Solve the System of Equations 2x+y-z=5 x+3z=14 -2x-3y+2z=2

$2 x + y - z = 5$ $x + 3 z = 14$ $- 2 x - 3 y + 2 z = 2$

Step 1

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

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

$y - z = 5 - 2 x$

$x + 3 z = 14$

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

Add $z$ to both sides of the equation.

$y = 5 - 2 x + z$

$x + 3 z = 14$

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

$y = 5 - 2 x + z$

$x + 3 z = 14$

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

Step 2

Replace all occurrences of $y$ with $5 - 2 x + z$ in each equation.

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

$- 2 x - 3 (5 - 2 x + z) + 2 z = 2$

$y = 5 - 2 x + z$

$x + 3 z = 14$

Simplify the left side.

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

Simplify each term.

Apply the distributive property.

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

$y = 5 - 2 x + z$

$x + 3 z = 14$

Simplify.

Multiply $- 3$ by $5$ .

$- 2 x - 15 - 3 (- 2 x) - 3 z + 2 z = 2$

$y = 5 - 2 x + z$

$x + 3 z = 14$

Multiply $- 2$ by $- 3$ .

$- 2 x - 15 + 6 x - 3 z + 2 z = 2$

$y = 5 - 2 x + z$

$x + 3 z = 14$

$- 2 x - 15 + 6 x - 3 z + 2 z = 2$

$y = 5 - 2 x + z$

$x + 3 z = 14$

$- 2 x - 15 + 6 x - 3 z + 2 z = 2$

$y = 5 - 2 x + z$

$x + 3 z = 14$

Simplify by adding terms.

Add $- 2 x$ and $6 x$ .

$4 x - 15 - 3 z + 2 z = 2$

$y = 5 - 2 x + z$

$x + 3 z = 14$

Add $- 3 z$ and $2 z$ .

$4 x - 15 - z = 2$

$y = 5 - 2 x + z$

$x + 3 z = 14$

$4 x - 15 - z = 2$

$y = 5 - 2 x + z$

$x + 3 z = 14$

$4 x - 15 - z = 2$

$y = 5 - 2 x + z$

$x + 3 z = 14$

$4 x - 15 - z = 2$

$y = 5 - 2 x + z$

$x + 3 z = 14$

$4 x - 15 - z = 2$

$y = 5 - 2 x + z$

$x + 3 z = 14$

Step 3

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

$x = 14 - 3 z$

$4 x - 15 - z = 2$

$y = 5 - 2 x + z$

Step 4

Replace all occurrences of $x$ with $14 - 3 z$ in each equation.

Replace all occurrences of $x$ in $4 x - 15 - z = 2$ with $14 - 3 z$ .

$4 (14 - 3 z) - 15 - z = 2$

$x = 14 - 3 z$

$y = 5 - 2 x + z$

Simplify the left side.

Simplify $4 (14 - 3 z) - 15 - z$ .

Simplify each term.

Apply the distributive property.

$4 \cdot 14 + 4 (- 3 z) - 15 - z = 2$

$x = 14 - 3 z$

$y = 5 - 2 x + z$

Multiply $4$ by $14$ .

$56 + 4 (- 3 z) - 15 - z = 2$

$x = 14 - 3 z$

$y = 5 - 2 x + z$

Multiply $- 3$ by $4$ .

$56 - 12 z - 15 - z = 2$

$x = 14 - 3 z$

$y = 5 - 2 x + z$

$56 - 12 z - 15 - z = 2$

$x = 14 - 3 z$

$y = 5 - 2 x + z$

Simplify by adding terms.

Subtract $15$ from $56$ .

$- 12 z + 41 - z = 2$

$x = 14 - 3 z$

$y = 5 - 2 x + z$

Subtract $z$ from $- 12 z$ .

$- 13 z + 41 = 2$

$x = 14 - 3 z$

$y = 5 - 2 x + z$

$- 13 z + 41 = 2$

$x = 14 - 3 z$

$y = 5 - 2 x + z$

$- 13 z + 41 = 2$

$x = 14 - 3 z$

$y = 5 - 2 x + z$

$- 13 z + 41 = 2$

$x = 14 - 3 z$

$y = 5 - 2 x + z$

Replace all occurrences of $x$ in $y = 5 - 2 x + z$ with $14 - 3 z$ .

$y = 5 - 2 (14 - 3 z) + z$

$- 13 z + 41 = 2$

$x = 14 - 3 z$

Simplify the right side.

Simplify $5 - 2 (14 - 3 z) + z$ .

Simplify each term.

