Solve the System of Equations 2x+2y+6z=16 2x-y+4z=28 -5x-3y-z=-10

$2 x + 2 y + 6 z = 16$ $2 x - y + 4 z = 28$ $- 5 x - 3 y - z = - 10$

Solve for $x$ in $2 x + 2 y + 6 z = 16$ .

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

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

$2 x + 6 z = 16 - 2 y$

$2 x - y + 4 z = 28$

$- 5 x - 3 y - z = - 10$

Subtract $6 z$ from both sides of the equation.

$2 x = 16 - 2 y - 6 z$

$2 x - y + 4 z = 28$

$- 5 x - 3 y - z = - 10$

$2 x = 16 - 2 y - 6 z$

$2 x - y + 4 z = 28$

$- 5 x - 3 y - z = - 10$

Divide each term in $2 x = 16 - 2 y - 6 z$ by $2$ and simplify.

Divide each term in $2 x = 16 - 2 y - 6 z$ by $2$ .

$\frac{2 x}{2} = \frac{16}{2} + \frac{- 2 y}{2} + \frac{- 6 z}{2}$

$2 x - y + 4 z = 28$

$- 5 x - 3 y - z = - 10$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{16}{2} + \frac{- 2 y}{2} + \frac{- 6 z}{2}$

$2 x - y + 4 z = 28$

$- 5 x - 3 y - z = - 10$

Divide $x$ by $1$ .

$x = \frac{16}{2} + \frac{- 2 y}{2} + \frac{- 6 z}{2}$

$2 x - y + 4 z = 28$

$- 5 x - 3 y - z = - 10$

$x = \frac{16}{2} + \frac{- 2 y}{2} + \frac{- 6 z}{2}$

$2 x - y + 4 z = 28$

$- 5 x - 3 y - z = - 10$

$x = \frac{16}{2} + \frac{- 2 y}{2} + \frac{- 6 z}{2}$

$2 x - y + 4 z = 28$

$- 5 x - 3 y - z = - 10$

Simplify the right side.

Simplify each term.

Divide $16$ by $2$ .

$x = 8 + \frac{- 2 y}{2} + \frac{- 6 z}{2}$

$2 x - y + 4 z = 28$

$- 5 x - 3 y - z = - 10$

Cancel the common factor of $- 2$ and $2$ .

Factor $2$ out of $- 2 y$ .

$x = 8 + \frac{2 (- y)}{2} + \frac{- 6 z}{2}$

$2 x - y + 4 z = 28$

$- 5 x - 3 y - z = - 10$

Cancel the common factors.

Factor $2$ out of $2$ .

$x = 8 + \frac{2 (- y)}{2 (1)} + \frac{- 6 z}{2}$

$2 x - y + 4 z = 28$

$- 5 x - 3 y - z = - 10$

Cancel the common factor.

$x = 8 + \frac{2 (- y)}{2 \cdot 1} + \frac{- 6 z}{2}$

$2 x - y + 4 z = 28$

$- 5 x - 3 y - z = - 10$

Rewrite the expression.

$x = 8 + \frac{- y}{1} + \frac{- 6 z}{2}$

$2 x - y + 4 z = 28$

$- 5 x - 3 y - z = - 10$

Divide $- y$ by $1$ .

$x = 8 - y + \frac{- 6 z}{2}$

$2 x - y + 4 z = 28$

$- 5 x - 3 y - z = - 10$

$x = 8 - y + \frac{- 6 z}{2}$

$2 x - y + 4 z = 28$

$- 5 x - 3 y - z = - 10$

$x = 8 - y + \frac{- 6 z}{2}$

$2 x - y + 4 z = 28$

$- 5 x - 3 y - z = - 10$

Cancel the common factor of $- 6$ and $2$ .

Factor $2$ out of $- 6 z$ .

$x = 8 - y + \frac{2 (- 3 z)}{2}$

$2 x - y + 4 z = 28$

$- 5 x - 3 y - z = - 10$

Cancel the common factors.

Factor $2$ out of $2$ .

$x = 8 - y + \frac{2 (- 3 z)}{2 (1)}$

$2 x - y + 4 z = 28$

$- 5 x - 3 y - z = - 10$

Cancel the common factor.

