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

$x + 5 y + 4 z = 3$ $x + 3 y + 4 z = - 3$ $- x + 7 y + 4 z = - 7$

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

Subtract $5 y$ from both sides of the equation.

$x + 4 z = 3 - 5 y$

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

$- x + 7 y + 4 z = - 7$

Subtract $4 z$ from both sides of the equation.

$x = 3 - 5 y - 4 z$

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

$- x + 7 y + 4 z = - 7$

$x = 3 - 5 y - 4 z$

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

$- x + 7 y + 4 z = - 7$

Replace all occurrences of $x$ with $3 - 5 y - 4 z$ in each equation.

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

$(3 - 5 y - 4 z) + 3 y + 4 z = - 3$

$x = 3 - 5 y - 4 z$

$- x + 7 y + 4 z = - 7$

Simplify the left side.

Simplify $(3 - 5 y - 4 z) + 3 y + 4 z$ .

Combine the opposite terms in $(3 - 5 y - 4 z) + 3 y + 4 z$ .

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

$(3 - 5 y) + 3 y + 0 = - 3$

$x = 3 - 5 y - 4 z$

$- x + 7 y + 4 z = - 7$

Add $(3 - 5 y) + 3 y$ and $0$ .

$(3 - 5 y) + 3 y = - 3$

$x = 3 - 5 y - 4 z$

$- x + 7 y + 4 z = - 7$

$(3 - 5 y) + 3 y = - 3$

$x = 3 - 5 y - 4 z$

$- x + 7 y + 4 z = - 7$

Add $- 5 y$ and $3 y$ .

$3 - 2 y = - 3$

$x = 3 - 5 y - 4 z$

$- x + 7 y + 4 z = - 7$

$3 - 2 y = - 3$

$x = 3 - 5 y - 4 z$

$- x + 7 y + 4 z = - 7$

$3 - 2 y = - 3$

$x = 3 - 5 y - 4 z$

$- x + 7 y + 4 z = - 7$

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

$- (3 - 5 y - 4 z) + 7 y + 4 z = - 7$

$3 - 2 y = - 3$

$x = 3 - 5 y - 4 z$

Simplify the left side.

Simplify $- (3 - 5 y - 4 z) + 7 y + 4 z$ .

Simplify each term.

Apply the distributive property.

$- 1 \cdot 3 - (- 5 y) - (- 4 z) + 7 y + 4 z = - 7$

$3 - 2 y = - 3$

$x = 3 - 5 y - 4 z$

Simplify.

Multiply $- 1$ by $3$ .

$- 3 - (- 5 y) - (- 4 z) + 7 y + 4 z = - 7$

$3 - 2 y = - 3$

$x = 3 - 5 y - 4 z$

Multiply $- 5$ by $- 1$ .

$- 3 + 5 y - (- 4 z) + 7 y + 4 z = - 7$

$3 - 2 y = - 3$

$x = 3 - 5 y - 4 z$

Multiply $- 4$ by $- 1$ .

$- 3 + 5 y + 4 z + 7 y + 4 z = - 7$

$3 - 2 y = - 3$

$x = 3 - 5 y - 4 z$

$- 3 + 5 y + 4 z + 7 y + 4 z = - 7$

$3 - 2 y = - 3$

$x = 3 - 5 y - 4 z$

$- 3 + 5 y + 4 z + 7 y + 4 z = - 7$

$3 - 2 y = - 3$

$x = 3 - 5 y - 4 z$

Simplify by adding terms.

Add $5 y$ and $7 y$ .

$- 3 + 12 y + 4 z + 4 z = - 7$

$3 - 2 y = - 3$

$x = 3 - 5 y - 4 z$

Add $4 z$ and $4 z$ .

$- 3 + 12 y + 8 z = - 7$

$3 - 2 y = - 3$

$x = 3 - 5 y - 4 z$

$- 3 + 12 y + 8 z = - 7$

$3 - 2 y = - 3$

$x = 3 - 5 y - 4 z$

$- 3 + 12 y + 8 z = - 7$

$3 - 2 y = - 3$

$x = 3 - 5 y - 4 z$

$- 3 + 12 y + 8 z = - 7$

$3 - 2 y = - 3$

$x = 3 - 5 y - 4 z$

$- 3 + 12 y + 8 z = - 7$

$3 - 2 y = - 3$

$x = 3 - 5 y - 4 z$

Solve for $y$ in $3 - 2 y = - 3$ .

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

Subtract $3$ from both sides of the equation.

$- 2 y = - 3 - 3$

$- 3 + 12 y + 8 z = - 7$

$x = 3 - 5 y - 4 z$

Subtract $3$ from $- 3$ .

$- 2 y = - 6$

$- 3 + 12 y + 8 z = - 7$

$x = 3 - 5 y - 4 z$

$- 2 y = - 6$

$- 3 + 12 y + 8 z = - 7$

$x = 3 - 5 y - 4 z$

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

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

$\frac{- 2 y}{- 2} = \frac{- 6}{- 2}$

$- 3 + 12 y + 8 z = - 7$

$x = 3 - 5 y - 4 z$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 y}{- 2} = \frac{- 6}{- 2}$

$- 3 + 12 y + 8 z = - 7$

$x = 3 - 5 y - 4 z$

Divide $y$ by $1$ .

