Solve the System of Equations 3x+2y+z=20 x-4y-z=-10 2x+y+2z=15

$3 x + 2 y + z = 20$ $x - 4 y - z = - 10$ $2 x + y + 2 z = 15$

Step 1

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

Subtract $3 x$ from both sides of the equation.

$2 y + z = 20 - 3 x$

$x - 4 y - z = - 10$

$2 x + y + 2 z = 15$

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

$z = 20 - 3 x - 2 y$

$x - 4 y - z = - 10$

$2 x + y + 2 z = 15$

$z = 20 - 3 x - 2 y$

$x - 4 y - z = - 10$

$2 x + y + 2 z = 15$

Step 2

Replace all occurrences of $z$ with $20 - 3 x - 2 y$ in each equation.

Replace all occurrences of $z$ in $x - 4 y - z = - 10$ with $20 - 3 x - 2 y$ .

$x - 4 y - (20 - 3 x - 2 y) = - 10$

$z = 20 - 3 x - 2 y$

$2 x + y + 2 z = 15$

Simplify the left side.

Simplify $x - 4 y - (20 - 3 x - 2 y)$ .

Simplify each term.

Apply the distributive property.

$x - 4 y - 1 \cdot 20 - (- 3 x) - (- 2 y) = - 10$

$z = 20 - 3 x - 2 y$

$2 x + y + 2 z = 15$

Simplify.

Multiply $- 1$ by $20$ .

$x - 4 y - 20 - (- 3 x) - (- 2 y) = - 10$

$z = 20 - 3 x - 2 y$

$2 x + y + 2 z = 15$

Multiply $- 3$ by $- 1$ .

$x - 4 y - 20 + 3 x - (- 2 y) = - 10$

$z = 20 - 3 x - 2 y$

$2 x + y + 2 z = 15$

Multiply $- 2$ by $- 1$ .

$x - 4 y - 20 + 3 x + 2 y = - 10$

$z = 20 - 3 x - 2 y$

$2 x + y + 2 z = 15$

$x - 4 y - 20 + 3 x + 2 y = - 10$

$z = 20 - 3 x - 2 y$

$2 x + y + 2 z = 15$

$x - 4 y - 20 + 3 x + 2 y = - 10$

$z = 20 - 3 x - 2 y$

$2 x + y + 2 z = 15$

Simplify by adding terms.

Add $x$ and $3 x$ .

$4 x - 4 y - 20 + 2 y = - 10$

$z = 20 - 3 x - 2 y$

$2 x + y + 2 z = 15$

Add $- 4 y$ and $2 y$ .

$4 x - 2 y - 20 = - 10$

$z = 20 - 3 x - 2 y$

$2 x + y + 2 z = 15$

$4 x - 2 y - 20 = - 10$

$z = 20 - 3 x - 2 y$

$2 x + y + 2 z = 15$

$4 x - 2 y - 20 = - 10$

$z = 20 - 3 x - 2 y$

$2 x + y + 2 z = 15$

$4 x - 2 y - 20 = - 10$

$z = 20 - 3 x - 2 y$

$2 x + y + 2 z = 15$

Replace all occurrences of $z$ in $2 x + y + 2 z = 15$ with $20 - 3 x - 2 y$ .

$2 x + y + 2 (20 - 3 x - 2 y) = 15$

$4 x - 2 y - 20 = - 10$

$z = 20 - 3 x - 2 y$

Simplify the left side.

Simplify $2 x + y + 2 (20 - 3 x - 2 y)$ .

Simplify each term.

Apply the distributive property.

$2 x + y + 2 \cdot 20 + 2 (- 3 x) + 2 (- 2 y) = 15$

$4 x - 2 y - 20 = - 10$

$z = 20 - 3 x - 2 y$

Simplify.

Multiply $2$ by $20$ .

$2 x + y + 40 + 2 (- 3 x) + 2 (- 2 y) = 15$

$4 x - 2 y - 20 = - 10$

$z = 20 - 3 x - 2 y$

Multiply $- 3$ by $2$ .

$2 x + y + 40 - 6 x + 2 (- 2 y) = 15$

$4 x - 2 y - 20 = - 10$

$z = 20 - 3 x - 2 y$

Multiply $- 2$ by $2$ .

$2 x + y + 40 - 6 x - 4 y = 15$

$4 x - 2 y - 20 = - 10$

$z = 20 - 3 x - 2 y$

$2 x + y + 40 - 6 x - 4 y = 15$

$4 x - 2 y - 20 = - 10$

$z = 20 - 3 x - 2 y$

$2 x + y + 40 - 6 x - 4 y = 15$

$4 x - 2 y - 20 = - 10$

$z = 20 - 3 x - 2 y$

Simplify by adding terms.

