Solve the System of Equations 2x+y-z=9 x-2y+z=-1 -x+2y+2z=-5

$2 x + y - z = 9$ $x - 2 y + z = - 1$ $- x + 2 y + 2 z = - 5$

Step 1

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

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

$y - z = 9 - 2 x$

$x - 2 y + z = - 1$

$- x + 2 y + 2 z = - 5$

Add $z$ to both sides of the equation.

$y = 9 - 2 x + z$

$x - 2 y + z = - 1$

$- x + 2 y + 2 z = - 5$

$y = 9 - 2 x + z$

$x - 2 y + z = - 1$

$- x + 2 y + 2 z = - 5$

Step 2

Replace all occurrences of $y$ with $9 - 2 x + z$ in each equation.

Replace all occurrences of $y$ in $x - 2 y + z = - 1$ with $9 - 2 x + z$ .

$x - 2 (9 - 2 x + z) + z = - 1$

$y = 9 - 2 x + z$

$- x + 2 y + 2 z = - 5$

Simplify the left side.

Simplify $x - 2 (9 - 2 x + z) + z$ .

Simplify each term.

Apply the distributive property.

$x - 2 \cdot 9 - 2 (- 2 x) - 2 z + z = - 1$

$y = 9 - 2 x + z$

$- x + 2 y + 2 z = - 5$

Simplify.

Multiply $- 2$ by $9$ .

$x - 18 - 2 (- 2 x) - 2 z + z = - 1$

$y = 9 - 2 x + z$

$- x + 2 y + 2 z = - 5$

Multiply $- 2$ by $- 2$ .

$x - 18 + 4 x - 2 z + z = - 1$

$y = 9 - 2 x + z$

$- x + 2 y + 2 z = - 5$

$x - 18 + 4 x - 2 z + z = - 1$

$y = 9 - 2 x + z$

$- x + 2 y + 2 z = - 5$

$x - 18 + 4 x - 2 z + z = - 1$

$y = 9 - 2 x + z$

$- x + 2 y + 2 z = - 5$

Simplify by adding terms.

Add $x$ and $4 x$ .

$5 x - 18 - 2 z + z = - 1$

$y = 9 - 2 x + z$

$- x + 2 y + 2 z = - 5$

Add $- 2 z$ and $z$ .

$5 x - 18 - z = - 1$

$y = 9 - 2 x + z$

$- x + 2 y + 2 z = - 5$

$5 x - 18 - z = - 1$

$y = 9 - 2 x + z$

$- x + 2 y + 2 z = - 5$

$5 x - 18 - z = - 1$

$y = 9 - 2 x + z$

$- x + 2 y + 2 z = - 5$

$5 x - 18 - z = - 1$

$y = 9 - 2 x + z$

$- x + 2 y + 2 z = - 5$

Replace all occurrences of $y$ in $- x + 2 y + 2 z = - 5$ with $9 - 2 x + z$ .

$- x + 2 (9 - 2 x + z) + 2 z = - 5$

$5 x - 18 - z = - 1$

$y = 9 - 2 x + z$

Simplify the left side.

Simplify $- x + 2 (9 - 2 x + z) + 2 z$ .

Simplify each term.

Apply the distributive property.

$- x + 2 \cdot 9 + 2 (- 2 x) + 2 z + 2 z = - 5$

$5 x - 18 - z = - 1$

$y = 9 - 2 x + z$

Simplify.

Multiply $2$ by $9$ .

$- x + 18 + 2 (- 2 x) + 2 z + 2 z = - 5$

$5 x - 18 - z = - 1$

$y = 9 - 2 x + z$

Multiply $- 2$ by $2$ .

$- x + 18 - 4 x + 2 z + 2 z = - 5$

$5 x - 18 - z = - 1$

$y = 9 - 2 x + z$

$- x + 18 - 4 x + 2 z + 2 z = - 5$

$5 x - 18 - z = - 1$

$y = 9 - 2 x + z$

$- x + 18 - 4 x + 2 z + 2 z = - 5$

$5 x - 18 - z = - 1$

$y = 9 - 2 x + z$

Simplify by adding terms.

Subtract $4 x$ from $- x$ .

$- 5 x + 18 + 2 z + 2 z = - 5$

$5 x - 18 - z = - 1$

$y = 9 - 2 x + z$

Add $2 z$ and $2 z$ .