$y = \frac{- 6}{- 2}$

$- 3 + 12 y + 8 z = - 7$

$x = 3 - 5 y - 4 z$

$y = \frac{- 6}{- 2}$

$- 3 + 12 y + 8 z = - 7$

$x = 3 - 5 y - 4 z$

$y = \frac{- 6}{- 2}$

$- 3 + 12 y + 8 z = - 7$

$x = 3 - 5 y - 4 z$

Simplify the right side.

Divide $- 6$ by $- 2$ .

$y = 3$

$- 3 + 12 y + 8 z = - 7$

$x = 3 - 5 y - 4 z$

$y = 3$

$- 3 + 12 y + 8 z = - 7$

$x = 3 - 5 y - 4 z$

$y = 3$

$- 3 + 12 y + 8 z = - 7$

$x = 3 - 5 y - 4 z$

$y = 3$

$- 3 + 12 y + 8 z = - 7$

$x = 3 - 5 y - 4 z$

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

Replace all occurrences of $y$ in $- 3 + 12 y + 8 z = - 7$ with $3$ .

$- 3 + 12 (3) + 8 z = - 7$

$y = 3$

$x = 3 - 5 y - 4 z$

Simplify the left side.

Simplify $- 3 + 12 (3) + 8 z$ .

Multiply $12$ by $3$ .

$- 3 + 36 + 8 z = - 7$

$y = 3$

$x = 3 - 5 y - 4 z$

Add $- 3$ and $36$ .

$33 + 8 z = - 7$

$y = 3$

$x = 3 - 5 y - 4 z$

$33 + 8 z = - 7$

$y = 3$

$x = 3 - 5 y - 4 z$

$33 + 8 z = - 7$

$y = 3$

$x = 3 - 5 y - 4 z$

Replace all occurrences of $y$ in $x = 3 - 5 y - 4 z$ with $3$ .

$x = 3 - 5 \cdot 3 - 4 z$

$33 + 8 z = - 7$

$y = 3$

Simplify the right side.

Simplify $3 - 5 \cdot 3 - 4 z$ .

Multiply $- 5$ by $3$ .

$x = 3 - 15 - 4 z$

$33 + 8 z = - 7$

$y = 3$

Subtract $15$ from $3$ .

$x = - 12 - 4 z$

$33 + 8 z = - 7$

$y = 3$

$x = - 12 - 4 z$

$33 + 8 z = - 7$

$y = 3$

$x = - 12 - 4 z$

$33 + 8 z = - 7$

$y = 3$

$x = - 12 - 4 z$

$33 + 8 z = - 7$

$y = 3$

Solve for $z$ in $33 + 8 z = - 7$ .

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

Subtract $33$ from both sides of the equation.

$8 z = - 7 - 33$

$x = - 12 - 4 z$

$y = 3$

Subtract $33$ from $- 7$ .

$8 z = - 40$

$x = - 12 - 4 z$

$y = 3$

$8 z = - 40$

$x = - 12 - 4 z$

$y = 3$

Divide each term in $8 z = - 40$ by $8$ and simplify.

Divide each term in $8 z = - 40$ by $8$ .

$\frac{8 z}{8} = \frac{- 40}{8}$

$x = - 12 - 4 z$

$y = 3$

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{8 z}{8} = \frac{- 40}{8}$

$x = - 12 - 4 z$

$y = 3$

Divide $z$ by $1$ .

$z = \frac{- 40}{8}$

$x = - 12 - 4 z$

$y = 3$

$z = \frac{- 40}{8}$

$x = - 12 - 4 z$

$y = 3$

$z = \frac{- 40}{8}$

$x = - 12 - 4 z$

$y = 3$

Simplify the right side.

Divide $- 40$ by $8$ .

$z = - 5$

$x = - 12 - 4 z$

$y = 3$

$z = - 5$

$x = - 12 - 4 z$

$y = 3$

$z = - 5$

$x = - 12 - 4 z$

$y = 3$

$z = - 5$

$x = - 12 - 4 z$

$y = 3$

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

Replace all occurrences of $z$ in $x = - 12 - 4 z$ with $- 5$ .

$x = - 12 - 4 \cdot - 5$

$z = - 5$

$y = 3$

Simplify the right side.

Simplify $- 12 - 4 \cdot - 5$ .

Multiply $- 4$ by $- 5$ .

$x = - 12 + 20$

$z = - 5$

$y = 3$

Add $- 12$ and $20$ .

$x = 8$

$z = - 5$

$y = 3$

$x = 8$

$z = - 5$

$y = 3$

$x = 8$

$z = - 5$

$y = 3$

$x = 8$

$z = - 5$

$y = 3$

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

$(8, 3, - 5)$

The result can be shown in multiple forms.

Point Form:

$(8, 3, - 5)$

Equation Form:

$x = 8, y = 3, z = - 5$

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

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