Subtract $6 x$ from $2 x$ .

$- 4 x + y + 40 - 4 y = 15$

$4 x - 2 y - 20 = - 10$

$z = 20 - 3 x - 2 y$

Subtract $4 y$ from $y$ .

$- 4 x - 3 y + 40 = 15$

$4 x - 2 y - 20 = - 10$

$z = 20 - 3 x - 2 y$

$- 4 x - 3 y + 40 = 15$

$4 x - 2 y - 20 = - 10$

$z = 20 - 3 x - 2 y$

$- 4 x - 3 y + 40 = 15$

$4 x - 2 y - 20 = - 10$

$z = 20 - 3 x - 2 y$

$- 4 x - 3 y + 40 = 15$

$4 x - 2 y - 20 = - 10$

$z = 20 - 3 x - 2 y$

$- 4 x - 3 y + 40 = 15$

$4 x - 2 y - 20 = - 10$

$z = 20 - 3 x - 2 y$

Step 3

Solve for $x$ in $- 4 x - 3 y + 40 = 15$ .

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

Add $3 y$ to both sides of the equation.

$- 4 x + 40 = 15 + 3 y$

$4 x - 2 y - 20 = - 10$

$z = 20 - 3 x - 2 y$

Subtract $40$ from both sides of the equation.

$- 4 x = 15 + 3 y - 40$

$4 x - 2 y - 20 = - 10$

$z = 20 - 3 x - 2 y$

Subtract $40$ from $15$ .

$- 4 x = 3 y - 25$

$4 x - 2 y - 20 = - 10$

$z = 20 - 3 x - 2 y$

$- 4 x = 3 y - 25$

$4 x - 2 y - 20 = - 10$

$z = 20 - 3 x - 2 y$

Divide each term in $- 4 x = 3 y - 25$ by $- 4$ and simplify.

Divide each term in $- 4 x = 3 y - 25$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{3 y}{- 4} + \frac{- 25}{- 4}$

$4 x - 2 y - 20 = - 10$

$z = 20 - 3 x - 2 y$

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{3 y}{- 4} + \frac{- 25}{- 4}$

$4 x - 2 y - 20 = - 10$

$z = 20 - 3 x - 2 y$

Divide $x$ by $1$ .

$x = \frac{3 y}{- 4} + \frac{- 25}{- 4}$

$4 x - 2 y - 20 = - 10$

$z = 20 - 3 x - 2 y$

$x = \frac{3 y}{- 4} + \frac{- 25}{- 4}$

$4 x - 2 y - 20 = - 10$

$z = 20 - 3 x - 2 y$

$x = \frac{3 y}{- 4} + \frac{- 25}{- 4}$

$4 x - 2 y - 20 = - 10$

$z = 20 - 3 x - 2 y$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$x = - \frac{3 y}{4} + \frac{- 25}{- 4}$

$4 x - 2 y - 20 = - 10$

$z = 20 - 3 x - 2 y$

Dividing two negative values results in a positive value.

$x = - \frac{3 y}{4} + \frac{25}{4}$

$4 x - 2 y - 20 = - 10$

$z = 20 - 3 x - 2 y$

$x = - \frac{3 y}{4} + \frac{25}{4}$

$4 x - 2 y - 20 = - 10$

$z = 20 - 3 x - 2 y$

$x = - \frac{3 y}{4} + \frac{25}{4}$

$4 x - 2 y - 20 = - 10$

$z = 20 - 3 x - 2 y$

$x = - \frac{3 y}{4} + \frac{25}{4}$

$4 x - 2 y - 20 = - 10$

$z = 20 - 3 x - 2 y$

$x = - \frac{3 y}{4} + \frac{25}{4}$

$4 x - 2 y - 20 = - 10$

$z = 20 - 3 x - 2 y$

Step 4

Replace all occurrences of $x$ with $- \frac{3 y}{4} + \frac{25}{4}$ in each equation.

Replace all occurrences of $x$ in $4 x - 2 y - 20 = - 10$ with $- \frac{3 y}{4} + \frac{25}{4}$ .

$4 (- \frac{3 y}{4} + \frac{25}{4}) - 2 y - 20 = - 10$

$x = - \frac{3 y}{4} + \frac{25}{4}$

$z = 20 - 3 x - 2 y$

Simplify the left side.

Simplify $4 (- \frac{3 y}{4} + \frac{25}{4}) - 2 y - 20$ .

