Solve the System of Equations 2x+y+3z=20 x+2y-z=-11 3x-2z=-3

$2 x + y + 3 z = 20$ $x + 2 y - z = - 11$ $3 x - 2 z = - 3$

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

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

$y + 3 z = 20 - 2 x$

$x + 2 y - z = - 11$

$3 x - 2 z = - 3$

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

$y = 20 - 2 x - 3 z$

$x + 2 y - z = - 11$

$3 x - 2 z = - 3$

$y = 20 - 2 x - 3 z$

$x + 2 y - z = - 11$

$3 x - 2 z = - 3$

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

Replace all occurrences of $y$ in $x + 2 y - z = - 11$ with $20 - 2 x - 3 z$ .

$x + 2 (20 - 2 x - 3 z) - z = - 11$

$y = 20 - 2 x - 3 z$

$3 x - 2 z = - 3$

Simplify the left side.

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

Simplify each term.

Apply the distributive property.

$x + 2 \cdot 20 + 2 (- 2 x) + 2 (- 3 z) - z = - 11$

$y = 20 - 2 x - 3 z$

$3 x - 2 z = - 3$

Simplify.

Multiply $2$ by $20$ .

$x + 40 + 2 (- 2 x) + 2 (- 3 z) - z = - 11$

$y = 20 - 2 x - 3 z$

$3 x - 2 z = - 3$

Multiply $- 2$ by $2$ .

$x + 40 - 4 x + 2 (- 3 z) - z = - 11$

$y = 20 - 2 x - 3 z$

$3 x - 2 z = - 3$

Multiply $- 3$ by $2$ .

$x + 40 - 4 x - 6 z - z = - 11$

$y = 20 - 2 x - 3 z$

$3 x - 2 z = - 3$

$x + 40 - 4 x - 6 z - z = - 11$

$y = 20 - 2 x - 3 z$

$3 x - 2 z = - 3$

$x + 40 - 4 x - 6 z - z = - 11$

$y = 20 - 2 x - 3 z$

$3 x - 2 z = - 3$

Simplify by adding terms.

Subtract $4 x$ from $x$ .

$- 3 x + 40 - 6 z - z = - 11$

$y = 20 - 2 x - 3 z$

$3 x - 2 z = - 3$

Subtract $z$ from $- 6 z$ .

$- 3 x + 40 - 7 z = - 11$

$y = 20 - 2 x - 3 z$

$3 x - 2 z = - 3$

$- 3 x + 40 - 7 z = - 11$

$y = 20 - 2 x - 3 z$

$3 x - 2 z = - 3$

$- 3 x + 40 - 7 z = - 11$

$y = 20 - 2 x - 3 z$

$3 x - 2 z = - 3$

$- 3 x + 40 - 7 z = - 11$

$y = 20 - 2 x - 3 z$

$3 x - 2 z = - 3$

$- 3 x + 40 - 7 z = - 11$

$y = 20 - 2 x - 3 z$

$3 x - 2 z = - 3$

Solve for $x$ in $- 3 x + 40 - 7 z = - 11$ .

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

Subtract $40$ from both sides of the equation.

$- 3 x - 7 z = - 11 - 40$

$y = 20 - 2 x - 3 z$

$3 x - 2 z = - 3$

Add $7 z$ to both sides of the equation.

$- 3 x = - 11 - 40 + 7 z$

$y = 20 - 2 x - 3 z$

$3 x - 2 z = - 3$

Subtract $40$ from $- 11$ .

$- 3 x = - 51 + 7 z$

$y = 20 - 2 x - 3 z$

$3 x - 2 z = - 3$

$- 3 x = - 51 + 7 z$

$y = 20 - 2 x - 3 z$

$3 x - 2 z = - 3$

Divide each term in $- 3 x = - 51 + 7 z$ by $- 3$ and simplify.

Divide each term in $- 3 x = - 51 + 7 z$ by $- 3$ .

$\frac{- 3 x}{- 3} = \frac{- 51}{- 3} + \frac{7 z}{- 3}$

$y = 20 - 2 x - 3 z$

$3 x - 2 z = - 3$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 x}{- 3} = \frac{- 51}{- 3} + \frac{7 z}{- 3}$

$y = 20 - 2 x - 3 z$

$3 x - 2 z = - 3$

Divide $x$ by $1$ .

$x = \frac{- 51}{- 3} + \frac{7 z}{- 3}$

$y = 20 - 2 x - 3 z$

$3 x - 2 z = - 3$

$x = \frac{- 51}{- 3} + \frac{7 z}{- 3}$

$y = 20 - 2 x - 3 z$

$3 x - 2 z = - 3$

$x = \frac{- 51}{- 3} + \frac{7 z}{- 3}$

$y = 20 - 2 x - 3 z$

$3 x - 2 z = - 3$

Simplify the right side.