$- 5 x + 18 + 4 z = - 5$

$5 x - 18 - z = - 1$

$y = 9 - 2 x + z$

$- 5 x + 18 + 4 z = - 5$

$5 x - 18 - z = - 1$

$y = 9 - 2 x + z$

$- 5 x + 18 + 4 z = - 5$

$5 x - 18 - z = - 1$

$y = 9 - 2 x + z$

$- 5 x + 18 + 4 z = - 5$

$5 x - 18 - z = - 1$

$y = 9 - 2 x + z$

$- 5 x + 18 + 4 z = - 5$

$5 x - 18 - z = - 1$

$y = 9 - 2 x + z$

Step 3

Solve for $x$ in $- 5 x + 18 + 4 z = - 5$ .

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

Subtract $18$ from both sides of the equation.

$- 5 x + 4 z = - 5 - 18$

$5 x - 18 - z = - 1$

$y = 9 - 2 x + z$

Subtract $4 z$ from both sides of the equation.

$- 5 x = - 5 - 18 - 4 z$

$5 x - 18 - z = - 1$

$y = 9 - 2 x + z$

Subtract $18$ from $- 5$ .

$- 5 x = - 23 - 4 z$

$5 x - 18 - z = - 1$

$y = 9 - 2 x + z$

$- 5 x = - 23 - 4 z$

$5 x - 18 - z = - 1$

$y = 9 - 2 x + z$

Divide each term in $- 5 x = - 23 - 4 z$ by $- 5$ and simplify.

Divide each term in $- 5 x = - 23 - 4 z$ by $- 5$ .

$\frac{- 5 x}{- 5} = \frac{- 23}{- 5} + \frac{- 4 z}{- 5}$

$5 x - 18 - z = - 1$

$y = 9 - 2 x + z$

Simplify the left side.

Cancel the common factor of $- 5$ .

Cancel the common factor.

$\frac{- 5 x}{- 5} = \frac{- 23}{- 5} + \frac{- 4 z}{- 5}$

$5 x - 18 - z = - 1$

$y = 9 - 2 x + z$

Divide $x$ by $1$ .

$x = \frac{- 23}{- 5} + \frac{- 4 z}{- 5}$

$5 x - 18 - z = - 1$

$y = 9 - 2 x + z$

$x = \frac{- 23}{- 5} + \frac{- 4 z}{- 5}$

$5 x - 18 - z = - 1$

$y = 9 - 2 x + z$

$x = \frac{- 23}{- 5} + \frac{- 4 z}{- 5}$

$5 x - 18 - z = - 1$

$y = 9 - 2 x + z$

Simplify the right side.

Simplify each term.

Dividing two negative values results in a positive value.

$x = \frac{23}{5} + \frac{- 4 z}{- 5}$

$5 x - 18 - z = - 1$

$y = 9 - 2 x + z$

Dividing two negative values results in a positive value.

$x = \frac{23}{5} + \frac{4 z}{5}$

$5 x - 18 - z = - 1$

$y = 9 - 2 x + z$

$x = \frac{23}{5} + \frac{4 z}{5}$

$5 x - 18 - z = - 1$

$y = 9 - 2 x + z$

$x = \frac{23}{5} + \frac{4 z}{5}$

$5 x - 18 - z = - 1$

$y = 9 - 2 x + z$

$x = \frac{23}{5} + \frac{4 z}{5}$

$5 x - 18 - z = - 1$

$y = 9 - 2 x + z$

$x = \frac{23}{5} + \frac{4 z}{5}$

$5 x - 18 - z = - 1$

$y = 9 - 2 x + z$

Step 4

Replace all occurrences of $x$ with $\frac{23}{5} + \frac{4 z}{5}$ in each equation.

Replace all occurrences of $x$ in $5 x - 18 - z = - 1$ with $\frac{23}{5} + \frac{4 z}{5}$ .

$5 (\frac{23}{5} + \frac{4 z}{5}) - 18 - z = - 1$

$x = \frac{23}{5} + \frac{4 z}{5}$

$y = 9 - 2 x + z$

Simplify the left side.

Simplify $5 (\frac{23}{5} + \frac{4 z}{5}) - 18 - z$ .

Simplify each term.

Apply the distributive property.

$5 (\frac{23}{5}) + 5 (\frac{4 z}{5}) - 18 - z = - 1$

$x = \frac{23}{5} + \frac{4 z}{5}$

$y = 9 - 2 x + z$

Cancel the common factor of $5$ .

Cancel the common factor.

$5 (\frac{23}{5}) + 5 (\frac{4 z}{5}) - 18 - z = - 1$

$x = \frac{23}{5} + \frac{4 z}{5}$

$y = 9 - 2 x + z$

Rewrite the expression.