$x = 8 - y + \frac{2 (- 3 z)}{2 \cdot 1}$

$2 x - y + 4 z = 28$

$- 5 x - 3 y - z = - 10$

Rewrite the expression.

$x = 8 - y + \frac{- 3 z}{1}$

$2 x - y + 4 z = 28$

$- 5 x - 3 y - z = - 10$

Divide $- 3 z$ by $1$ .

$x = 8 - y - 3 z$

$2 x - y + 4 z = 28$

$- 5 x - 3 y - z = - 10$

$x = 8 - y - 3 z$

$2 x - y + 4 z = 28$

$- 5 x - 3 y - z = - 10$

$x = 8 - y - 3 z$

$2 x - y + 4 z = 28$

$- 5 x - 3 y - z = - 10$

$x = 8 - y - 3 z$

$2 x - y + 4 z = 28$

$- 5 x - 3 y - z = - 10$

$x = 8 - y - 3 z$

$2 x - y + 4 z = 28$

$- 5 x - 3 y - z = - 10$

$x = 8 - y - 3 z$

$2 x - y + 4 z = 28$

$- 5 x - 3 y - z = - 10$

$x = 8 - y - 3 z$

$2 x - y + 4 z = 28$

$- 5 x - 3 y - z = - 10$

Replace all occurrences of $x$ with $8 - y - 3 z$ in each equation.

Replace all occurrences of $x$ in $2 x - y + 4 z = 28$ with $8 - y - 3 z$ .

$2 (8 - y - 3 z) - y + 4 z = 28$

$x = 8 - y - 3 z$

$- 5 x - 3 y - z = - 10$

Simplify the left side.

Simplify $2 (8 - y - 3 z) - y + 4 z$ .

Simplify each term.

Apply the distributive property.

$2 \cdot 8 + 2 (- y) + 2 (- 3 z) - y + 4 z = 28$

$x = 8 - y - 3 z$

$- 5 x - 3 y - z = - 10$

Simplify.

Multiply $2$ by $8$ .

$16 + 2 (- y) + 2 (- 3 z) - y + 4 z = 28$

$x = 8 - y - 3 z$

$- 5 x - 3 y - z = - 10$

Multiply $- 1$ by $2$ .

$16 - 2 y + 2 (- 3 z) - y + 4 z = 28$

$x = 8 - y - 3 z$

$- 5 x - 3 y - z = - 10$

Multiply $- 3$ by $2$ .

$16 - 2 y - 6 z - y + 4 z = 28$

$x = 8 - y - 3 z$

$- 5 x - 3 y - z = - 10$

$16 - 2 y - 6 z - y + 4 z = 28$

$x = 8 - y - 3 z$

$- 5 x - 3 y - z = - 10$

$16 - 2 y - 6 z - y + 4 z = 28$

$x = 8 - y - 3 z$

$- 5 x - 3 y - z = - 10$

Simplify by adding terms.

Subtract $y$ from $- 2 y$ .

$16 - 3 y - 6 z + 4 z = 28$

$x = 8 - y - 3 z$

$- 5 x - 3 y - z = - 10$

Add $- 6 z$ and $4 z$ .

$16 - 3 y - 2 z = 28$

$x = 8 - y - 3 z$

$- 5 x - 3 y - z = - 10$

$16 - 3 y - 2 z = 28$

$x = 8 - y - 3 z$

$- 5 x - 3 y - z = - 10$

$16 - 3 y - 2 z = 28$

$x = 8 - y - 3 z$

$- 5 x - 3 y - z = - 10$

$16 - 3 y - 2 z = 28$

$x = 8 - y - 3 z$

$- 5 x - 3 y - z = - 10$

Replace all occurrences of $x$ in $- 5 x - 3 y - z = - 10$ with $8 - y - 3 z$ .

$- 5 (8 - y - 3 z) - 3 y - z = - 10$

$16 - 3 y - 2 z = 28$

$x = 8 - y - 3 z$

Simplify the left side.

Simplify $- 5 (8 - y - 3 z) - 3 y - z$ .