Simplify each term.

Apply the distributive property.

$4 (- \frac{3 y}{4}) + 4 (\frac{25}{4}) - 2 y - 20 = - 10$

$x = - \frac{3 y}{4} + \frac{25}{4}$

$z = 20 - 3 x - 2 y$

Cancel the common factor of $4$ .

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

$4 (\frac{- 3 y}{4}) + 4 (\frac{25}{4}) - 2 y - 20 = - 10$

$x = - \frac{3 y}{4} + \frac{25}{4}$

$z = 20 - 3 x - 2 y$

Cancel the common factor.

$4 (\frac{- 3 y}{4}) + 4 (\frac{25}{4}) - 2 y - 20 = - 10$

$x = - \frac{3 y}{4} + \frac{25}{4}$

$z = 20 - 3 x - 2 y$

Rewrite the expression.

$- 3 y + 4 (\frac{25}{4}) - 2 y - 20 = - 10$

$x = - \frac{3 y}{4} + \frac{25}{4}$

$z = 20 - 3 x - 2 y$

$- 3 y + 4 (\frac{25}{4}) - 2 y - 20 = - 10$

$x = - \frac{3 y}{4} + \frac{25}{4}$

$z = 20 - 3 x - 2 y$

Cancel the common factor of $4$ .

Cancel the common factor.

$- 3 y + 4 (\frac{25}{4}) - 2 y - 20 = - 10$

$x = - \frac{3 y}{4} + \frac{25}{4}$

$z = 20 - 3 x - 2 y$

Rewrite the expression.

$- 3 y + 25 - 2 y - 20 = - 10$

$x = - \frac{3 y}{4} + \frac{25}{4}$

$z = 20 - 3 x - 2 y$

$- 3 y + 25 - 2 y - 20 = - 10$

$x = - \frac{3 y}{4} + \frac{25}{4}$

$z = 20 - 3 x - 2 y$

$- 3 y + 25 - 2 y - 20 = - 10$

$x = - \frac{3 y}{4} + \frac{25}{4}$

$z = 20 - 3 x - 2 y$

Simplify by adding terms.

Subtract $2 y$ from $- 3 y$ .

$- 5 y + 25 - 20 = - 10$

$x = - \frac{3 y}{4} + \frac{25}{4}$

$z = 20 - 3 x - 2 y$

Subtract $20$ from $25$ .

$- 5 y + 5 = - 10$

$x = - \frac{3 y}{4} + \frac{25}{4}$

$z = 20 - 3 x - 2 y$

$- 5 y + 5 = - 10$

$x = - \frac{3 y}{4} + \frac{25}{4}$

$z = 20 - 3 x - 2 y$

$- 5 y + 5 = - 10$

$x = - \frac{3 y}{4} + \frac{25}{4}$

$z = 20 - 3 x - 2 y$

$- 5 y + 5 = - 10$

$x = - \frac{3 y}{4} + \frac{25}{4}$

$z = 20 - 3 x - 2 y$

Replace all occurrences of $x$ in $z = 20 - 3 x - 2 y$ with $- \frac{3 y}{4} + \frac{25}{4}$ .

$z = 20 - 3 (- \frac{3 y}{4} + \frac{25}{4}) - 2 y$

$- 5 y + 5 = - 10$

$x = - \frac{3 y}{4} + \frac{25}{4}$

Simplify the right side.

Simplify $20 - 3 (- \frac{3 y}{4} + \frac{25}{4}) - 2 y$ .

Simplify each term.

Apply the distributive property.

$z = 20 - 3 (- \frac{3 y}{4}) - 3 (\frac{25}{4}) - 2 y$

$- 5 y + 5 = - 10$

$x = - \frac{3 y}{4} + \frac{25}{4}$

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

Multiply $- 1$ by $- 3$ .

$z = 20 + 3 (\frac{3 y}{4}) - 3 (\frac{25}{4}) - 2 y$

$- 5 y + 5 = - 10$

$x = - \frac{3 y}{4} + \frac{25}{4}$

Combine $3$ and $\frac{3 y}{4}$ .

$z = 20 + \frac{3 (3 y)}{4} - 3 (\frac{25}{4}) - 2 y$

$- 5 y + 5 = - 10$

$x = - \frac{3 y}{4} + \frac{25}{4}$

Multiply $3$ by $3$ .