Simplify each term.

Divide $- 51$ by $- 3$ .

$x = 17 + \frac{7 z}{- 3}$

$y = 20 - 2 x - 3 z$

$3 x - 2 z = - 3$

Move the negative in front of the fraction.

$x = 17 - \frac{7 z}{3}$

$y = 20 - 2 x - 3 z$

$3 x - 2 z = - 3$

$x = 17 - \frac{7 z}{3}$

$y = 20 - 2 x - 3 z$

$3 x - 2 z = - 3$

$x = 17 - \frac{7 z}{3}$

$y = 20 - 2 x - 3 z$

$3 x - 2 z = - 3$

$x = 17 - \frac{7 z}{3}$

$y = 20 - 2 x - 3 z$

$3 x - 2 z = - 3$

$x = 17 - \frac{7 z}{3}$

$y = 20 - 2 x - 3 z$

$3 x - 2 z = - 3$

Replace all occurrences of $x$ with $17 - \frac{7 z}{3}$ in each equation.

Replace all occurrences of $x$ in $y = 20 - 2 x - 3 z$ with $17 - \frac{7 z}{3}$ .

$y = 20 - 2 (17 - \frac{7 z}{3}) - 3 z$

$x = 17 - \frac{7 z}{3}$

$3 x - 2 z = - 3$

Simplify the right side.

Simplify $20 - 2 (17 - \frac{7 z}{3}) - 3 z$ .

Simplify each term.

Apply the distributive property.

$y = 20 - 2 \cdot 17 - 2 (- \frac{7 z}{3}) - 3 z$

$x = 17 - \frac{7 z}{3}$

$3 x - 2 z = - 3$

Multiply $- 2$ by $17$ .

$y = 20 - 34 - 2 (- \frac{7 z}{3}) - 3 z$

$x = 17 - \frac{7 z}{3}$

$3 x - 2 z = - 3$

Multiply $- 2 (- \frac{7 z}{3})$ .

Multiply $- 1$ by $- 2$ .

$y = 20 - 34 + 2 (\frac{7 z}{3}) - 3 z$

$x = 17 - \frac{7 z}{3}$

$3 x - 2 z = - 3$

Combine $2$ and $\frac{7 z}{3}$ .

$y = 20 - 34 + \frac{2 (7 z)}{3} - 3 z$

$x = 17 - \frac{7 z}{3}$

$3 x - 2 z = - 3$

Multiply $7$ by $2$ .

$y = 20 - 34 + \frac{14 z}{3} - 3 z$

$x = 17 - \frac{7 z}{3}$

$3 x - 2 z = - 3$

$y = 20 - 34 + \frac{14 z}{3} - 3 z$

$x = 17 - \frac{7 z}{3}$

$3 x - 2 z = - 3$

$y = 20 - 34 + \frac{14 z}{3} - 3 z$

$x = 17 - \frac{7 z}{3}$

$3 x - 2 z = - 3$

Subtract $34$ from $20$ .

$y = - 14 + \frac{14 z}{3} - 3 z$

$x = 17 - \frac{7 z}{3}$

$3 x - 2 z = - 3$

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

$y = - 14 + \frac{14 z}{3} - 3 z \cdot \frac{3}{3}$

$x = 17 - \frac{7 z}{3}$

$3 x - 2 z = - 3$

Simplify terms.

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

$y = - 14 + \frac{14 z}{3} + \frac{- 3 z \cdot 3}{3}$

$x = 17 - \frac{7 z}{3}$

$3 x - 2 z = - 3$

Combine the numerators over the common denominator.

$y = - 14 + \frac{14 z - 3 z \cdot 3}{3}$

$x = 17 - \frac{7 z}{3}$

$3 x - 2 z = - 3$

$y = - 14 + \frac{14 z - 3 z \cdot 3}{3}$

$x = 17 - \frac{7 z}{3}$

$3 x - 2 z = - 3$

Simplify each term.

Simplify the numerator.

Factor $z$ out of $14 z - 3 z \cdot 3$ .

Factor $z$ out of $14 z$ .

$y = - 14 + \frac{z \cdot 14 - 3 z \cdot 3}{3}$

$x = 17 - \frac{7 z}{3}$

$3 x - 2 z = - 3$

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

$y = - 14 + \frac{z \cdot 14 + z (- 3 \cdot 3)}{3}$

$x = 17 - \frac{7 z}{3}$

$3 x - 2 z = - 3$

Factor $z$ out of $z \cdot 14 + z (- 3 \cdot 3)$ .