$23 + 5 (\frac{4 z}{5}) - 18 - z = - 1$

$x = \frac{23}{5} + \frac{4 z}{5}$

$y = 9 - 2 x + z$

$23 + 5 (\frac{4 z}{5}) - 18 - z = - 1$

$x = \frac{23}{5} + \frac{4 z}{5}$

$y = 9 - 2 x + z$

Cancel the common factor of $5$ .

Cancel the common factor.

$23 + 5 (\frac{4 z}{5}) - 18 - z = - 1$

$x = \frac{23}{5} + \frac{4 z}{5}$

$y = 9 - 2 x + z$

Rewrite the expression.

$23 + 4 z - 18 - z = - 1$

$x = \frac{23}{5} + \frac{4 z}{5}$

$y = 9 - 2 x + z$

$23 + 4 z - 18 - z = - 1$

$x = \frac{23}{5} + \frac{4 z}{5}$

$y = 9 - 2 x + z$

$23 + 4 z - 18 - z = - 1$

$x = \frac{23}{5} + \frac{4 z}{5}$

$y = 9 - 2 x + z$

Simplify by adding terms.

Subtract $18$ from $23$ .

$4 z + 5 - z = - 1$

$x = \frac{23}{5} + \frac{4 z}{5}$

$y = 9 - 2 x + z$

Subtract $z$ from $4 z$ .

$3 z + 5 = - 1$

$x = \frac{23}{5} + \frac{4 z}{5}$

$y = 9 - 2 x + z$

$3 z + 5 = - 1$

$x = \frac{23}{5} + \frac{4 z}{5}$

$y = 9 - 2 x + z$

$3 z + 5 = - 1$

$x = \frac{23}{5} + \frac{4 z}{5}$

$y = 9 - 2 x + z$

$3 z + 5 = - 1$

$x = \frac{23}{5} + \frac{4 z}{5}$

$y = 9 - 2 x + z$

Replace all occurrences of $x$ in $y = 9 - 2 x + z$ with $\frac{23}{5} + \frac{4 z}{5}$ .

$y = 9 - 2 (\frac{23}{5} + \frac{4 z}{5}) + z$

$3 z + 5 = - 1$

$x = \frac{23}{5} + \frac{4 z}{5}$

Simplify the right side.

Simplify $9 - 2 (\frac{23}{5} + \frac{4 z}{5}) + z$ .

Simplify each term.

Apply the distributive property.

$y = 9 - 2 (\frac{23}{5}) - 2 \frac{4 z}{5} + z$

$3 z + 5 = - 1$

$x = \frac{23}{5} + \frac{4 z}{5}$

Multiply $- 2 (\frac{23}{5})$ .

Combine $- 2$ and $\frac{23}{5}$ .

$y = 9 + \frac{- 2 \cdot 23}{5} - 2 \frac{4 z}{5} + z$

$3 z + 5 = - 1$

$x = \frac{23}{5} + \frac{4 z}{5}$

Multiply $- 2$ by $23$ .

$y = 9 + \frac{- 46}{5} - 2 \frac{4 z}{5} + z$

$3 z + 5 = - 1$

$x = \frac{23}{5} + \frac{4 z}{5}$

$y = 9 + \frac{- 46}{5} - 2 \frac{4 z}{5} + z$

$3 z + 5 = - 1$

$x = \frac{23}{5} + \frac{4 z}{5}$

Multiply $- 2 \frac{4 z}{5}$ .

Combine $- 2$ and $\frac{4 z}{5}$ .

$y = 9 + \frac{- 46}{5} + \frac{- 2 (4 z)}{5} + z$

$3 z + 5 = - 1$

$x = \frac{23}{5} + \frac{4 z}{5}$

Multiply $4$ by $- 2$ .

$y = 9 + \frac{- 46}{5} + \frac{- 8 z}{5} + z$

$3 z + 5 = - 1$

$x = \frac{23}{5} + \frac{4 z}{5}$

$y = 9 + \frac{- 46}{5} + \frac{- 8 z}{5} + z$

$3 z + 5 = - 1$

$x = \frac{23}{5} + \frac{4 z}{5}$

Simplify each term.

Move the negative in front of the fraction.

$y = 9 - \frac{46}{5} + \frac{- 8 z}{5} + z$

$3 z + 5 = - 1$

$x = \frac{23}{5} + \frac{4 z}{5}$

Move the negative in front of the fraction.