Simplify each term.

Apply the distributive property.

$- 5 \cdot 8 - 5 (- y) - 5 (- 3 z) - 3 y - z = - 10$

$16 - 3 y - 2 z = 28$

$x = 8 - y - 3 z$

Simplify.

Multiply $- 5$ by $8$ .

$- 40 - 5 (- y) - 5 (- 3 z) - 3 y - z = - 10$

$16 - 3 y - 2 z = 28$

$x = 8 - y - 3 z$

Multiply $- 1$ by $- 5$ .

$- 40 + 5 y - 5 (- 3 z) - 3 y - z = - 10$

$16 - 3 y - 2 z = 28$

$x = 8 - y - 3 z$

Multiply $- 3$ by $- 5$ .

$- 40 + 5 y + 15 z - 3 y - z = - 10$

$16 - 3 y - 2 z = 28$

$x = 8 - y - 3 z$

$- 40 + 5 y + 15 z - 3 y - z = - 10$

$16 - 3 y - 2 z = 28$

$x = 8 - y - 3 z$

$- 40 + 5 y + 15 z - 3 y - z = - 10$

$16 - 3 y - 2 z = 28$

$x = 8 - y - 3 z$

Simplify by adding terms.

Subtract $3 y$ from $5 y$ .

$- 40 + 2 y + 15 z - z = - 10$

$16 - 3 y - 2 z = 28$

$x = 8 - y - 3 z$

Subtract $z$ from $15 z$ .

$- 40 + 2 y + 14 z = - 10$

$16 - 3 y - 2 z = 28$

$x = 8 - y - 3 z$

$- 40 + 2 y + 14 z = - 10$

$16 - 3 y - 2 z = 28$

$x = 8 - y - 3 z$

$- 40 + 2 y + 14 z = - 10$

$16 - 3 y - 2 z = 28$

$x = 8 - y - 3 z$

$- 40 + 2 y + 14 z = - 10$

$16 - 3 y - 2 z = 28$

$x = 8 - y - 3 z$

$- 40 + 2 y + 14 z = - 10$

$16 - 3 y - 2 z = 28$

$x = 8 - y - 3 z$

Solve for $y$ in $- 40 + 2 y + 14 z = - 10$ .

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

Add $40$ to both sides of the equation.

$2 y + 14 z = - 10 + 40$

$16 - 3 y - 2 z = 28$

$x = 8 - y - 3 z$

Subtract $14 z$ from both sides of the equation.

$2 y = - 10 + 40 - 14 z$

$16 - 3 y - 2 z = 28$

$x = 8 - y - 3 z$

Add $- 10$ and $40$ .

$2 y = 30 - 14 z$

$16 - 3 y - 2 z = 28$

$x = 8 - y - 3 z$

$2 y = 30 - 14 z$

$16 - 3 y - 2 z = 28$

$x = 8 - y - 3 z$

Divide each term in $2 y = 30 - 14 z$ by $2$ and simplify.

Divide each term in $2 y = 30 - 14 z$ by $2$ .

$\frac{2 y}{2} = \frac{30}{2} + \frac{- 14 z}{2}$

$16 - 3 y - 2 z = 28$

$x = 8 - y - 3 z$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 y}{2} = \frac{30}{2} + \frac{- 14 z}{2}$

$16 - 3 y - 2 z = 28$

$x = 8 - y - 3 z$

Divide $y$ by $1$ .

$y = \frac{30}{2} + \frac{- 14 z}{2}$

$16 - 3 y - 2 z = 28$

$x = 8 - y - 3 z$

$y = \frac{30}{2} + \frac{- 14 z}{2}$

$16 - 3 y - 2 z = 28$

$x = 8 - y - 3 z$

$y = \frac{30}{2} + \frac{- 14 z}{2}$

$16 - 3 y - 2 z = 28$

$x = 8 - y - 3 z$

Simplify the right side.

Simplify each term.

Divide $30$ by $2$ .

$y = 15 + \frac{- 14 z}{2}$

$16 - 3 y - 2 z = 28$

$x = 8 - y - 3 z$

Cancel the common factor of $- 14$ and $2$ .