$z = 20 + \frac{9 y}{4} - 3 (\frac{25}{4}) - 2 y$

$- 5 y + 5 = - 10$

$x = - \frac{3 y}{4} + \frac{25}{4}$

$z = 20 + \frac{9 y}{4} - 3 (\frac{25}{4}) - 2 y$

$- 5 y + 5 = - 10$

$x = - \frac{3 y}{4} + \frac{25}{4}$

Multiply $- 3 (\frac{25}{4})$ .

Combine $- 3$ and $\frac{25}{4}$ .

$z = 20 + \frac{9 y}{4} + \frac{- 3 \cdot 25}{4} - 2 y$

$- 5 y + 5 = - 10$

$x = - \frac{3 y}{4} + \frac{25}{4}$

Multiply $- 3$ by $25$ .

$z = 20 + \frac{9 y}{4} + \frac{- 75}{4} - 2 y$

$- 5 y + 5 = - 10$

$x = - \frac{3 y}{4} + \frac{25}{4}$

$z = 20 + \frac{9 y}{4} + \frac{- 75}{4} - 2 y$

$- 5 y + 5 = - 10$

$x = - \frac{3 y}{4} + \frac{25}{4}$

Move the negative in front of the fraction.

$z = 20 + \frac{9 y}{4} - \frac{75}{4} - 2 y$

$- 5 y + 5 = - 10$

$x = - \frac{3 y}{4} + \frac{25}{4}$

$z = 20 + \frac{9 y}{4} - \frac{75}{4} - 2 y$

$- 5 y + 5 = - 10$

$x = - \frac{3 y}{4} + \frac{25}{4}$

To write $20$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$z = \frac{9 y}{4} + 20 \cdot \frac{4}{4} - \frac{75}{4} - 2 y$

$- 5 y + 5 = - 10$

$x = - \frac{3 y}{4} + \frac{25}{4}$

Combine $20$ and $\frac{4}{4}$ .

$z = \frac{9 y}{4} + \frac{20 \cdot 4}{4} - \frac{75}{4} - 2 y$

$- 5 y + 5 = - 10$

$x = - \frac{3 y}{4} + \frac{25}{4}$

Combine the numerators over the common denominator.

$z = \frac{9 y}{4} + \frac{20 \cdot 4 - 75}{4} - 2 y$

$- 5 y + 5 = - 10$

$x = - \frac{3 y}{4} + \frac{25}{4}$

Simplify the numerator.

Multiply $20$ by $4$ .

$z = \frac{9 y}{4} + \frac{80 - 75}{4} - 2 y$

$- 5 y + 5 = - 10$

$x = - \frac{3 y}{4} + \frac{25}{4}$

Subtract $75$ from $80$ .

$z = \frac{9 y}{4} + \frac{5}{4} - 2 y$

$- 5 y + 5 = - 10$

$x = - \frac{3 y}{4} + \frac{25}{4}$

$z = \frac{9 y}{4} + \frac{5}{4} - 2 y$

$- 5 y + 5 = - 10$

$x = - \frac{3 y}{4} + \frac{25}{4}$

To write $- 2 y$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$z = \frac{9 y}{4} - 2 y \cdot \frac{4}{4} + \frac{5}{4}$

$- 5 y + 5 = - 10$

$x = - \frac{3 y}{4} + \frac{25}{4}$

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

$z = \frac{9 y}{4} + \frac{- 2 y \cdot 4}{4} + \frac{5}{4}$

$- 5 y + 5 = - 10$

$x = - \frac{3 y}{4} + \frac{25}{4}$

Combine the numerators over the common denominator.

$z = \frac{9 y - 2 y \cdot 4}{4} + \frac{5}{4}$

$- 5 y + 5 = - 10$

$x = - \frac{3 y}{4} + \frac{25}{4}$

Combine the numerators over the common denominator.

$z = \frac{9 y - 2 y \cdot 4 + 5}{4}$

$- 5 y + 5 = - 10$

$x = - \frac{3 y}{4} + \frac{25}{4}$

Multiply $4$ by $- 2$ .

$z = \frac{9 y - 8 y + 5}{4}$

$- 5 y + 5 = - 10$

$x = - \frac{3 y}{4} + \frac{25}{4}$

Subtract $8 y$ from $9 y$ .

$z = \frac{y + 5}{4}$

$- 5 y + 5 = - 10$

$x = - \frac{3 y}{4} + \frac{25}{4}$

$z = \frac{y + 5}{4}$

$- 5 y + 5 = - 10$

$x = - \frac{3 y}{4} + \frac{25}{4}$

$z = \frac{y + 5}{4}$

$- 5 y + 5 = - 10$

$x = - \frac{3 y}{4} + \frac{25}{4}$

$z = \frac{y + 5}{4}$

$- 5 y + 5 = - 10$

$x = - \frac{3 y}{4} + \frac{25}{4}$

Step 5

Solve for $y$ in $- 5 y + 5 = - 10$ .

