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

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

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

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

$y + 3 z = 3 - 7 x$

$- 4 x - y - 2 z = - 4$

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

Subtract $3 z$ from both sides of the equation.

$y = 3 - 7 x - 3 z$

$- 4 x - y - 2 z = - 4$

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

$y = 3 - 7 x - 3 z$

$- 4 x - y - 2 z = - 4$

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

Replace all occurrences of $y$ with $3 - 7 x - 3 z$ in each equation.

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

$- 4 x - (3 - 7 x - 3 z) - 2 z = - 4$

$y = 3 - 7 x - 3 z$

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

Simplify the left side.

Simplify $- 4 x - (3 - 7 x - 3 z) - 2 z$ .

Simplify each term.

Apply the distributive property.

$- 4 x - 1 \cdot 3 - (- 7 x) - (- 3 z) - 2 z = - 4$

$y = 3 - 7 x - 3 z$

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

Simplify.

Multiply $- 1$ by $3$ .

$- 4 x - 3 - (- 7 x) - (- 3 z) - 2 z = - 4$

$y = 3 - 7 x - 3 z$

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

Multiply $- 7$ by $- 1$ .

$- 4 x - 3 + 7 x - (- 3 z) - 2 z = - 4$

$y = 3 - 7 x - 3 z$

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

Multiply $- 3$ by $- 1$ .

$- 4 x - 3 + 7 x + 3 z - 2 z = - 4$

$y = 3 - 7 x - 3 z$

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

$- 4 x - 3 + 7 x + 3 z - 2 z = - 4$

$y = 3 - 7 x - 3 z$

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

$- 4 x - 3 + 7 x + 3 z - 2 z = - 4$

$y = 3 - 7 x - 3 z$

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

Simplify by adding terms.

Add $- 4 x$ and $7 x$ .

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

$y = 3 - 7 x - 3 z$

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

Subtract $2 z$ from $3 z$ .

$3 x - 3 + z = - 4$

$y = 3 - 7 x - 3 z$

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

$3 x - 3 + z = - 4$

$y = 3 - 7 x - 3 z$

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

$3 x - 3 + z = - 4$

$y = 3 - 7 x - 3 z$

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

$3 x - 3 + z = - 4$

$y = 3 - 7 x - 3 z$

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

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

$- 5 x - (3 - 7 x - 3 z) - 4 z = - 2$

$3 x - 3 + z = - 4$

$y = 3 - 7 x - 3 z$

Simplify the left side.

Simplify $- 5 x - (3 - 7 x - 3 z) - 4 z$ .

Simplify each term.

Apply the distributive property.

$- 5 x - 1 \cdot 3 - (- 7 x) - (- 3 z) - 4 z = - 2$

$3 x - 3 + z = - 4$

$y = 3 - 7 x - 3 z$

Simplify.

Multiply $- 1$ by $3$ .

$- 5 x - 3 - (- 7 x) - (- 3 z) - 4 z = - 2$

$3 x - 3 + z = - 4$

$y = 3 - 7 x - 3 z$

Multiply $- 7$ by $- 1$ .

$- 5 x - 3 + 7 x - (- 3 z) - 4 z = - 2$

$3 x - 3 + z = - 4$

$y = 3 - 7 x - 3 z$

Multiply $- 3$ by $- 1$ .

$- 5 x - 3 + 7 x + 3 z - 4 z = - 2$

$3 x - 3 + z = - 4$

$y = 3 - 7 x - 3 z$

$- 5 x - 3 + 7 x + 3 z - 4 z = - 2$

$3 x - 3 + z = - 4$

$y = 3 - 7 x - 3 z$

$- 5 x - 3 + 7 x + 3 z - 4 z = - 2$

$3 x - 3 + z = - 4$

$y = 3 - 7 x - 3 z$

Simplify by adding terms.

Add $- 5 x$ and $7 x$ .

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

$3 x - 3 + z = - 4$

$y = 3 - 7 x - 3 z$

Subtract $4 z$ from $3 z$ .

$2 x - 3 - z = - 2$

$3 x - 3 + z = - 4$

$y = 3 - 7 x - 3 z$

$2 x - 3 - z = - 2$

$3 x - 3 + z = - 4$

$y = 3 - 7 x - 3 z$

$2 x - 3 - z = - 2$

$3 x - 3 + z = - 4$

$y = 3 - 7 x - 3 z$

$2 x - 3 - z = - 2$

$3 x - 3 + z = - 4$

$y = 3 - 7 x - 3 z$

$2 x - 3 - z = - 2$

$3 x - 3 + z = - 4$

$y = 3 - 7 x - 3 z$

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

Subtract $3 x$ from both sides of the equation.

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

$2 x - 3 - z = - 2$

$y = 3 - 7 x - 3 z$

Add $3$ to both sides of the equation.

$z = - 4 - 3 x + 3$

$2 x - 3 - z = - 2$

$y = 3 - 7 x - 3 z$

Add $- 4$ and $3$ .