Factor $2$ out of $- 14 z$ .

$y = 15 + \frac{2 (- 7 z)}{2}$

$16 - 3 y - 2 z = 28$

$x = 8 - y - 3 z$

Cancel the common factors.

Factor $2$ out of $2$ .

$y = 15 + \frac{2 (- 7 z)}{2 (1)}$

$16 - 3 y - 2 z = 28$

$x = 8 - y - 3 z$

Cancel the common factor.

$y = 15 + \frac{2 (- 7 z)}{2 \cdot 1}$

$16 - 3 y - 2 z = 28$

$x = 8 - y - 3 z$

Rewrite the expression.

$y = 15 + \frac{- 7 z}{1}$

$16 - 3 y - 2 z = 28$

$x = 8 - y - 3 z$

Divide $- 7 z$ by $1$ .

$y = 15 - 7 z$

$16 - 3 y - 2 z = 28$

$x = 8 - y - 3 z$

$y = 15 - 7 z$

$16 - 3 y - 2 z = 28$

$x = 8 - y - 3 z$

$y = 15 - 7 z$

$16 - 3 y - 2 z = 28$

$x = 8 - y - 3 z$

$y = 15 - 7 z$

$16 - 3 y - 2 z = 28$

$x = 8 - y - 3 z$

$y = 15 - 7 z$

$16 - 3 y - 2 z = 28$

$x = 8 - y - 3 z$

$y = 15 - 7 z$

$16 - 3 y - 2 z = 28$

$x = 8 - y - 3 z$

$y = 15 - 7 z$

$16 - 3 y - 2 z = 28$

$x = 8 - y - 3 z$

Replace all occurrences of $y$ with $15 - 7 z$ in each equation.

Replace all occurrences of $y$ in $16 - 3 y - 2 z = 28$ with $15 - 7 z$ .

$16 - 3 (15 - 7 z) - 2 z = 28$

$y = 15 - 7 z$

$x = 8 - y - 3 z$

Simplify the left side.

Simplify $16 - 3 (15 - 7 z) - 2 z$ .

Simplify each term.

Apply the distributive property.

$16 - 3 \cdot 15 - 3 (- 7 z) - 2 z = 28$

$y = 15 - 7 z$

$x = 8 - y - 3 z$

Multiply $- 3$ by $15$ .

$16 - 45 - 3 (- 7 z) - 2 z = 28$

$y = 15 - 7 z$

$x = 8 - y - 3 z$

Multiply $- 7$ by $- 3$ .

$16 - 45 + 21 z - 2 z = 28$

$y = 15 - 7 z$

$x = 8 - y - 3 z$

$16 - 45 + 21 z - 2 z = 28$

$y = 15 - 7 z$

$x = 8 - y - 3 z$

Simplify by adding terms.

Subtract $45$ from $16$ .

$- 29 + 21 z - 2 z = 28$

$y = 15 - 7 z$

$x = 8 - y - 3 z$

Subtract $2 z$ from $21 z$ .

$- 29 + 19 z = 28$

$y = 15 - 7 z$

$x = 8 - y - 3 z$

$- 29 + 19 z = 28$

$y = 15 - 7 z$

$x = 8 - y - 3 z$

$- 29 + 19 z = 28$

$y = 15 - 7 z$

$x = 8 - y - 3 z$

$- 29 + 19 z = 28$

$y = 15 - 7 z$

$x = 8 - y - 3 z$

Replace all occurrences of $y$ in $x = 8 - y - 3 z$ with $15 - 7 z$ .

$x = 8 - (15 - 7 z) - 3 z$

$- 29 + 19 z = 28$

$y = 15 - 7 z$

Simplify the right side.

Simplify $8 - (15 - 7 z) - 3 z$ .

Simplify each term.

Apply the distributive property.

$x = 8 - 1 \cdot 15 - (- 7 z) - 3 z$

$- 29 + 19 z = 28$

$y = 15 - 7 z$

Multiply $- 1$ by $15$ .

$x = 8 - 15 - (- 7 z) - 3 z$

$- 29 + 19 z = 28$

$y = 15 - 7 z$

Multiply $- 7$ by $- 1$ .

