Solve the System of Equations -2x+3y+4z=5 6x+7y+z=9 3x+7y+z=-9

$- 2 x + 3 y + 4 z = 5$ $6 x + 7 y + z = 9$ $3 x + 7 y + z = - 9$

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

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

$7 y + z = 9 - 6 x$

$- 2 x + 3 y + 4 z = 5$

$3 x + 7 y + z = - 9$

Subtract $7 y$ from both sides of the equation.

$z = 9 - 6 x - 7 y$

$- 2 x + 3 y + 4 z = 5$

$3 x + 7 y + z = - 9$

$z = 9 - 6 x - 7 y$

$- 2 x + 3 y + 4 z = 5$

$3 x + 7 y + z = - 9$

Replace all occurrences of $z$ with $9 - 6 x - 7 y$ in each equation.

Replace all occurrences of $z$ in $- 2 x + 3 y + 4 z = 5$ with $9 - 6 x - 7 y$ .

$- 2 x + 3 y + 4 (9 - 6 x - 7 y) = 5$

$z = 9 - 6 x - 7 y$

$3 x + 7 y + z = - 9$

Simplify the left side.

Simplify $- 2 x + 3 y + 4 (9 - 6 x - 7 y)$ .

Simplify each term.

Apply the distributive property.

$- 2 x + 3 y + 4 \cdot 9 + 4 (- 6 x) + 4 (- 7 y) = 5$

$z = 9 - 6 x - 7 y$

$3 x + 7 y + z = - 9$

Simplify.

Multiply $4$ by $9$ .

$- 2 x + 3 y + 36 + 4 (- 6 x) + 4 (- 7 y) = 5$

$z = 9 - 6 x - 7 y$

$3 x + 7 y + z = - 9$

Multiply $- 6$ by $4$ .

$- 2 x + 3 y + 36 - 24 x + 4 (- 7 y) = 5$

$z = 9 - 6 x - 7 y$

$3 x + 7 y + z = - 9$

Multiply $- 7$ by $4$ .

$- 2 x + 3 y + 36 - 24 x - 28 y = 5$

$z = 9 - 6 x - 7 y$

$3 x + 7 y + z = - 9$

$- 2 x + 3 y + 36 - 24 x - 28 y = 5$

$z = 9 - 6 x - 7 y$

$3 x + 7 y + z = - 9$

$- 2 x + 3 y + 36 - 24 x - 28 y = 5$

$z = 9 - 6 x - 7 y$

$3 x + 7 y + z = - 9$

Simplify by adding terms.

Subtract $24 x$ from $- 2 x$ .

$- 26 x + 3 y + 36 - 28 y = 5$

$z = 9 - 6 x - 7 y$

$3 x + 7 y + z = - 9$

Subtract $28 y$ from $3 y$ .

$- 26 x - 25 y + 36 = 5$

$z = 9 - 6 x - 7 y$

$3 x + 7 y + z = - 9$

$- 26 x - 25 y + 36 = 5$

$z = 9 - 6 x - 7 y$

$3 x + 7 y + z = - 9$

$- 26 x - 25 y + 36 = 5$

$z = 9 - 6 x - 7 y$

$3 x + 7 y + z = - 9$

$- 26 x - 25 y + 36 = 5$

$z = 9 - 6 x - 7 y$

$3 x + 7 y + z = - 9$

Replace all occurrences of $z$ in $3 x + 7 y + z = - 9$ with $9 - 6 x - 7 y$ .

$3 x + 7 y + 9 - 6 x - 7 y = - 9$

$- 26 x - 25 y + 36 = 5$

$z = 9 - 6 x - 7 y$

Simplify the left side.

Simplify $3 x + 7 y + 9 - 6 x - 7 y$ .

Remove parentheses.

$3 x + 7 y + 9 - 6 x - 7 y = - 9$

$- 26 x - 25 y + 36 = 5$

$z = 9 - 6 x - 7 y$

Combine the opposite terms in $3 x + 7 y + 9 - 6 x - 7 y$ .

Subtract $7 y$ from $7 y$ .

$3 x + 9 - 6 x + 0 = - 9$

$- 26 x - 25 y + 36 = 5$

$z = 9 - 6 x - 7 y$

Add $3 x + 9 - 6 x$ and $0$ .

$3 x + 9 - 6 x = - 9$

$- 26 x - 25 y + 36 = 5$

$z = 9 - 6 x - 7 y$

$3 x + 9 - 6 x = - 9$

$- 26 x - 25 y + 36 = 5$

$z = 9 - 6 x - 7 y$

Subtract $6 x$ from $3 x$ .

$- 3 x + 9 = - 9$

$- 26 x - 25 y + 36 = 5$

$z = 9 - 6 x - 7 y$

$- 3 x + 9 = - 9$

$- 26 x - 25 y + 36 = 5$

$z = 9 - 6 x - 7 y$

$- 3 x + 9 = - 9$

$- 26 x - 25 y + 36 = 5$

$z = 9 - 6 x - 7 y$

$- 3 x + 9 = - 9$

$- 26 x - 25 y + 36 = 5$

$z = 9 - 6 x - 7 y$

Solve for $x$ in $- 3 x + 9 = - 9$ .

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

Subtract $9$ from both sides of the equation.

$- 3 x = - 9 - 9$

$- 26 x - 25 y + 36 = 5$

$z = 9 - 6 x - 7 y$

Subtract $9$ from $- 9$ .

$- 3 x = - 18$

$- 26 x - 25 y + 36 = 5$

$z = 9 - 6 x - 7 y$

$- 3 x = - 18$

$- 26 x - 25 y + 36 = 5$

$z = 9 - 6 x - 7 y$

Divide each term in $- 3 x = - 18$ by $- 3$ and simplify.