$y = - 14 + \frac{z (14 - 3 \cdot 3)}{3}$

$x = 17 - \frac{7 z}{3}$

$3 x - 2 z = - 3$

$y = - 14 + \frac{z (14 - 3 \cdot 3)}{3}$

$x = 17 - \frac{7 z}{3}$

$3 x - 2 z = - 3$

Multiply $- 3$ by $3$ .

$y = - 14 + \frac{z (14 - 9)}{3}$

$x = 17 - \frac{7 z}{3}$

$3 x - 2 z = - 3$

Subtract $9$ from $14$ .

$y = - 14 + \frac{z \cdot 5}{3}$

$x = 17 - \frac{7 z}{3}$

$3 x - 2 z = - 3$

$y = - 14 + \frac{z \cdot 5}{3}$

$x = 17 - \frac{7 z}{3}$

$3 x - 2 z = - 3$

Move $5$ to the left of $z$ .

$y = - 14 + \frac{5 z}{3}$

$x = 17 - \frac{7 z}{3}$

$3 x - 2 z = - 3$

$y = - 14 + \frac{5 z}{3}$

$x = 17 - \frac{7 z}{3}$

$3 x - 2 z = - 3$

$y = - 14 + \frac{5 z}{3}$

$x = 17 - \frac{7 z}{3}$

$3 x - 2 z = - 3$

$y = - 14 + \frac{5 z}{3}$

$x = 17 - \frac{7 z}{3}$

$3 x - 2 z = - 3$

Replace all occurrences of $x$ in $3 x - 2 z = - 3$ with $17 - \frac{7 z}{3}$ .

$3 (17 - \frac{7 z}{3}) - 2 z = - 3$

$y = - 14 + \frac{5 z}{3}$

$x = 17 - \frac{7 z}{3}$

Simplify the left side.

Simplify $3 (17 - \frac{7 z}{3}) - 2 z$ .

Simplify each term.

Apply the distributive property.

$3 \cdot 17 + 3 (- \frac{7 z}{3}) - 2 z = - 3$

$y = - 14 + \frac{5 z}{3}$

$x = 17 - \frac{7 z}{3}$

Multiply $3$ by $17$ .

$51 + 3 (- \frac{7 z}{3}) - 2 z = - 3$

$y = - 14 + \frac{5 z}{3}$

$x = 17 - \frac{7 z}{3}$

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{7 z}{3}$ into the numerator.

$51 + 3 (\frac{- 7 z}{3}) - 2 z = - 3$

$y = - 14 + \frac{5 z}{3}$

$x = 17 - \frac{7 z}{3}$

Cancel the common factor.

$51 + 3 (\frac{- 7 z}{3}) - 2 z = - 3$

$y = - 14 + \frac{5 z}{3}$

$x = 17 - \frac{7 z}{3}$

Rewrite the expression.

$51 - 7 z - 2 z = - 3$

$y = - 14 + \frac{5 z}{3}$

$x = 17 - \frac{7 z}{3}$

$51 - 7 z - 2 z = - 3$

$y = - 14 + \frac{5 z}{3}$

$x = 17 - \frac{7 z}{3}$

$51 - 7 z - 2 z = - 3$

$y = - 14 + \frac{5 z}{3}$

$x = 17 - \frac{7 z}{3}$

Subtract $2 z$ from $- 7 z$ .

$51 - 9 z = - 3$

$y = - 14 + \frac{5 z}{3}$

$x = 17 - \frac{7 z}{3}$

$51 - 9 z = - 3$

$y = - 14 + \frac{5 z}{3}$

$x = 17 - \frac{7 z}{3}$

$51 - 9 z = - 3$

$y = - 14 + \frac{5 z}{3}$

$x = 17 - \frac{7 z}{3}$

$51 - 9 z = - 3$

$y = - 14 + \frac{5 z}{3}$

$x = 17 - \frac{7 z}{3}$

Solve for $z$ in $51 - 9 z = - 3$ .

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

Subtract $51$ from both sides of the equation.

$- 9 z = - 3 - 51$

$y = - 14 + \frac{5 z}{3}$

$x = 17 - \frac{7 z}{3}$

Subtract $51$ from $- 3$ .

$- 9 z = - 54$

$y = - 14 + \frac{5 z}{3}$

$x = 17 - \frac{7 z}{3}$

$- 9 z = - 54$

$y = - 14 + \frac{5 z}{3}$

$x = 17 - \frac{7 z}{3}$

Divide each term in $- 9 z = - 54$ by $- 9$ and simplify.

