Solve the System of Equations -7x+9y+10z=10 -x+y+z=3 x-2y-z=1

$- 7 x + 9 y + 10 z = 10$ $- x + y + z = 3$ $x - 2 y - z = 1$

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

Add $x$ to both sides of the equation.

$y + z = 3 + x$

$- 7 x + 9 y + 10 z = 10$

$x - 2 y - z = 1$

Subtract $z$ from both sides of the equation.

$y = 3 + x - z$

$- 7 x + 9 y + 10 z = 10$

$x - 2 y - z = 1$

$y = 3 + x - z$

$- 7 x + 9 y + 10 z = 10$

$x - 2 y - z = 1$

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

Replace all occurrences of $y$ in $- 7 x + 9 y + 10 z = 10$ with $3 + x - z$ .

$- 7 x + 9 (3 + x - z) + 10 z = 10$

$y = 3 + x - z$

$x - 2 y - z = 1$

Simplify the left side.

Simplify $- 7 x + 9 (3 + x - z) + 10 z$ .

Simplify each term.

Apply the distributive property.

$- 7 x + 9 \cdot 3 + 9 x + 9 (- z) + 10 z = 10$

$y = 3 + x - z$

$x - 2 y - z = 1$

Simplify.

Multiply $9$ by $3$ .

$- 7 x + 27 + 9 x + 9 (- z) + 10 z = 10$

$y = 3 + x - z$

$x - 2 y - z = 1$

Multiply $- 1$ by $9$ .

$- 7 x + 27 + 9 x - 9 z + 10 z = 10$

$y = 3 + x - z$

$x - 2 y - z = 1$

$- 7 x + 27 + 9 x - 9 z + 10 z = 10$

$y = 3 + x - z$

$x - 2 y - z = 1$

$- 7 x + 27 + 9 x - 9 z + 10 z = 10$

$y = 3 + x - z$

$x - 2 y - z = 1$

Simplify by adding terms.

Add $- 7 x$ and $9 x$ .

$2 x + 27 - 9 z + 10 z = 10$

$y = 3 + x - z$

$x - 2 y - z = 1$

Add $- 9 z$ and $10 z$ .

$2 x + 27 + z = 10$

$y = 3 + x - z$

$x - 2 y - z = 1$

$2 x + 27 + z = 10$

$y = 3 + x - z$

$x - 2 y - z = 1$

$2 x + 27 + z = 10$

$y = 3 + x - z$

$x - 2 y - z = 1$

$2 x + 27 + z = 10$

$y = 3 + x - z$

$x - 2 y - z = 1$

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

$x - 2 (3 + x - z) - z = 1$

$2 x + 27 + z = 10$

$y = 3 + x - z$

Simplify the left side.

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

Simplify each term.

Apply the distributive property.

$x - 2 \cdot 3 - 2 x - 2 (- z) - z = 1$

$2 x + 27 + z = 10$

$y = 3 + x - z$

Simplify.

Multiply $- 2$ by $3$ .

$x - 6 - 2 x - 2 (- z) - z = 1$

$2 x + 27 + z = 10$

$y = 3 + x - z$

Multiply $- 1$ by $- 2$ .

$x - 6 - 2 x + 2 z - z = 1$

$2 x + 27 + z = 10$

$y = 3 + x - z$

$x - 6 - 2 x + 2 z - z = 1$

$2 x + 27 + z = 10$

$y = 3 + x - z$

$x - 6 - 2 x + 2 z - z = 1$

$2 x + 27 + z = 10$

$y = 3 + x - z$

Simplify by adding terms.

Subtract $2 x$ from $x$ .

$- x - 6 + 2 z - z = 1$

$2 x + 27 + z = 10$

$y = 3 + x - z$

Subtract $z$ from $2 z$ .

$- x - 6 + z = 1$

$2 x + 27 + z = 10$

$y = 3 + x - z$

$- x - 6 + z = 1$

$2 x + 27 + z = 10$

$y = 3 + x - z$

$- x - 6 + z = 1$

$2 x + 27 + z = 10$

$y = 3 + x - z$

$- x - 6 + z = 1$

$2 x + 27 + z = 10$

$y = 3 + x - z$

$- x - 6 + z = 1$

$2 x + 27 + z = 10$

$y = 3 + x - z$

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

Add $x$ to both sides of the equation.

$- 6 + z = 1 + x$

$2 x + 27 + z = 10$

$y = 3 + x - z$

Add $6$ to both sides of the equation.

