Solve the System of Equations x+y+z=7 x-y+2z=7 5x+y+z=11

$x + y + z = 7$ $x - y + 2 z = 7$ $5 x + y + z = 11$

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

Subtract $y$ from both sides of the equation.

$x + z = 7 - y$

$x - y + 2 z = 7$

$5 x + y + z = 11$

Subtract $z$ from both sides of the equation.

$x = 7 - y - z$

$x - y + 2 z = 7$

$5 x + y + z = 11$

$x = 7 - y - z$

$x - y + 2 z = 7$

$5 x + y + z = 11$

Replace all occurrences of $x$ with $7 - y - z$ in each equation.

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

$(7 - y - z) - y + 2 z = 7$

$x = 7 - y - z$

$5 x + y + z = 11$

Simplify the left side.

Simplify $(7 - y - z) - y + 2 z$ .

Subtract $y$ from $- y$ .

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

$x = 7 - y - z$

$5 x + y + z = 11$

Add $- z$ and $2 z$ .

$7 - 2 y + z = 7$

$x = 7 - y - z$

$5 x + y + z = 11$

$7 - 2 y + z = 7$

$x = 7 - y - z$

$5 x + y + z = 11$

$7 - 2 y + z = 7$

$x = 7 - y - z$

$5 x + y + z = 11$

Replace all occurrences of $x$ in $5 x + y + z = 11$ with $7 - y - z$ .

$5 (7 - y - z) + y + z = 11$

$7 - 2 y + z = 7$

$x = 7 - y - z$

Simplify the left side.

Simplify $5 (7 - y - z) + y + z$ .

Simplify each term.

Apply the distributive property.

$5 \cdot 7 + 5 (- y) + 5 (- z) + y + z = 11$

$7 - 2 y + z = 7$

$x = 7 - y - z$

Simplify.

Multiply $5$ by $7$ .

$35 + 5 (- y) + 5 (- z) + y + z = 11$

$7 - 2 y + z = 7$

$x = 7 - y - z$

Multiply $- 1$ by $5$ .

$35 - 5 y + 5 (- z) + y + z = 11$

$7 - 2 y + z = 7$

$x = 7 - y - z$

Multiply $- 1$ by $5$ .

$35 - 5 y - 5 z + y + z = 11$

$7 - 2 y + z = 7$

$x = 7 - y - z$

$35 - 5 y - 5 z + y + z = 11$

$7 - 2 y + z = 7$

$x = 7 - y - z$

$35 - 5 y - 5 z + y + z = 11$

$7 - 2 y + z = 7$

$x = 7 - y - z$

Simplify by adding terms.

Add $- 5 y$ and $y$ .

$35 - 4 y - 5 z + z = 11$

$7 - 2 y + z = 7$

$x = 7 - y - z$

Add $- 5 z$ and $z$ .

$35 - 4 y - 4 z = 11$

$7 - 2 y + z = 7$

$x = 7 - y - z$

$35 - 4 y - 4 z = 11$

$7 - 2 y + z = 7$

$x = 7 - y - z$

$35 - 4 y - 4 z = 11$

$7 - 2 y + z = 7$

$x = 7 - y - z$

$35 - 4 y - 4 z = 11$

$7 - 2 y + z = 7$

$x = 7 - y - z$

$35 - 4 y - 4 z = 11$

$7 - 2 y + z = 7$

$x = 7 - y - z$

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

Subtract $7$ from both sides of the equation.

$- 2 y + z = 7 - 7$

$35 - 4 y - 4 z = 11$

$x = 7 - y - z$

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

$z = 7 - 7 + 2 y$

$35 - 4 y - 4 z = 11$

$x = 7 - y - z$

Subtract $7$ from $7$ .

$z = 0 + 2 y$

$35 - 4 y - 4 z = 11$

$x = 7 - y - z$

Add $0$ and $2 y$ .

$z = 2 y$

$35 - 4 y - 4 z = 11$

$x = 7 - y - z$

$z = 2 y$

$35 - 4 y - 4 z = 11$

$x = 7 - y - z$

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

Replace all occurrences of $z$ in $35 - 4 y - 4 z = 11$ with $2 y$ .