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

Subtract $5$ from both sides of the equation.

$- 5 y = - 10 - 5$

$z = \frac{y + 5}{4}$

$x = - \frac{3 y}{4} + \frac{25}{4}$

Subtract $5$ from $- 10$ .

$- 5 y = - 15$

$z = \frac{y + 5}{4}$

$x = - \frac{3 y}{4} + \frac{25}{4}$

$- 5 y = - 15$

$z = \frac{y + 5}{4}$

$x = - \frac{3 y}{4} + \frac{25}{4}$

Divide each term in $- 5 y = - 15$ by $- 5$ and simplify.

Divide each term in $- 5 y = - 15$ by $- 5$ .

$\frac{- 5 y}{- 5} = \frac{- 15}{- 5}$

$z = \frac{y + 5}{4}$

$x = - \frac{3 y}{4} + \frac{25}{4}$

Simplify the left side.

Cancel the common factor of $- 5$ .

Cancel the common factor.

$\frac{- 5 y}{- 5} = \frac{- 15}{- 5}$

$z = \frac{y + 5}{4}$

$x = - \frac{3 y}{4} + \frac{25}{4}$

Divide $y$ by $1$ .

$y = \frac{- 15}{- 5}$

$z = \frac{y + 5}{4}$

$x = - \frac{3 y}{4} + \frac{25}{4}$

$y = \frac{- 15}{- 5}$

$z = \frac{y + 5}{4}$

$x = - \frac{3 y}{4} + \frac{25}{4}$

$y = \frac{- 15}{- 5}$

$z = \frac{y + 5}{4}$

$x = - \frac{3 y}{4} + \frac{25}{4}$

Simplify the right side.

Divide $- 15$ by $- 5$ .

$y = 3$

$z = \frac{y + 5}{4}$

$x = - \frac{3 y}{4} + \frac{25}{4}$

$y = 3$

$z = \frac{y + 5}{4}$

$x = - \frac{3 y}{4} + \frac{25}{4}$

$y = 3$

$z = \frac{y + 5}{4}$

$x = - \frac{3 y}{4} + \frac{25}{4}$

$y = 3$

$z = \frac{y + 5}{4}$

$x = - \frac{3 y}{4} + \frac{25}{4}$

Step 6

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

Replace all occurrences of $y$ in $z = \frac{y + 5}{4}$ with $3$ .

$z = \frac{(3) + 5}{4}$

$y = 3$

$x = - \frac{3 y}{4} + \frac{25}{4}$

Simplify the right side.

Simplify $\frac{(3) + 5}{4}$ .

Add $3$ and $5$ .

$z = \frac{8}{4}$

$y = 3$

$x = - \frac{3 y}{4} + \frac{25}{4}$

Divide $8$ by $4$ .

$z = 2$

$y = 3$

$x = - \frac{3 y}{4} + \frac{25}{4}$

$z = 2$

$y = 3$

$x = - \frac{3 y}{4} + \frac{25}{4}$

$z = 2$

$y = 3$

$x = - \frac{3 y}{4} + \frac{25}{4}$

Replace all occurrences of $y$ in $x = - \frac{3 y}{4} + \frac{25}{4}$ with $3$ .

$x = - \frac{3 (3)}{4} + \frac{25}{4}$

$z = 2$

$y = 3$

Simplify the right side.

Simplify $- \frac{3 (3)}{4} + \frac{25}{4}$ .

Combine the numerators over the common denominator.

$x = \frac{- 3 \cdot 3 + 25}{4}$

$z = 2$

$y = 3$

Simplify the expression.

Multiply $- 3$ by $3$ .

$x = \frac{- 9 + 25}{4}$

$z = 2$

$y = 3$

Add $- 9$ and $25$ .

$x = \frac{16}{4}$

$z = 2$

$y = 3$

Divide $16$ by $4$ .

$x = 4$

$z = 2$

$y = 3$

$x = 4$

$z = 2$

$y = 3$

$x = 4$

$z = 2$

$y = 3$

$x = 4$

$z = 2$

$y = 3$

$x = 4$

$z = 2$

$y = 3$

Step 7

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

$(4, 3, 2)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(4, 3, 2)$

Equation Form:

$x = 4, y = 3, z = 2$

Solve the System of Equations 3x+2y+z=20 x-4y-z=-10 2x+y+2z=15

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