Solve the System of Equations 2x-3y+z=10 -x+y-z=1 5x+y+z=-7

$2 x - 3 y + z = 10$ $- x + y - z = 1$ $5 x + y + z = - 7$

Step 1

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

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

$- 3 y + z = 10 - 2 x$

$- x + y - z = 1$

$5 x + y + z = - 7$

Add $3 y$ to both sides of the equation.

$z = 10 - 2 x + 3 y$

$- x + y - z = 1$

$5 x + y + z = - 7$

$z = 10 - 2 x + 3 y$

$- x + y - z = 1$

$5 x + y + z = - 7$

Step 2

Replace all occurrences of $z$ with $10 - 2 x + 3 y$ in each equation.

Replace all occurrences of $z$ in $- x + y - z = 1$ with $10 - 2 x + 3 y$ .

$- x + y - (10 - 2 x + 3 y) = 1$

$z = 10 - 2 x + 3 y$

$5 x + y + z = - 7$

Simplify the left side.

Simplify $- x + y - (10 - 2 x + 3 y)$ .

Simplify each term.

Apply the distributive property.

$- x + y - 1 \cdot 10 - (- 2 x) - (3 y) = 1$

$z = 10 - 2 x + 3 y$

$5 x + y + z = - 7$

Simplify.

Multiply $- 1$ by $10$ .

$- x + y - 10 - (- 2 x) - (3 y) = 1$

$z = 10 - 2 x + 3 y$

$5 x + y + z = - 7$

Multiply $- 2$ by $- 1$ .

$- x + y - 10 + 2 x - (3 y) = 1$

$z = 10 - 2 x + 3 y$

$5 x + y + z = - 7$

Multiply $3$ by $- 1$ .

$- x + y - 10 + 2 x - 3 y = 1$

$z = 10 - 2 x + 3 y$

$5 x + y + z = - 7$

$- x + y - 10 + 2 x - 3 y = 1$

$z = 10 - 2 x + 3 y$

$5 x + y + z = - 7$

$- x + y - 10 + 2 x - 3 y = 1$

$z = 10 - 2 x + 3 y$

$5 x + y + z = - 7$

Simplify by adding terms.

Add $- x$ and $2 x$ .

$x + y - 10 - 3 y = 1$

$z = 10 - 2 x + 3 y$

$5 x + y + z = - 7$

Subtract $3 y$ from $y$ .

$x - 2 y - 10 = 1$

$z = 10 - 2 x + 3 y$

$5 x + y + z = - 7$

$x - 2 y - 10 = 1$

$z = 10 - 2 x + 3 y$

$5 x + y + z = - 7$

$x - 2 y - 10 = 1$

$z = 10 - 2 x + 3 y$

$5 x + y + z = - 7$

$x - 2 y - 10 = 1$

$z = 10 - 2 x + 3 y$

$5 x + y + z = - 7$

Replace all occurrences of $z$ in $5 x + y + z = - 7$ with $10 - 2 x + 3 y$ .

$5 x + y + 10 - 2 x + 3 y = - 7$

$x - 2 y - 10 = 1$

$z = 10 - 2 x + 3 y$

Simplify the left side.

Simplify $5 x + y + 10 - 2 x + 3 y$ .

Remove parentheses.

$5 x + y + 10 - 2 x + 3 y = - 7$

$x - 2 y - 10 = 1$

$z = 10 - 2 x + 3 y$

Subtract $2 x$ from $5 x$ .

$3 x + y + 10 + 3 y = - 7$

$x - 2 y - 10 = 1$

$z = 10 - 2 x + 3 y$

Add $y$ and $3 y$ .

$3 x + 4 y + 10 = - 7$

$x - 2 y - 10 = 1$

$z = 10 - 2 x + 3 y$

$3 x + 4 y + 10 = - 7$

$x - 2 y - 10 = 1$

$z = 10 - 2 x + 3 y$

$3 x + 4 y + 10 = - 7$

$x - 2 y - 10 = 1$

$z = 10 - 2 x + 3 y$

$3 x + 4 y + 10 = - 7$

$x - 2 y - 10 = 1$

$z = 10 - 2 x + 3 y$

Step 3

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

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

$x - 10 = 1 + 2 y$

$3 x + 4 y + 10 = - 7$

$z = 10 - 2 x + 3 y$

Add $10$ to both sides of the equation.

$x = 1 + 2 y + 10$

$3 x + 4 y + 10 = - 7$

$z = 10 - 2 x + 3 y$

Add $1$ and $10$ .

$x = 2 y + 11$

$3 x + 4 y + 10 = - 7$

$z = 10 - 2 x + 3 y$

$x = 2 y + 11$

$3 x + 4 y + 10 = - 7$

$z = 10 - 2 x + 3 y$

Step 4

Replace all occurrences of $x$ with $2 y + 11$ in each equation.

Replace all occurrences of $x$ in $3 x + 4 y + 10 = - 7$ with $2 y + 11$ .

$3 (2 y + 11) + 4 y + 10 = - 7$

$x = 2 y + 11$

$z = 10 - 2 x + 3 y$

Simplify the left side.

Simplify $3 (2 y + 11) + 4 y + 10$ .

Simplify each term.

Apply the distributive property.

$3 (2 y) + 3 \cdot 11 + 4 y + 10 = - 7$

$x = 2 y + 11$

$z = 10 - 2 x + 3 y$

Multiply $2$ by $3$ .

$6 y + 3 \cdot 11 + 4 y + 10 = - 7$

$x = 2 y + 11$

$z = 10 - 2 x + 3 y$

Multiply $3$ by $11$ .