Divide each term in $- 3 x = - 18$ by $- 3$ .

$\frac{- 3 x}{- 3} = \frac{- 18}{- 3}$

$- 26 x - 25 y + 36 = 5$

$z = 9 - 6 x - 7 y$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 x}{- 3} = \frac{- 18}{- 3}$

$- 26 x - 25 y + 36 = 5$

$z = 9 - 6 x - 7 y$

Divide $x$ by $1$ .

$x = \frac{- 18}{- 3}$

$- 26 x - 25 y + 36 = 5$

$z = 9 - 6 x - 7 y$

$x = \frac{- 18}{- 3}$

$- 26 x - 25 y + 36 = 5$

$z = 9 - 6 x - 7 y$

$x = \frac{- 18}{- 3}$

$- 26 x - 25 y + 36 = 5$

$z = 9 - 6 x - 7 y$

Simplify the right side.

Divide $- 18$ by $- 3$ .

$x = 6$

$- 26 x - 25 y + 36 = 5$

$z = 9 - 6 x - 7 y$

$x = 6$

$- 26 x - 25 y + 36 = 5$

$z = 9 - 6 x - 7 y$

$x = 6$

$- 26 x - 25 y + 36 = 5$

$z = 9 - 6 x - 7 y$

$x = 6$

$- 26 x - 25 y + 36 = 5$

$z = 9 - 6 x - 7 y$

Replace all occurrences of $x$ with $6$ in each equation.

Replace all occurrences of $x$ in $- 26 x - 25 y + 36 = 5$ with $6$ .

$- 26 \cdot 6 - 25 y + 36 = 5$

$x = 6$

$z = 9 - 6 x - 7 y$

Simplify the left side.

Simplify $- 26 \cdot 6 - 25 y + 36$ .

Multiply $- 26$ by $6$ .

$- 156 - 25 y + 36 = 5$

$x = 6$

$z = 9 - 6 x - 7 y$

Add $- 156$ and $36$ .

$- 25 y - 120 = 5$

$x = 6$

$z = 9 - 6 x - 7 y$

$- 25 y - 120 = 5$

$x = 6$

$z = 9 - 6 x - 7 y$

$- 25 y - 120 = 5$

$x = 6$

$z = 9 - 6 x - 7 y$

Replace all occurrences of $x$ in $z = 9 - 6 x - 7 y$ with $6$ .

$z = 9 - 6 \cdot 6 - 7 y$

$- 25 y - 120 = 5$

$x = 6$

Simplify the right side.

Simplify $9 - 6 \cdot 6 - 7 y$ .

Multiply $- 6$ by $6$ .

$z = 9 - 36 - 7 y$

$- 25 y - 120 = 5$

$x = 6$

Subtract $36$ from $9$ .

$z = - 27 - 7 y$

$- 25 y - 120 = 5$

$x = 6$

$z = - 27 - 7 y$

$- 25 y - 120 = 5$

$x = 6$

$z = - 27 - 7 y$

$- 25 y - 120 = 5$

$x = 6$

$z = - 27 - 7 y$

$- 25 y - 120 = 5$

$x = 6$

Solve for $y$ in $- 25 y - 120 = 5$ .

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

Add $120$ to both sides of the equation.

$- 25 y = 5 + 120$

$z = - 27 - 7 y$

$x = 6$

Add $5$ and $120$ .

$- 25 y = 125$

$z = - 27 - 7 y$

$x = 6$

$- 25 y = 125$

$z = - 27 - 7 y$

$x = 6$

Divide each term in $- 25 y = 125$ by $- 25$ and simplify.

Divide each term in $- 25 y = 125$ by $- 25$ .

$\frac{- 25 y}{- 25} = \frac{125}{- 25}$

$z = - 27 - 7 y$

$x = 6$

Simplify the left side.

Cancel the common factor of $- 25$ .

Cancel the common factor.

$\frac{- 25 y}{- 25} = \frac{125}{- 25}$

$z = - 27 - 7 y$

$x = 6$

Divide $y$ by $1$ .

$y = \frac{125}{- 25}$

$z = - 27 - 7 y$

$x = 6$

$y = \frac{125}{- 25}$

$z = - 27 - 7 y$

$x = 6$

$y = \frac{125}{- 25}$

$z = - 27 - 7 y$

$x = 6$

Simplify the right side.

Divide $125$ by $- 25$ .

$y = - 5$

$z = - 27 - 7 y$

$x = 6$

$y = - 5$

$z = - 27 - 7 y$

$x = 6$

$y = - 5$

$z = - 27 - 7 y$

$x = 6$

$y = - 5$

$z = - 27 - 7 y$

$x = 6$

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

Replace all occurrences of $y$ in $z = - 27 - 7 y$ with $- 5$ .

$z = - 27 - 7 \cdot - 5$

$y = - 5$

$x = 6$

Simplify the right side.

Simplify $- 27 - 7 \cdot - 5$ .

Multiply $- 7$ by $- 5$ .

$z = - 27 + 35$

$y = - 5$

$x = 6$

Add $- 27$ and $35$ .

$z = 8$

$y = - 5$

$x = 6$

$z = 8$

$y = - 5$

$x = 6$

$z = 8$

$y = - 5$

$x = 6$

$z = 8$

$y = - 5$

$x = 6$

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

$(6, - 5, 8)$

The result can be shown in multiple forms.

Point Form:

$(6, - 5, 8)$

Equation Form:

$x = 6, y = - 5, z = 8$

Solve the System of Equations -2x+3y+4z=5 6x+7y+z=9 3x+7y+z=-9

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