$y = 9 - \frac{46}{5} - \frac{8 z}{5} + z$

$3 z + 5 = - 1$

$x = \frac{23}{5} + \frac{4 z}{5}$

$y = 9 - \frac{46}{5} - \frac{8 z}{5} + z$

$3 z + 5 = - 1$

$x = \frac{23}{5} + \frac{4 z}{5}$

$y = 9 - \frac{46}{5} - \frac{8 z}{5} + z$

$3 z + 5 = - 1$

$x = \frac{23}{5} + \frac{4 z}{5}$

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

$y = 9 \cdot \frac{5}{5} - \frac{46}{5} - \frac{8 z}{5} + z$

$3 z + 5 = - 1$

$x = \frac{23}{5} + \frac{4 z}{5}$

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

$y = \frac{9 \cdot 5}{5} - \frac{46}{5} - \frac{8 z}{5} + z$

$3 z + 5 = - 1$

$x = \frac{23}{5} + \frac{4 z}{5}$

Combine the numerators over the common denominator.

$y = \frac{9 \cdot 5 - 46}{5} - \frac{8 z}{5} + z$

$3 z + 5 = - 1$

$x = \frac{23}{5} + \frac{4 z}{5}$

Simplify the numerator.

Multiply $9$ by $5$ .

$y = \frac{45 - 46}{5} - \frac{8 z}{5} + z$

$3 z + 5 = - 1$

$x = \frac{23}{5} + \frac{4 z}{5}$

Subtract $46$ from $45$ .

$y = \frac{- 1}{5} - \frac{8 z}{5} + z$

$3 z + 5 = - 1$

$x = \frac{23}{5} + \frac{4 z}{5}$

$y = \frac{- 1}{5} - \frac{8 z}{5} + z$

$3 z + 5 = - 1$

$x = \frac{23}{5} + \frac{4 z}{5}$

Move the negative in front of the fraction.

$y = - \frac{1}{5} - \frac{8 z}{5} + z$

$3 z + 5 = - 1$

$x = \frac{23}{5} + \frac{4 z}{5}$

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

$y = - \frac{1}{5} - \frac{8 z}{5} + z \cdot \frac{5}{5}$

$3 z + 5 = - 1$

$x = \frac{23}{5} + \frac{4 z}{5}$

Simplify terms.

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

$y = - \frac{1}{5} - \frac{8 z}{5} + \frac{z \cdot 5}{5}$

$3 z + 5 = - 1$

$x = \frac{23}{5} + \frac{4 z}{5}$

Combine the numerators over the common denominator.

$y = - \frac{1}{5} + \frac{- 8 z + z \cdot 5}{5}$

$3 z + 5 = - 1$

$x = \frac{23}{5} + \frac{4 z}{5}$

Combine the numerators over the common denominator.

$y = \frac{- 1 - 8 z + z \cdot 5}{5}$

$3 z + 5 = - 1$

$x = \frac{23}{5} + \frac{4 z}{5}$

$y = \frac{- 1 - 8 z + z \cdot 5}{5}$

$3 z + 5 = - 1$

$x = \frac{23}{5} + \frac{4 z}{5}$

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

$y = \frac{- 1 - 8 z + 5 z}{5}$

$3 z + 5 = - 1$

$x = \frac{23}{5} + \frac{4 z}{5}$

Simplify terms.

Add $- 8 z$ and $5 z$ .

$y = \frac{- 1 - 3 z}{5}$

$3 z + 5 = - 1$

$x = \frac{23}{5} + \frac{4 z}{5}$

Rewrite $- 1$ as $- 1 (1)$ .

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

$3 z + 5 = - 1$

$x = \frac{23}{5} + \frac{4 z}{5}$

Factor $- 1$ out of $- 3 z$ .

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

$3 z + 5 = - 1$

$x = \frac{23}{5} + \frac{4 z}{5}$

Factor $- 1$ out of $- 1 (1) - (3 z)$ .

$y = \frac{- 1 (1 + 3 z)}{5}$

$3 z + 5 = - 1$

$x = \frac{23}{5} + \frac{4 z}{5}$

Move the negative in front of the fraction.

$y = - \frac{1 + 3 z}{5}$

$3 z + 5 = - 1$

$x = \frac{23}{5} + \frac{4 z}{5}$

$y = - \frac{1 + 3 z}{5}$

$3 z + 5 = - 1$

$x = \frac{23}{5} + \frac{4 z}{5}$

$y = - \frac{1 + 3 z}{5}$

$3 z + 5 = - 1$

$x = \frac{23}{5} + \frac{4 z}{5}$

$y = - \frac{1 + 3 z}{5}$

$3 z + 5 = - 1$

$x = \frac{23}{5} + \frac{4 z}{5}$

$y = - \frac{1 + 3 z}{5}$

$3 z + 5 = - 1$

$x = \frac{23}{5} + \frac{4 z}{5}$

Step 5

Solve for $z$ in $3 z + 5 = - 1$ .