$z = 1 + x + 6$

$2 x + 27 + z = 10$

$y = 3 + x - z$

Add $1$ and $6$ .

$z = x + 7$

$2 x + 27 + z = 10$

$y = 3 + x - z$

$z = x + 7$

$2 x + 27 + z = 10$

$y = 3 + x - z$

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

Replace all occurrences of $z$ in $2 x + 27 + z = 10$ with $x + 7$ .

$2 x + 27 + x + 7 = 10$

$z = x + 7$

$y = 3 + x - z$

Simplify the left side.

Simplify $2 x + 27 + x + 7$ .

Remove parentheses.

$2 x + 27 + x + 7 = 10$

$z = x + 7$

$y = 3 + x - z$

Add $2 x$ and $x$ .

$3 x + 27 + 7 = 10$

$z = x + 7$

$y = 3 + x - z$

Add $27$ and $7$ .

$3 x + 34 = 10$

$z = x + 7$

$y = 3 + x - z$

$3 x + 34 = 10$

$z = x + 7$

$y = 3 + x - z$

$3 x + 34 = 10$

$z = x + 7$

$y = 3 + x - z$

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

$y = 3 + x - (x + 7)$

$3 x + 34 = 10$

$z = x + 7$

Simplify the right side.

Simplify $3 + x - (x + 7)$ .

Simplify each term.

Apply the distributive property.

$y = 3 + x - x - 1 \cdot 7$

$3 x + 34 = 10$

$z = x + 7$

Multiply $- 1$ by $7$ .

$y = 3 + x - x - 7$

$3 x + 34 = 10$

$z = x + 7$

$y = 3 + x - x - 7$

$3 x + 34 = 10$

$z = x + 7$

Simplify by adding terms.

Combine the opposite terms in $3 + x - x - 7$ .

Subtract $x$ from $x$ .

$y = 3 + 0 - 7$

$3 x + 34 = 10$

$z = x + 7$

Add $3$ and $0$ .

$y = 3 - 7$

$3 x + 34 = 10$

$z = x + 7$

$y = 3 - 7$

$3 x + 34 = 10$

$z = x + 7$

Subtract $7$ from $3$ .

$y = - 4$

$3 x + 34 = 10$

$z = x + 7$

$y = - 4$

$3 x + 34 = 10$

$z = x + 7$

$y = - 4$

$3 x + 34 = 10$

$z = x + 7$

$y = - 4$

$3 x + 34 = 10$

$z = x + 7$

$y = - 4$

$3 x + 34 = 10$

$z = x + 7$

Solve for $x$ in $3 x + 34 = 10$ .

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

Subtract $34$ from both sides of the equation.

$3 x = 10 - 34$

$y = - 4$

$z = x + 7$

Subtract $34$ from $10$ .

$3 x = - 24$

$y = - 4$

$z = x + 7$

$3 x = - 24$

$y = - 4$

$z = x + 7$

Divide each term in $3 x = - 24$ by $3$ and simplify.

Divide each term in $3 x = - 24$ by $3$ .

$\frac{3 x}{3} = \frac{- 24}{3}$

$y = - 4$

$z = x + 7$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 24}{3}$

$y = - 4$

$z = x + 7$

Divide $x$ by $1$ .

$x = \frac{- 24}{3}$

$y = - 4$

$z = x + 7$

$x = \frac{- 24}{3}$

$y = - 4$

$z = x + 7$

$x = \frac{- 24}{3}$

$y = - 4$

$z = x + 7$

Simplify the right side.

Divide $- 24$ by $3$ .

$x = - 8$

$y = - 4$

$z = x + 7$

$x = - 8$

$y = - 4$

$z = x + 7$

$x = - 8$

$y = - 4$

$z = x + 7$

$x = - 8$

$y = - 4$

$z = x + 7$

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

Replace all occurrences of $x$ in $z = x + 7$ with $- 8$ .

$z = (- 8) + 7$

$x = - 8$

$y = - 4$

Simplify the right side.

Add $- 8$ and $7$ .

$z = - 1$

$x = - 8$

$y = - 4$

$z = - 1$

$x = - 8$

$y = - 4$

$z = - 1$

$x = - 8$

$y = - 4$

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

$(- 8, - 4, - 1)$

The result can be shown in multiple forms.

Point Form:

$(- 8, - 4, - 1)$

Equation Form:

$x = - 8, y = - 4, z = - 1$

Solve the System of Equations -7x+9y+10z=10 -x+y+z=3 x-2y-z=1

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