3 z = - 12$

$y = \frac{6}{5}$

$x = 3 + 5 y - z$

Step 4.2.1.1.2

Multiply $7$ by $6$ .

$3 + \frac{42}{5} - 3 z = - 12$

$y = \frac{6}{5}$

$x = 3 + 5 y - z$

$3 + \frac{42}{5} - 3 z = - 12$

$y = \frac{6}{5}$

$x = 3 + 5 y - z$

Step 4.2.1.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$3 \cdot \frac{5}{5} + \frac{42}{5} - 3 z = - 12$

$y = \frac{6}{5}$

$x = 3 + 5 y - z$

Step 4.2.1.3

Combine $3$ and $\frac{5}{5}$ .

$\frac{3 \cdot 5}{5} + \frac{42}{5} - 3 z = - 12$

$y = \frac{6}{5}$

$x = 3 + 5 y - z$

Step 4.2.1.4

Combine the numerators over the common denominator.

$\frac{3 \cdot 5 + 42}{5} - 3 z = - 12$

$y = \frac{6}{5}$

$x = 3 + 5 y - z$

Step 4.2.1.5

Simplify the numerator.

Tap for more steps...

Step 4.2.1.5.1

Multiply $3$ by $5$ .

$\frac{15 + 42}{5} - 3 z = - 12$

$y = \frac{6}{5}$

$x = 3 + 5 y - z$

Step 4.2.1.5.2

Add $15$ and $42$ .

$\frac{57}{5} - 3 z = - 12$

$y = \frac{6}{5}$

$x = 3 + 5 y - z$

$\frac{57}{5} - 3 z = - 12$

$y = \frac{6}{5}$

$x = 3 + 5 y - z$

$\frac{57}{5} - 3 z = - 12$

$y = \frac{6}{5}$

$x = 3 + 5 y - z$

$\frac{57}{5} - 3 z = - 12$

$y = \frac{6}{5}$

$x = 3 + 5 y - z$

Step 4.3

Replace all occurrences of $y$ in $x = 3 + 5 y - z$ with $\frac{6}{5}$ .

$x = 3 + 5 (\frac{6}{5}) - z$

$\frac{57}{5} - 3 z = - 12$

$y = \frac{6}{5}$

Step 4.4

Simplify the right side.

Tap for more steps...

Step 4.4.1

Simplify $3 + 5 (\frac{6}{5}) - z$ .

Tap for more steps...

Step 4.4.1.1

Cancel the common factor of $5$ .

Tap for more steps...

Step 4.4.1.1.1

Cancel the common factor.

$x = 3 + 5 (\frac{6}{5}) - z$

$\frac{57}{5} - 3 z = - 12$

$y = \frac{6}{5}$

Step 4.4.1.1.2

Rewrite the expression.

$x = 3 + 6 - z$

$\frac{57}{5} - 3 z = - 12$

$y = \frac{6}{5}$

$x = 3 + 6 - z$

$\frac{57}{5} - 3 z = - 12$

$y = \frac{6}{5}$

Step 4.4.1.2

Add $3$ and $6$ .

$x = 9 - z$

$\frac{57}{5} - 3 z = - 12$

$y = \frac{6}{5}$

$x = 9 - z$

$\frac{57}{5} - 3 z = - 12$

$y = \frac{6}{5}$

$x = 9 - z$

$\frac{57}{5} - 3 z = - 12$

$y = \frac{6}{5}$

$x = 9 - z$

$\frac{57}{5} - 3 z = - 12$

$y = \frac{6}{5}$

Step 5

Solve for $z$ in $\frac{57}{5} - 3 z = - 12$ .

Tap for more steps...

Step 5.1

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

Tap for more steps...

Step 5.1.1

Subtract $\frac{57}{5}$ from both sides of the equation.

$- 3 z = - 12 - \frac{57}{5}$

$x = 9 - z$

$y = \frac{6}{5}$

Step 5.1.2

To write $- 12$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- 3 z = - 12 \cdot \frac{5}{5} - \frac{57}{5}$

$x = 9 - z$

$y = \frac{6}{5}$

Step 5.1.3

Combine $- 12$ and $\frac{5}{5}$ .

$- 3 z = \frac{- 12 \cdot 5}{5} - \frac{57}{5}$

$x = 9 - z$

$y = \frac{6}{5}$

Step 5.1.4

Combine the numerators over the common denominator.

$- 3 z = \frac{- 12 \cdot 5 - 57}{5}$

$x = 9 - z$

$y = \frac{6}{5}$

Step 5.1.5

Simplify the numerator.

Tap for more steps...

Step 5.1.5.1

Multiply $- 12$ by $5$ .

$- 3 z = \frac{- 60 - 57}{5}$

$x = 9 - z$

$y = \frac{6}{5}$

Step 5.1.5.2

Subtract $57$ from $- 60$ .

$- 3 z = \frac{- 117}{5}$

$x = 9 - z$

$y = \frac{6}{5}$

$- 3 z = \frac{- 117}{5}$

$x = 9 - z$

$y = \frac{6}{5}$

Step 5.1.6

Move the negative in front of the fraction.

$- 3 z = - \frac{117}{5}$

$x = 9 - z$

$y = \frac{6}{5}$

$- 3 z = - \frac{117}{5}$

$x = 9 - z$

$y = \frac{6}{5}$

Step 5.2

Divide each term in $- 3 z = - \frac{117}{5}$ by $- 3$ and simplify.

Tap for more steps...

Step 5.2.1

Divide each term in $- 3 z = - \frac{117}{5}$ by $- 3$ .

