Solve the System of Equations 2x+3y-2z=-7 x-2y+4z=15 2y+z=1

$2 x + 3 y - 2 z = - 7$ $x - 2 y + 4 z = 15$ $2 y + z = 1$

Step 1

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

Add $2 y$ to both sides of the equation.

$x + 4 z = 15 + 2 y$

$2 x + 3 y - 2 z = - 7$

$2 y + z = 1$

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

$x = 15 + 2 y - 4 z$

$2 x + 3 y - 2 z = - 7$

$2 y + z = 1$

$x = 15 + 2 y - 4 z$

$2 x + 3 y - 2 z = - 7$

$2 y + z = 1$

Step 2

Replace all occurrences of $x$ with $15 + 2 y - 4 z$ in each equation.

Replace all occurrences of $x$ in $2 x + 3 y - 2 z = - 7$ with $15 + 2 y - 4 z$ .

$2 (15 + 2 y - 4 z) + 3 y - 2 z = - 7$

$x = 15 + 2 y - 4 z$

$2 y + z = 1$

Simplify the left side.

Simplify $2 (15 + 2 y - 4 z) + 3 y - 2 z$ .

Simplify each term.

Apply the distributive property.

$2 \cdot 15 + 2 (2 y) + 2 (- 4 z) + 3 y - 2 z = - 7$

$x = 15 + 2 y - 4 z$

$2 y + z = 1$

Simplify.

Multiply $2$ by $15$ .

$30 + 2 (2 y) + 2 (- 4 z) + 3 y - 2 z = - 7$

$x = 15 + 2 y - 4 z$

$2 y + z = 1$

Multiply $2$ by $2$ .

$30 + 4 y + 2 (- 4 z) + 3 y - 2 z = - 7$

$x = 15 + 2 y - 4 z$

$2 y + z = 1$

Multiply $- 4$ by $2$ .

$30 + 4 y - 8 z + 3 y - 2 z = - 7$

$x = 15 + 2 y - 4 z$

$2 y + z = 1$

$30 + 4 y - 8 z + 3 y - 2 z = - 7$

$x = 15 + 2 y - 4 z$

$2 y + z = 1$

$30 + 4 y - 8 z + 3 y - 2 z = - 7$

$x = 15 + 2 y - 4 z$

$2 y + z = 1$

Simplify by adding terms.

Add $4 y$ and $3 y$ .

$30 + 7 y - 8 z - 2 z = - 7$

$x = 15 + 2 y - 4 z$

$2 y + z = 1$

Subtract $2 z$ from $- 8 z$ .

$30 + 7 y - 10 z = - 7$

$x = 15 + 2 y - 4 z$

$2 y + z = 1$

$30 + 7 y - 10 z = - 7$

$x = 15 + 2 y - 4 z$

$2 y + z = 1$

$30 + 7 y - 10 z = - 7$

$x = 15 + 2 y - 4 z$

$2 y + z = 1$

$30 + 7 y - 10 z = - 7$

$x = 15 + 2 y - 4 z$

$2 y + z = 1$

$30 + 7 y - 10 z = - 7$

$x = 15 + 2 y - 4 z$

$2 y + z = 1$

Step 3

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

$z = 1 - 2 y$

$30 + 7 y - 10 z = - 7$

$x = 15 + 2 y - 4 z$

Step 4

Replace all occurrences of $z$ with $1 - 2 y$ in each equation.

Replace all occurrences of $z$ in $30 + 7 y - 10 z = - 7$ with $1 - 2 y$ .

$30 + 7 y - 10 (1 - 2 y) = - 7$

$z = 1 - 2 y$

$x = 15 + 2 y - 4 z$

Simplify the left side.

Simplify $30 + 7 y - 10 (1 - 2 y)$ .

Simplify each term.

Apply the distributive property.

$30 + 7 y - 10 \cdot 1 - 10 (- 2 y) = - 7$

$z = 1 - 2 y$

$x = 15 + 2 y - 4 z$

Multiply $- 10$ by $1$ .

$30 + 7 y - 10 - 10 (- 2 y) = - 7$

$z = 1 - 2 y$

$x = 15 + 2 y - 4 z$

Multiply $- 2$ by $- 10$ .

$30 + 7 y - 10 + 20 y = - 7$

$z = 1 - 2 y$

$x = 15 + 2 y - 4 z$

$30 + 7 y - 10 + 20 y = - 7$

$z = 1 - 2 y$

$x = 15 + 2 y - 4 z$

Simplify by adding terms.

Subtract $10$ from $30$ .

$7 y + 20 + 20 y = - 7$

$z = 1 - 2 y$

$x = 15 + 2 y - 4 z$

Add $7 y$ and $20 y$ .

$27 y + 20 = - 7$

$z = 1 - 2 y$

$x = 15 + 2 y - 4 z$

$27 y + 20 = - 7$

$z = 1 - 2 y$

$x = 15 + 2 y - 4 z$

$27 y + 20 = - 7$

$z = 1 - 2 y$

$x = 15 + 2 y - 4 z$

$27 y + 20 = - 7$

$z = 1 - 2 y$

$x = 15 + 2 y - 4 z$

Replace all occurrences of $z$ in $x = 15 + 2 y - 4 z$ with $1 - 2 y$ .