$6 y + 33 + 4 y + 10 = - 7$

$x = 2 y + 11$

$z = 10 - 2 x + 3 y$

$6 y + 33 + 4 y + 10 = - 7$

$x = 2 y + 11$

$z = 10 - 2 x + 3 y$

Simplify by adding terms.

Add $6 y$ and $4 y$ .

$10 y + 33 + 10 = - 7$

$x = 2 y + 11$

$z = 10 - 2 x + 3 y$

Add $33$ and $10$ .

$10 y + 43 = - 7$

$x = 2 y + 11$

$z = 10 - 2 x + 3 y$

$10 y + 43 = - 7$

$x = 2 y + 11$

$z = 10 - 2 x + 3 y$

$10 y + 43 = - 7$

$x = 2 y + 11$

$z = 10 - 2 x + 3 y$

$10 y + 43 = - 7$

$x = 2 y + 11$

$z = 10 - 2 x + 3 y$

Replace all occurrences of $x$ in $z = 10 - 2 x + 3 y$ with $2 y + 11$ .

$z = 10 - 2 (2 y + 11) + 3 y$

$10 y + 43 = - 7$

$x = 2 y + 11$

Simplify the right side.

Simplify $10 - 2 (2 y + 11) + 3 y$ .

Simplify each term.

Apply the distributive property.

$z = 10 - 2 (2 y) - 2 \cdot 11 + 3 y$

$10 y + 43 = - 7$

$x = 2 y + 11$

Multiply $2$ by $- 2$ .

$z = 10 - 4 y - 2 \cdot 11 + 3 y$

$10 y + 43 = - 7$

$x = 2 y + 11$

Multiply $- 2$ by $11$ .

$z = 10 - 4 y - 22 + 3 y$

$10 y + 43 = - 7$

$x = 2 y + 11$

$z = 10 - 4 y - 22 + 3 y$

$10 y + 43 = - 7$

$x = 2 y + 11$

Simplify by adding terms.

Subtract $22$ from $10$ .

$z = - 4 y - 12 + 3 y$

$10 y + 43 = - 7$

$x = 2 y + 11$

Add $- 4 y$ and $3 y$ .

$z = - y - 12$

$10 y + 43 = - 7$

$x = 2 y + 11$

$z = - y - 12$

$10 y + 43 = - 7$

$x = 2 y + 11$

$z = - y - 12$

$10 y + 43 = - 7$

$x = 2 y + 11$

$z = - y - 12$

$10 y + 43 = - 7$

$x = 2 y + 11$

$z = - y - 12$

$10 y + 43 = - 7$

$x = 2 y + 11$

Step 5

Solve for $y$ in $10 y + 43 = - 7$ .

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

Subtract $43$ from both sides of the equation.

$10 y = - 7 - 43$

$z = - y - 12$

$x = 2 y + 11$

Subtract $43$ from $- 7$ .

$10 y = - 50$

$z = - y - 12$

$x = 2 y + 11$

$10 y = - 50$

$z = - y - 12$

$x = 2 y + 11$

Divide each term in $10 y = - 50$ by $10$ and simplify.

Divide each term in $10 y = - 50$ by $10$ .

$\frac{10 y}{10} = \frac{- 50}{10}$

$z = - y - 12$

$x = 2 y + 11$

Simplify the left side.

Cancel the common factor of $10$ .

Cancel the common factor.

$\frac{10 y}{10} = \frac{- 50}{10}$

$z = - y - 12$

$x = 2 y + 11$

Divide $y$ by $1$ .

$y = \frac{- 50}{10}$

$z = - y - 12$

$x = 2 y + 11$

$y = \frac{- 50}{10}$

$z = - y - 12$

$x = 2 y + 11$

$y = \frac{- 50}{10}$

$z = - y - 12$

$x = 2 y + 11$

Simplify the right side.

Divide $- 50$ by $10$ .

$y = - 5$

$z = - y - 12$

$x = 2 y + 11$

$y = - 5$

$z = - y - 12$

$x = 2 y + 11$

$y = - 5$

$z = - y - 12$

$x = 2 y + 11$

$y = - 5$

$z = - y - 12$

$x = 2 y + 11$

Step 6

Replace all occurrences of $y$ with $- 5$ in each equation.

Replace all occurrences of $y$ in $z = - y - 12$ with $- 5$ .

$z = - (- 5) - 12$

$y = - 5$

$x = 2 y + 11$

Simplify the right side.

Simplify $- (- 5) - 12$ .

Multiply $- 1$ by $- 5$ .

$z = 5 - 12$

$y = - 5$

$x = 2 y + 11$

Subtract $12$ from $5$ .

$z = - 7$

$y = - 5$

$x = 2 y + 11$

$z = - 7$

$y = - 5$

$x = 2 y + 11$

$z = - 7$

$y = - 5$

$x = 2 y + 11$

Replace all occurrences of $y$ in $x = 2 y + 11$ with $- 5$ .

$x = 2 (- 5) + 11$

$z = - 7$

$y = - 5$

Simplify the right side.

Simplify $2 (- 5) + 11$ .

Multiply $2$ by $- 5$ .

$x = - 10 + 11$

$z = - 7$

$y = - 5$

Add $- 10$ and $11$ .

$x = 1$

$z = - 7$

$y = - 5$

$x = 1$

$z = - 7$

$y = - 5$

$x = 1$

$z = - 7$

$y = - 5$

$x = 1$

$z = - 7$

$y = - 5$

Step 7

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

$(1, - 5, - 7)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(1, - 5, - 7)$

Equation Form:

$x = 1, y = - 5, z = - 7$

Solve the System of Equations 2x-3y+z=10 -x+y-z=1 5x+y+z=-7

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