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

Subtract $5$ from both sides of the equation.

$3 z = - 1 - 5$

$y = - \frac{1 + 3 z}{5}$

$x = \frac{23}{5} + \frac{4 z}{5}$

Subtract $5$ from $- 1$ .

$3 z = - 6$

$y = - \frac{1 + 3 z}{5}$

$x = \frac{23}{5} + \frac{4 z}{5}$

$3 z = - 6$

$y = - \frac{1 + 3 z}{5}$

$x = \frac{23}{5} + \frac{4 z}{5}$

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

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

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

$y = - \frac{1 + 3 z}{5}$

$x = \frac{23}{5} + \frac{4 z}{5}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

$y = - \frac{1 + 3 z}{5}$

$x = \frac{23}{5} + \frac{4 z}{5}$

Divide $z$ by $1$ .

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

$y = - \frac{1 + 3 z}{5}$

$x = \frac{23}{5} + \frac{4 z}{5}$

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

$y = - \frac{1 + 3 z}{5}$

$x = \frac{23}{5} + \frac{4 z}{5}$

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

$y = - \frac{1 + 3 z}{5}$

$x = \frac{23}{5} + \frac{4 z}{5}$

Simplify the right side.

Divide $- 6$ by $3$ .

$z = - 2$

$y = - \frac{1 + 3 z}{5}$

$x = \frac{23}{5} + \frac{4 z}{5}$

$z = - 2$

$y = - \frac{1 + 3 z}{5}$

$x = \frac{23}{5} + \frac{4 z}{5}$

$z = - 2$

$y = - \frac{1 + 3 z}{5}$

$x = \frac{23}{5} + \frac{4 z}{5}$

$z = - 2$

$y = - \frac{1 + 3 z}{5}$

$x = \frac{23}{5} + \frac{4 z}{5}$

Step 6

Replace all occurrences of $z$ with $- 2$ in each equation.

Replace all occurrences of $z$ in $y = - \frac{1 + 3 z}{5}$ with $- 2$ .

$y = - \frac{1 + 3 (- 2)}{5}$

$z = - 2$

$x = \frac{23}{5} + \frac{4 z}{5}$

Simplify the right side.

Simplify $- \frac{1 + 3 (- 2)}{5}$ .

Simplify the numerator.

Multiply $3$ by $- 2$ .

$y = - \frac{1 - 6}{5}$

$z = - 2$

$x = \frac{23}{5} + \frac{4 z}{5}$

Subtract $6$ from $1$ .

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

$z = - 2$

$x = \frac{23}{5} + \frac{4 z}{5}$

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

$z = - 2$

$x = \frac{23}{5} + \frac{4 z}{5}$

Simplify the expression.

Divide $- 5$ by $5$ .

$y = 1$

$z = - 2$

$x = \frac{23}{5} + \frac{4 z}{5}$

Multiply $- 1$ by $- 1$ .

$y = 1$

$z = - 2$

$x = \frac{23}{5} + \frac{4 z}{5}$

$y = 1$

$z = - 2$

$x = \frac{23}{5} + \frac{4 z}{5}$

$y = 1$

$z = - 2$

$x = \frac{23}{5} + \frac{4 z}{5}$

$y = 1$

$z = - 2$

$x = \frac{23}{5} + \frac{4 z}{5}$

Replace all occurrences of $z$ in $x = \frac{23}{5} + \frac{4 z}{5}$ with $- 2$ .

$x = \frac{23}{5} + \frac{4 (- 2)}{5}$

$y = 1$

$z = - 2$

Simplify the right side.

Simplify $\frac{23}{5} + \frac{4 (- 2)}{5}$ .

Combine the numerators over the common denominator.

$x = \frac{23 + 4 (- 2)}{5}$

$y = 1$

$z = - 2$

Simplify the expression.

Multiply $4$ by $- 2$ .

$x = \frac{23 - 8}{5}$

$y = 1$

$z = - 2$

Subtract $8$ from $23$ .

$x = \frac{15}{5}$

$y = 1$

$z = - 2$

Divide $15$ by $5$ .

$x = 3$

$y = 1$

$z = - 2$

$x = 3$

$y = 1$

$z = - 2$

$x = 3$

$y = 1$

$z = - 2$

$x = 3$

$y = 1$

$z = - 2$

$x = 3$

$y = 1$

$z = - 2$

Step 7

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

$(3, 1, - 2)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(3, 1, - 2)$

Equation Form:

$x = 3, y = 1, z = - 2$

Solve the System of Equations 2x+y-z=9 x-2y+z=-1 -x+2y+2z=-5

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