Solve the System of Equations 2x-y+2z=15 -x+y+z=3 3x-y+2z=18

$2 x - y + 2 z = 15$ $- x + y + z = 3$ $3 x - y + 2 z = 18$

Step 1

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

Add $x$ to both sides of the equation.

$y + z = 3 + x$

$2 x - y + 2 z = 15$

$3 x - y + 2 z = 18$

Subtract $z$ from both sides of the equation.

$y = 3 + x - z$

$2 x - y + 2 z = 15$

$3 x - y + 2 z = 18$

$y = 3 + x - z$

$2 x - y + 2 z = 15$

$3 x - y + 2 z = 18$

Step 2

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

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

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

$y = 3 + x - z$

$3 x - y + 2 z = 18$

Simplify the left side.

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

Simplify each term.

Apply the distributive property.

$2 x - 1 \cdot 3 - x + z + 2 z = 15$

$y = 3 + x - z$

$3 x - y + 2 z = 18$

Simplify.

Multiply $- 1$ by $3$ .

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

$y = 3 + x - z$

$3 x - y + 2 z = 18$

Multiply $- - z$ .

Multiply $- 1$ by $- 1$ .

$2 x - 3 - x + 1 z + 2 z = 15$

$y = 3 + x - z$

$3 x - y + 2 z = 18$

Multiply $z$ by $1$ .

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

$y = 3 + x - z$

$3 x - y + 2 z = 18$

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

$y = 3 + x - z$

$3 x - y + 2 z = 18$

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

$y = 3 + x - z$

$3 x - y + 2 z = 18$

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

$y = 3 + x - z$

$3 x - y + 2 z = 18$

Simplify by adding terms.

Subtract $x$ from $2 x$ .

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

$y = 3 + x - z$

$3 x - y + 2 z = 18$

Add $z$ and $2 z$ .

$x - 3 + 3 z = 15$

$y = 3 + x - z$

$3 x - y + 2 z = 18$

$x - 3 + 3 z = 15$

$y = 3 + x - z$

$3 x - y + 2 z = 18$

$x - 3 + 3 z = 15$

$y = 3 + x - z$

$3 x - y + 2 z = 18$

$x - 3 + 3 z = 15$

$y = 3 + x - z$

$3 x - y + 2 z = 18$

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

$3 x - (3 + x - z) + 2 z = 18$

$x - 3 + 3 z = 15$

$y = 3 + x - z$

Simplify the left side.

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

Simplify each term.

Apply the distributive property.

$3 x - 1 \cdot 3 - x + z + 2 z = 18$

$x - 3 + 3 z = 15$

$y = 3 + x - z$

Simplify.

Multiply $- 1$ by $3$ .

$3 x - 3 - x + z + 2 z = 18$

$x - 3 + 3 z = 15$

$y = 3 + x - z$

Multiply $- - z$ .

Multiply $- 1$ by $- 1$ .

$3 x - 3 - x + 1 z + 2 z = 18$

$x - 3 + 3 z = 15$

$y = 3 + x - z$

Multiply $z$ by $1$ .

$3 x - 3 - x + z + 2 z = 18$

$x - 3 + 3 z = 15$

$y = 3 + x - z$

$3 x - 3 - x + z + 2 z = 18$

$x - 3 + 3 z = 15$

$y = 3 + x - z$

$3 x - 3 - x + z + 2 z = 18$

$x - 3 + 3 z = 15$

$y = 3 + x - z$

$3 x - 3 - x + z + 2 z = 18$

$x - 3 + 3 z = 15$

$y = 3 + x - z$

Simplify by adding terms.

Subtract $x$ from $3 x$ .

$2 x - 3 + z + 2 z = 18$

$x - 3 + 3 z = 15$

$y = 3 + x - z$

Add $z$ and $2 z$ .

$2 x - 3 + 3 z = 18$

$x - 3 + 3 z = 15$

$y = 3 + x - z$

$2 x - 3 + 3 z = 18$

$x - 3 + 3 z = 15$

$y = 3 + x - z$

$2 x - 3 + 3 z = 18$

$x - 3 + 3 z = 15$

$y = 3 + x - z$

$2 x - 3 + 3 z = 18$

$x - 3 + 3 z = 15$

$y = 3 + x - z$

$2 x - 3 + 3 z = 18$

$x - 3 + 3 z = 15$

$y = 3 + x - z$

Step 3

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

Add $3$ to both sides of the equation.

$x + 3 z = 15 + 3$

$2 x - 3 + 3 z = 18$

$y = 3 + x - z$

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

$x = 15 + 3 - 3 z$

$2 x - 3 + 3 z = 18$

$y = 3 + x - z$

Add $15$ and $3$ .

$x = 18 - 3 z$

$2 x - 3 + 3 z = 18$

$y = 3 + x - z$

$x = 18 - 3 z$

$2 x - 3 + 3 z = 18$

$y = 3 + x - z$

Step 4

Replace all occurrences of $x$ with $18 - 3 z$ in each equation.

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

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

$x = 18 - 3 z$

$y = 3 + x - z$

Simplify the left side.

Simplify $2 (18 - 3 z) - 3 + 3 z$ .

Simplify each term.

Apply the distributive property.

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

$x = 18 - 3 z$

$y = 3 + x - z$

Multiply $2$ by $18$ .