$\frac{- 3 z}{- 3} = \frac{- \frac{117}{5}}{- 3}$

$x = 9 - z$

$y = \frac{6}{5}$

Step 5.2.2

Simplify the left side.

Tap for more steps...

Step 5.2.2.1

Cancel the common factor of $- 3$ .

Tap for more steps...

Step 5.2.2.1.1

Cancel the common factor.

$\frac{- 3 z}{- 3} = \frac{- \frac{117}{5}}{- 3}$

$x = 9 - z$

$y = \frac{6}{5}$

Step 5.2.2.1.2

Divide $z$ by $1$ .

$z = \frac{- \frac{117}{5}}{- 3}$

$x = 9 - z$

$y = \frac{6}{5}$

$z = \frac{- \frac{117}{5}}{- 3}$

$x = 9 - z$

$y = \frac{6}{5}$

$z = \frac{- \frac{117}{5}}{- 3}$

$x = 9 - z$

$y = \frac{6}{5}$

Step 5.2.3

Simplify the right side.

Tap for more steps...

Step 5.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$z = - \frac{117}{5} \cdot \frac{1}{- 3}$

$x = 9 - z$

$y = \frac{6}{5}$

Step 5.2.3.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 5.2.3.2.1

Move the leading negative in $- \frac{117}{5}$ into the numerator.

$z = \frac{- 117}{5} \cdot \frac{1}{- 3}$

$x = 9 - z$

$y = \frac{6}{5}$

Step 5.2.3.2.2

Factor $3$ out of $- 117$ .

$z = \frac{3 (- 39)}{5} \cdot \frac{1}{- 3}$

$x = 9 - z$

$y = \frac{6}{5}$

Step 5.2.3.2.3

Factor $3$ out of $- 3$ .

$z = \frac{3 \cdot - 39}{5} \cdot \frac{1}{3 \cdot - 1}$

$x = 9 - z$

$y = \frac{6}{5}$

Step 5.2.3.2.4

Cancel the common factor.

$z = \frac{3 \cdot - 39}{5} \cdot \frac{1}{3 \cdot - 1}$

$x = 9 - z$

$y = \frac{6}{5}$

Step 5.2.3.2.5

Rewrite the expression.

$z = \frac{- 39}{5} \cdot \frac{1}{- 1}$

$x = 9 - z$

$y = \frac{6}{5}$

$z = \frac{- 39}{5} \cdot \frac{1}{- 1}$

$x = 9 - z$

$y = \frac{6}{5}$

Step 5.2.3.3

Multiply $\frac{- 39}{5}$ by $\frac{1}{- 1}$ .

$z = \frac{- 39}{5 \cdot - 1}$

$x = 9 - z$

$y = \frac{6}{5}$

Step 5.2.3.4

Multiply $5$ by $- 1$ .

$z = \frac{- 39}{- 5}$

$x = 9 - z$

$y = \frac{6}{5}$

Step 5.2.3.5

Dividing two negative values results in a positive value.

$z = \frac{39}{5}$

$x = 9 - z$

$y = \frac{6}{5}$

$z = \frac{39}{5}$

$x = 9 - z$

$y = \frac{6}{5}$

$z = \frac{39}{5}$

$x = 9 - z$

$y = \frac{6}{5}$

$z = \frac{39}{5}$

$x = 9 - z$

$y = \frac{6}{5}$

Step 6

Replace all occurrences of $z$ with $\frac{39}{5}$ in each equation.

Tap for more steps...

Step 6.1

Replace all occurrences of $z$ in $x = 9 - z$ with $\frac{39}{5}$ .

$x = 9 - (\frac{39}{5})$

$z = \frac{39}{5}$

$y = \frac{6}{5}$

Step 6.2

Simplify the right side.

Tap for more steps...

Step 6.2.1

Simplify $9 - (\frac{39}{5})$ .

Tap for more steps...

Step 6.2.1.1

To write $9$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$x = 9 \cdot \frac{5}{5} - \frac{39}{5}$

$z = \frac{39}{5}$

$y = \frac{6}{5}$

Step 6.2.1.2

Combine $9$ and $\frac{5}{5}$ .

$x = \frac{9 \cdot 5}{5} - \frac{39}{5}$

$z = \frac{39}{5}$

$y = \frac{6}{5}$

Step 6.2.1.3

Combine the numerators over the common denominator.

$x = \frac{9 \cdot 5 - 39}{5}$

$z = \frac{39}{5}$

$y = \frac{6}{5}$

Step 6.2.1.4

Simplify the numerator.

Tap for more steps...

Step 6.2.1.4.1

Multiply $9$ by $5$ .

$x = \frac{45 - 39}{5}$

$z = \frac{39}{5}$

$y = \frac{6}{5}$

Step 6.2.1.4.2

Subtract $39$ from $45$ .

$x = \frac{6}{5}$

$z = \frac{39}{5}$

$y = \frac{6}{5}$

$x = \frac{6}{5}$

$z = \frac{39}{5}$

$y = \frac{6}{5}$

$x = \frac{6}{5}$

$z = \frac{39}{5}$

$y = \frac{6}{5}$

$x = \frac{6}{5}$

$z = \frac{39}{5}$

$y = \frac{6}{5}$

$x = \frac{6}{5}$

$z = \frac{39}{5}$

$y = \frac{6}{5}$

Step 7

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

$(\frac{6}{5}, \frac{6}{5}, \frac{39}{5})$

Step 8

The result can be shown in multiple forms.

Point Form:

$(\frac{6}{5}, \frac{6}{5}, \frac{39}{5})$

Equation Form:

$x = \frac{6}{5}, y = \frac{6}{5}, z = \frac{39}{5}$