Apply the distributive property.

$y = 5 - 2 \cdot 14 - 2 (- 3 z) + z$

$- 13 z + 41 = 2$

$x = 14 - 3 z$

Multiply $- 2$ by $14$ .

$y = 5 - 28 - 2 (- 3 z) + z$

$- 13 z + 41 = 2$

$x = 14 - 3 z$

Multiply $- 3$ by $- 2$ .

$y = 5 - 28 + 6 z + z$

$- 13 z + 41 = 2$

$x = 14 - 3 z$

$y = 5 - 28 + 6 z + z$

$- 13 z + 41 = 2$

$x = 14 - 3 z$

Simplify by adding terms.

Subtract $28$ from $5$ .

$y = - 23 + 6 z + z$

$- 13 z + 41 = 2$

$x = 14 - 3 z$

Add $6 z$ and $z$ .

$y = - 23 + 7 z$

$- 13 z + 41 = 2$

$x = 14 - 3 z$

$y = - 23 + 7 z$

$- 13 z + 41 = 2$

$x = 14 - 3 z$

$y = - 23 + 7 z$

$- 13 z + 41 = 2$

$x = 14 - 3 z$

$y = - 23 + 7 z$

$- 13 z + 41 = 2$

$x = 14 - 3 z$

$y = - 23 + 7 z$

$- 13 z + 41 = 2$

$x = 14 - 3 z$

Step 5

Solve for $z$ in $- 13 z + 41 = 2$ .

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

Subtract $41$ from both sides of the equation.

$- 13 z = 2 - 41$

$y = - 23 + 7 z$

$x = 14 - 3 z$

Subtract $41$ from $2$ .

$- 13 z = - 39$

$y = - 23 + 7 z$

$x = 14 - 3 z$

$- 13 z = - 39$

$y = - 23 + 7 z$

$x = 14 - 3 z$

Divide each term in $- 13 z = - 39$ by $- 13$ and simplify.

Divide each term in $- 13 z = - 39$ by $- 13$ .

$\frac{- 13 z}{- 13} = \frac{- 39}{- 13}$

$y = - 23 + 7 z$

$x = 14 - 3 z$

Simplify the left side.

Cancel the common factor of $- 13$ .

Cancel the common factor.

$\frac{- 13 z}{- 13} = \frac{- 39}{- 13}$

$y = - 23 + 7 z$

$x = 14 - 3 z$

Divide $z$ by $1$ .

$z = \frac{- 39}{- 13}$

$y = - 23 + 7 z$

$x = 14 - 3 z$

$z = \frac{- 39}{- 13}$

$y = - 23 + 7 z$

$x = 14 - 3 z$

$z = \frac{- 39}{- 13}$

$y = - 23 + 7 z$

$x = 14 - 3 z$

Simplify the right side.

Divide $- 39$ by $- 13$ .

$z = 3$

$y = - 23 + 7 z$

$x = 14 - 3 z$

$z = 3$

$y = - 23 + 7 z$

$x = 14 - 3 z$

$z = 3$

$y = - 23 + 7 z$

$x = 14 - 3 z$

$z = 3$

$y = - 23 + 7 z$

$x = 14 - 3 z$

Step 6

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

Replace all occurrences of $z$ in $y = - 23 + 7 z$ with $3$ .

$y = - 23 + 7 (3)$

$z = 3$

$x = 14 - 3 z$

Simplify the right side.

Simplify $- 23 + 7 (3)$ .

Multiply $7$ by $3$ .

$y = - 23 + 21$

$z = 3$

$x = 14 - 3 z$

Add $- 23$ and $21$ .

$y = - 2$

$z = 3$

$x = 14 - 3 z$

$y = - 2$

$z = 3$

$x = 14 - 3 z$

$y = - 2$

$z = 3$

$x = 14 - 3 z$

Replace all occurrences of $z$ in $x = 14 - 3 z$ with $3$ .

$x = 14 - 3 \cdot 3$

$y = - 2$

$z = 3$

Simplify the right side.

Simplify $14 - 3 \cdot 3$ .

Multiply $- 3$ by $3$ .

$x = 14 - 9$

$y = - 2$

$z = 3$

Subtract $9$ from $14$ .

$x = 5$

$y = - 2$

$z = 3$

$x = 5$

$y = - 2$

$z = 3$

$x = 5$

$y = - 2$

$z = 3$

$x = 5$

$y = - 2$

$z = 3$

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:

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

Solve the System of Equations 2x+y-z=5 x+3z=14 -2x-3y+2z=2

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