$x = 15 + 2 y - 4 (1 - 2 y)$

$27 y + 20 = - 7$

$z = 1 - 2 y$

Simplify the right side.

Simplify $15 + 2 y - 4 (1 - 2 y)$ .

Simplify each term.

Apply the distributive property.

$x = 15 + 2 y - 4 \cdot 1 - 4 (- 2 y)$

$27 y + 20 = - 7$

$z = 1 - 2 y$

Multiply $- 4$ by $1$ .

$x = 15 + 2 y - 4 - 4 (- 2 y)$

$27 y + 20 = - 7$

$z = 1 - 2 y$

Multiply $- 2$ by $- 4$ .

$x = 15 + 2 y - 4 + 8 y$

$27 y + 20 = - 7$

$z = 1 - 2 y$

$x = 15 + 2 y - 4 + 8 y$

$27 y + 20 = - 7$

$z = 1 - 2 y$

Simplify by adding terms.

Subtract $4$ from $15$ .

$x = 2 y + 11 + 8 y$

$27 y + 20 = - 7$

$z = 1 - 2 y$

Add $2 y$ and $8 y$ .

$x = 10 y + 11$

$27 y + 20 = - 7$

$z = 1 - 2 y$

$x = 10 y + 11$

$27 y + 20 = - 7$

$z = 1 - 2 y$

$x = 10 y + 11$

$27 y + 20 = - 7$

$z = 1 - 2 y$

$x = 10 y + 11$

$27 y + 20 = - 7$

$z = 1 - 2 y$

$x = 10 y + 11$

$27 y + 20 = - 7$

$z = 1 - 2 y$

Step 5

Solve for $y$ in $27 y + 20 = - 7$ .

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

Subtract $20$ from both sides of the equation.

$27 y = - 7 - 20$

$x = 10 y + 11$

$z = 1 - 2 y$

Subtract $20$ from $- 7$ .

$27 y = - 27$

$x = 10 y + 11$

$z = 1 - 2 y$

$27 y = - 27$

$x = 10 y + 11$

$z = 1 - 2 y$

Divide each term in $27 y = - 27$ by $27$ and simplify.

Divide each term in $27 y = - 27$ by $27$ .

$\frac{27 y}{27} = \frac{- 27}{27}$

$x = 10 y + 11$

$z = 1 - 2 y$

Simplify the left side.

Cancel the common factor of $27$ .

Cancel the common factor.

$\frac{27 y}{27} = \frac{- 27}{27}$

$x = 10 y + 11$

$z = 1 - 2 y$

Divide $y$ by $1$ .

$y = \frac{- 27}{27}$

$x = 10 y + 11$

$z = 1 - 2 y$

$y = \frac{- 27}{27}$

$x = 10 y + 11$

$z = 1 - 2 y$

$y = \frac{- 27}{27}$

$x = 10 y + 11$

$z = 1 - 2 y$

Simplify the right side.

Divide $- 27$ by $27$ .

$y = - 1$

$x = 10 y + 11$

$z = 1 - 2 y$

$y = - 1$

$x = 10 y + 11$

$z = 1 - 2 y$

$y = - 1$

$x = 10 y + 11$

$z = 1 - 2 y$

$y = - 1$

$x = 10 y + 11$

$z = 1 - 2 y$

Step 6

Replace all occurrences of $y$ with $- 1$ in each equation.

Replace all occurrences of $y$ in $x = 10 y + 11$ with $- 1$ .

$x = 10 (- 1) + 11$

$y = - 1$

$z = 1 - 2 y$

Simplify the right side.

Simplify $10 (- 1) + 11$ .

Multiply $10$ by $- 1$ .

$x = - 10 + 11$

$y = - 1$

$z = 1 - 2 y$

Add $- 10$ and $11$ .

$x = 1$

$y = - 1$

$z = 1 - 2 y$

$x = 1$

$y = - 1$

$z = 1 - 2 y$

$x = 1$

$y = - 1$

$z = 1 - 2 y$

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

$z = 1 - 2 \cdot - 1$

$x = 1$

$y = - 1$

Simplify the right side.

Simplify $1 - 2 \cdot - 1$ .

Multiply $- 2$ by $- 1$ .

$z = 1 + 2$

$x = 1$

$y = - 1$

Add $1$ and $2$ .

$z = 3$

$x = 1$

$y = - 1$

$z = 3$

$x = 1$

$y = - 1$

$z = 3$

$x = 1$

$y = - 1$

$z = 3$

$x = 1$

$y = - 1$

Step 7

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

$(1, - 1, 3)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(1, - 1, 3)$

Equation Form:

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

Solve the System of Equations 2x+3y-2z=-7 x-2y+4z=15 2y+z=1

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