$36 + 2 (- 3 z) - 3 + 3 z = 18$

$x = 18 - 3 z$

$y = 3 + x - z$

Multiply $- 3$ by $2$ .

$36 - 6 z - 3 + 3 z = 18$

$x = 18 - 3 z$

$y = 3 + x - z$

$36 - 6 z - 3 + 3 z = 18$

$x = 18 - 3 z$

$y = 3 + x - z$

Simplify by adding terms.

Subtract $3$ from $36$ .

$- 6 z + 33 + 3 z = 18$

$x = 18 - 3 z$

$y = 3 + x - z$

Add $- 6 z$ and $3 z$ .

$- 3 z + 33 = 18$

$x = 18 - 3 z$

$y = 3 + x - z$

$- 3 z + 33 = 18$

$x = 18 - 3 z$

$y = 3 + x - z$

$- 3 z + 33 = 18$

$x = 18 - 3 z$

$y = 3 + x - z$

$- 3 z + 33 = 18$

$x = 18 - 3 z$

$y = 3 + x - z$

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

$y = 3 + 18 - 3 z - z$

$- 3 z + 33 = 18$

$x = 18 - 3 z$

Simplify $y = 3 + 18 - 3 z - z$ .

Simplify the left side.

Remove parentheses.

$y = 3 + 18 - 3 z - z$

$- 3 z + 33 = 18$

$x = 18 - 3 z$

$y = 3 + 18 - 3 z - z$

$- 3 z + 33 = 18$

$x = 18 - 3 z$

Simplify the right side.

Simplify $3 + 18 - 3 z - z$ .

Add $3$ and $18$ .

$y = 21 - 3 z - z$

$- 3 z + 33 = 18$

$x = 18 - 3 z$

Subtract $z$ from $- 3 z$ .

$y = 21 - 4 z$

$- 3 z + 33 = 18$

$x = 18 - 3 z$

$y = 21 - 4 z$

$- 3 z + 33 = 18$

$x = 18 - 3 z$

$y = 21 - 4 z$

$- 3 z + 33 = 18$

$x = 18 - 3 z$

$y = 21 - 4 z$

$- 3 z + 33 = 18$

$x = 18 - 3 z$

$y = 21 - 4 z$

$- 3 z + 33 = 18$

$x = 18 - 3 z$

Step 5

Solve for $z$ in $- 3 z + 33 = 18$ .

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

Subtract $33$ from both sides of the equation.

$- 3 z = 18 - 33$

$y = 21 - 4 z$

$x = 18 - 3 z$

Subtract $33$ from $18$ .

$- 3 z = - 15$

$y = 21 - 4 z$

$x = 18 - 3 z$

$- 3 z = - 15$

$y = 21 - 4 z$

$x = 18 - 3 z$

Divide each term in $- 3 z = - 15$ by $- 3$ and simplify.

Divide each term in $- 3 z = - 15$ by $- 3$ .

$\frac{- 3 z}{- 3} = \frac{- 15}{- 3}$

$y = 21 - 4 z$

$x = 18 - 3 z$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 z}{- 3} = \frac{- 15}{- 3}$

$y = 21 - 4 z$

$x = 18 - 3 z$

Divide $z$ by $1$ .

$z = \frac{- 15}{- 3}$

$y = 21 - 4 z$

$x = 18 - 3 z$

$z = \frac{- 15}{- 3}$

$y = 21 - 4 z$

$x = 18 - 3 z$

$z = \frac{- 15}{- 3}$

$y = 21 - 4 z$

$x = 18 - 3 z$

Simplify the right side.

Divide $- 15$ by $- 3$ .

$z = 5$

$y = 21 - 4 z$

$x = 18 - 3 z$

$z = 5$

$y = 21 - 4 z$

$x = 18 - 3 z$

$z = 5$

$y = 21 - 4 z$

$x = 18 - 3 z$

$z = 5$

$y = 21 - 4 z$

$x = 18 - 3 z$

Step 6

Replace all occurrences of $z$ with $5$ in each equation.

Replace all occurrences of $z$ in $y = 21 - 4 z$ with $5$ .

$y = 21 - 4 \cdot 5$

$z = 5$

$x = 18 - 3 z$

Simplify the right side.

Simplify $21 - 4 \cdot 5$ .

Multiply $- 4$ by $5$ .

$y = 21 - 20$

$z = 5$

$x = 18 - 3 z$

Subtract $20$ from $21$ .

$y = 1$

$z = 5$

$x = 18 - 3 z$

$y = 1$

$z = 5$

$x = 18 - 3 z$

$y = 1$

$z = 5$

$x = 18 - 3 z$

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

$x = 18 - 3 \cdot 5$

$y = 1$

$z = 5$

Simplify the right side.

Simplify $18 - 3 \cdot 5$ .

Multiply $- 3$ by $5$ .

$x = 18 - 15$

$y = 1$

$z = 5$

Subtract $15$ from $18$ .

$x = 3$

$y = 1$

$z = 5$

$x = 3$

$y = 1$

$z = 5$

$x = 3$

$y = 1$

$z = 5$

$x = 3$

$y = 1$

$z = 5$

Step 7

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

$(3, 1, 5)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(3, 1, 5)$

Equation Form:

$x = 3, y = 1, z = 5$

Solve the System of Equations 2x-y+2z=15 -x+y+z=3 3x-y+2z=18

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