Divide each term in $- 9 z = - 54$ by $- 9$ .

$\frac{- 9 z}{- 9} = \frac{- 54}{- 9}$

$y = - 14 + \frac{5 z}{3}$

$x = 17 - \frac{7 z}{3}$

Simplify the left side.

Cancel the common factor of $- 9$ .

Cancel the common factor.

$\frac{- 9 z}{- 9} = \frac{- 54}{- 9}$

$y = - 14 + \frac{5 z}{3}$

$x = 17 - \frac{7 z}{3}$

Divide $z$ by $1$ .

$z = \frac{- 54}{- 9}$

$y = - 14 + \frac{5 z}{3}$

$x = 17 - \frac{7 z}{3}$

$z = \frac{- 54}{- 9}$

$y = - 14 + \frac{5 z}{3}$

$x = 17 - \frac{7 z}{3}$

$z = \frac{- 54}{- 9}$

$y = - 14 + \frac{5 z}{3}$

$x = 17 - \frac{7 z}{3}$

Simplify the right side.

Divide $- 54$ by $- 9$ .

$z = 6$

$y = - 14 + \frac{5 z}{3}$

$x = 17 - \frac{7 z}{3}$

$z = 6$

$y = - 14 + \frac{5 z}{3}$

$x = 17 - \frac{7 z}{3}$

$z = 6$

$y = - 14 + \frac{5 z}{3}$

$x = 17 - \frac{7 z}{3}$

$z = 6$

$y = - 14 + \frac{5 z}{3}$

$x = 17 - \frac{7 z}{3}$

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

Replace all occurrences of $z$ in $y = - 14 + \frac{5 z}{3}$ with $6$ .

$y = - 14 + \frac{5 (6)}{3}$

$z = 6$

$x = 17 - \frac{7 z}{3}$

Simplify the right side.

Simplify $- 14 + \frac{5 (6)}{3}$ .

Multiply $5$ by $6$ .

$y = - 14 + \frac{30}{3}$

$z = 6$

$x = 17 - \frac{7 z}{3}$

Divide $30$ by $3$ .

$y = - 14 + 10$

$z = 6$

$x = 17 - \frac{7 z}{3}$

Add $- 14$ and $10$ .

$y = - 4$

$z = 6$

$x = 17 - \frac{7 z}{3}$

$y = - 4$

$z = 6$

$x = 17 - \frac{7 z}{3}$

$y = - 4$

$z = 6$

$x = 17 - \frac{7 z}{3}$

Replace all occurrences of $z$ in $x = 17 - \frac{7 z}{3}$ with $6$ .

$x = 17 - \frac{7 (6)}{3}$

$y = - 4$

$z = 6$

Simplify the right side.

Simplify $17 - \frac{7 (6)}{3}$ .

Simplify each term.

Cancel the common factor of $6$ and $3$ .

Factor $3$ out of $7 (6)$ .

$x = 17 - \frac{3 (7 \cdot (2))}{3}$

$y = - 4$

$z = 6$

Cancel the common factors.

Factor $3$ out of $3$ .

$x = 17 - \frac{3 (7 \cdot (2))}{3 (1)}$

$y = - 4$

$z = 6$

Cancel the common factor.

$x = 17 - \frac{3 (7 \cdot (2))}{3 \cdot 1}$

$y = - 4$

$z = 6$

Rewrite the expression.

$x = 17 - \frac{7 \cdot (2)}{1}$

$y = - 4$

$z = 6$

Divide $7 \cdot (2)$ by $1$ .

$x = 17 - (7 \cdot (2))$

$y = - 4$

$z = 6$

$x = 17 - (7 \cdot (2))$

$y = - 4$

$z = 6$

$x = 17 - (7 \cdot (2))$

$y = - 4$

$z = 6$

Multiply $7$ by $2$ .

$x = 17 - 1 \cdot 14$

$y = - 4$

$z = 6$

Multiply $- 1$ by $14$ .

$x = 17 - 14$

$y = - 4$

$z = 6$

$x = 17 - 14$

$y = - 4$

$z = 6$

Subtract $14$ from $17$ .

$x = 3$

$y = - 4$

$z = 6$

$x = 3$

$y = - 4$

$z = 6$

$x = 3$

$y = - 4$

$z = 6$

$x = 3$

$y = - 4$

$z = 6$

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

$(3, - 4, 6)$

The result can be shown in multiple forms.

Point Form:

$(3, - 4, 6)$

Equation Form:

$x = 3, y = - 4, z = 6$

Solve the System of Equations 2x+y+3z=20 x+2y-z=-11 3x-2z=-3

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