$35 - 4 y - 4 (2 y) = 11$

$z = 2 y$

$x = 7 - y - z$

Simplify the left side.

Simplify $35 - 4 y - 4 (2 y)$ .

Multiply $2$ by $- 4$ .

$35 - 4 y - 8 y = 11$

$z = 2 y$

$x = 7 - y - z$

Subtract $8 y$ from $- 4 y$ .

$35 - 12 y = 11$

$z = 2 y$

$x = 7 - y - z$

$35 - 12 y = 11$

$z = 2 y$

$x = 7 - y - z$

$35 - 12 y = 11$

$z = 2 y$

$x = 7 - y - z$

Replace all occurrences of $z$ in $x = 7 - y - z$ with $2 y$ .

$x = 7 - y - (2 y)$

$35 - 12 y = 11$

$z = 2 y$

Simplify the right side.

Simplify $7 - y - (2 y)$ .

Multiply $2$ by $- 1$ .

$x = 7 - y - 2 y$

$35 - 12 y = 11$

$z = 2 y$

Subtract $2 y$ from $- y$ .

$x = 7 - 3 y$

$35 - 12 y = 11$

$z = 2 y$

$x = 7 - 3 y$

$35 - 12 y = 11$

$z = 2 y$

$x = 7 - 3 y$

$35 - 12 y = 11$

$z = 2 y$

$x = 7 - 3 y$

$35 - 12 y = 11$

$z = 2 y$

Solve for $y$ in $35 - 12 y = 11$ .

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

Subtract $35$ from both sides of the equation.

$- 12 y = 11 - 35$

$x = 7 - 3 y$

$z = 2 y$

Subtract $35$ from $11$ .

$- 12 y = - 24$

$x = 7 - 3 y$

$z = 2 y$

$- 12 y = - 24$

$x = 7 - 3 y$

$z = 2 y$

Divide each term in $- 12 y = - 24$ by $- 12$ and simplify.

Divide each term in $- 12 y = - 24$ by $- 12$ .

$\frac{- 12 y}{- 12} = \frac{- 24}{- 12}$

$x = 7 - 3 y$

$z = 2 y$

Simplify the left side.

Cancel the common factor of $- 12$ .

Cancel the common factor.

$\frac{- 12 y}{- 12} = \frac{- 24}{- 12}$

$x = 7 - 3 y$

$z = 2 y$

Divide $y$ by $1$ .

$y = \frac{- 24}{- 12}$

$x = 7 - 3 y$

$z = 2 y$

$y = \frac{- 24}{- 12}$

$x = 7 - 3 y$

$z = 2 y$

$y = \frac{- 24}{- 12}$

$x = 7 - 3 y$

$z = 2 y$

Simplify the right side.

Divide $- 24$ by $- 12$ .

$y = 2$

$x = 7 - 3 y$

$z = 2 y$

$y = 2$

$x = 7 - 3 y$

$z = 2 y$

$y = 2$

$x = 7 - 3 y$

$z = 2 y$

$y = 2$

$x = 7 - 3 y$

$z = 2 y$

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

Replace all occurrences of $y$ in $x = 7 - 3 y$ with $2$ .

$x = 7 - 3 \cdot 2$

$y = 2$

$z = 2 y$

Simplify the right side.

Simplify $7 - 3 \cdot 2$ .

Multiply $- 3$ by $2$ .

$x = 7 - 6$

$y = 2$

$z = 2 y$

Subtract $6$ from $7$ .

$x = 1$

$y = 2$

$z = 2 y$

$x = 1$

$y = 2$

$z = 2 y$

$x = 1$

$y = 2$

$z = 2 y$

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

$z = 2 (2)$

$x = 1$

$y = 2$

Simplify the right side.

Multiply $2$ by $2$ .

$z = 4$

$x = 1$

$y = 2$

$z = 4$

$x = 1$

$y = 2$

$z = 4$

$x = 1$

$y = 2$

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

$(1, 2, 4)$

The result can be shown in multiple forms.

Point Form:

$(1, 2, 4)$

Equation Form:

$x = 1, y = 2, z = 4$

Solve the System of Equations x+y+z=7 x-y+2z=7 5x+y+z=11

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