$x = 8 - 15 + 7 z - 3 z$

$- 29 + 19 z = 28$

$y = 15 - 7 z$

$x = 8 - 15 + 7 z - 3 z$

$- 29 + 19 z = 28$

$y = 15 - 7 z$

Simplify by adding terms.

Subtract $15$ from $8$ .

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

$- 29 + 19 z = 28$

$y = 15 - 7 z$

Subtract $3 z$ from $7 z$ .

$x = - 7 + 4 z$

$- 29 + 19 z = 28$

$y = 15 - 7 z$

$x = - 7 + 4 z$

$- 29 + 19 z = 28$

$y = 15 - 7 z$

$x = - 7 + 4 z$

$- 29 + 19 z = 28$

$y = 15 - 7 z$

$x = - 7 + 4 z$

$- 29 + 19 z = 28$

$y = 15 - 7 z$

$x = - 7 + 4 z$

$- 29 + 19 z = 28$

$y = 15 - 7 z$

Solve for $z$ in $- 29 + 19 z = 28$ .

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

Add $29$ to both sides of the equation.

$19 z = 28 + 29$

$x = - 7 + 4 z$

$y = 15 - 7 z$

Add $28$ and $29$ .

$19 z = 57$

$x = - 7 + 4 z$

$y = 15 - 7 z$

$19 z = 57$

$x = - 7 + 4 z$

$y = 15 - 7 z$

Divide each term in $19 z = 57$ by $19$ and simplify.

Divide each term in $19 z = 57$ by $19$ .

$\frac{19 z}{19} = \frac{57}{19}$

$x = - 7 + 4 z$

$y = 15 - 7 z$

Simplify the left side.

Cancel the common factor of $19$ .

Cancel the common factor.

$\frac{19 z}{19} = \frac{57}{19}$

$x = - 7 + 4 z$

$y = 15 - 7 z$

Divide $z$ by $1$ .

$z = \frac{57}{19}$

$x = - 7 + 4 z$

$y = 15 - 7 z$

$z = \frac{57}{19}$

$x = - 7 + 4 z$

$y = 15 - 7 z$

$z = \frac{57}{19}$

$x = - 7 + 4 z$

$y = 15 - 7 z$

Simplify the right side.

Divide $57$ by $19$ .

$z = 3$

$x = - 7 + 4 z$

$y = 15 - 7 z$

$z = 3$

$x = - 7 + 4 z$

$y = 15 - 7 z$

$z = 3$

$x = - 7 + 4 z$

$y = 15 - 7 z$

$z = 3$

$x = - 7 + 4 z$

$y = 15 - 7 z$

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

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

$x = - 7 + 4 (3)$

$z = 3$

$y = 15 - 7 z$

Simplify the right side.

Simplify $- 7 + 4 (3)$ .

Multiply $4$ by $3$ .

$x = - 7 + 12$

$z = 3$

$y = 15 - 7 z$

Add $- 7$ and $12$ .

$x = 5$

$z = 3$

$y = 15 - 7 z$

$x = 5$

$z = 3$

$y = 15 - 7 z$

$x = 5$

$z = 3$

$y = 15 - 7 z$

Replace all occurrences of $z$ in $y = 15 - 7 z$ with $3$ .

$y = 15 - 7 \cdot 3$

$x = 5$

$z = 3$

Simplify the right side.

Simplify $15 - 7 \cdot 3$ .

Multiply $- 7$ by $3$ .

$y = 15 - 21$

$x = 5$

$z = 3$

Subtract $21$ from $15$ .

$y = - 6$

$x = 5$

$z = 3$

$y = - 6$

$x = 5$

$z = 3$

$y = - 6$

$x = 5$

$z = 3$

$y = - 6$

$x = 5$

$z = 3$

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

$(5, - 6, 3)$

The result can be shown in multiple forms.

Point Form:

$(5, - 6, 3)$

Equation Form:

$x = 5, y = - 6, z = 3$

Solve the System of Equations 2x+2y+6z=16 2x-y+4z=28 -5x-3y-z=-10

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