$z = - 3 x - 1$

$2 x - 3 - z = - 2$

$y = 3 - 7 x - 3 z$

$z = - 3 x - 1$

$2 x - 3 - z = - 2$

$y = 3 - 7 x - 3 z$

Replace all occurrences of $z$ with $- 3 x - 1$ in each equation.

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

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

$z = - 3 x - 1$

$y = 3 - 7 x - 3 z$

Simplify the left side.

Simplify $2 x - 3 - (- 3 x - 1)$ .

Simplify each term.

Apply the distributive property.

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

$z = - 3 x - 1$

$y = 3 - 7 x - 3 z$

Multiply $- 3$ by $- 1$ .

$2 x - 3 + 3 x + 1 = - 2$

$z = - 3 x - 1$

$y = 3 - 7 x - 3 z$

Multiply $- 1$ by $- 1$ .

$2 x - 3 + 3 x + 1 = - 2$

$z = - 3 x - 1$

$y = 3 - 7 x - 3 z$

$2 x - 3 + 3 x + 1 = - 2$

$z = - 3 x - 1$

$y = 3 - 7 x - 3 z$

Simplify by adding terms.

Add $2 x$ and $3 x$ .

$5 x - 3 + 1 = - 2$

$z = - 3 x - 1$

$y = 3 - 7 x - 3 z$

Add $- 3$ and $1$ .

$5 x - 2 = - 2$

$z = - 3 x - 1$

$y = 3 - 7 x - 3 z$

$5 x - 2 = - 2$

$z = - 3 x - 1$

$y = 3 - 7 x - 3 z$

$5 x - 2 = - 2$

$z = - 3 x - 1$

$y = 3 - 7 x - 3 z$

$5 x - 2 = - 2$

$z = - 3 x - 1$

$y = 3 - 7 x - 3 z$

Replace all occurrences of $z$ in $y = 3 - 7 x - 3 z$ with $- 3 x - 1$ .

$y = 3 - 7 x - 3 (- 3 x - 1)$

$5 x - 2 = - 2$

$z = - 3 x - 1$

Simplify the right side.

Simplify $3 - 7 x - 3 (- 3 x - 1)$ .

Simplify each term.

Apply the distributive property.

$y = 3 - 7 x - 3 (- 3 x) - 3 \cdot - 1$

$5 x - 2 = - 2$

$z = - 3 x - 1$

Multiply $- 3$ by $- 3$ .

$y = 3 - 7 x + 9 x - 3 \cdot - 1$

$5 x - 2 = - 2$

$z = - 3 x - 1$

Multiply $- 3$ by $- 1$ .

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

$5 x - 2 = - 2$

$z = - 3 x - 1$

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

$5 x - 2 = - 2$

$z = - 3 x - 1$

Simplify by adding terms.

Add $3$ and $3$ .

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

$5 x - 2 = - 2$

$z = - 3 x - 1$

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

$y = 2 x + 6$

$5 x - 2 = - 2$

$z = - 3 x - 1$

$y = 2 x + 6$

$5 x - 2 = - 2$

$z = - 3 x - 1$

$y = 2 x + 6$

$5 x - 2 = - 2$

$z = - 3 x - 1$

$y = 2 x + 6$

$5 x - 2 = - 2$

$z = - 3 x - 1$

$y = 2 x + 6$

$5 x - 2 = - 2$

$z = - 3 x - 1$

Solve for $x$ in $5 x - 2 = - 2$ .

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

Add $2$ to both sides of the equation.

$5 x = - 2 + 2$

$y = 2 x + 6$

$z = - 3 x - 1$

Add $- 2$ and $2$ .

$5 x = 0$

$y = 2 x + 6$

$z = - 3 x - 1$

$5 x = 0$

$y = 2 x + 6$

$z = - 3 x - 1$

Divide each term in $5 x = 0$ by $5$ and simplify.

Divide each term in $5 x = 0$ by $5$ .

$\frac{5 x}{5} = \frac{0}{5}$

$y = 2 x + 6$

$z = - 3 x - 1$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} = \frac{0}{5}$

$y = 2 x + 6$

$z = - 3 x - 1$

Divide $x$ by $1$ .

$x = \frac{0}{5}$

$y = 2 x + 6$

$z = - 3 x - 1$

$x = \frac{0}{5}$

$y = 2 x + 6$

$z = - 3 x - 1$

$x = \frac{0}{5}$

$y = 2 x + 6$

$z = - 3 x - 1$

Simplify the right side.

Divide $0$ by $5$ .

$x = 0$

$y = 2 x + 6$

$z = - 3 x - 1$

$x = 0$

$y = 2 x + 6$

$z = - 3 x - 1$

$x = 0$

$y = 2 x + 6$

$z = - 3 x - 1$

$x = 0$

$y = 2 x + 6$

$z = - 3 x - 1$

Replace all occurrences of $x$ with $0$ in each equation.

Replace all occurrences of $x$ in $y = 2 x + 6$ with $0$ .

$y = 2 (0) + 6$

$x = 0$

$z = - 3 x - 1$

Simplify the right side.

Simplify $2 (0) + 6$ .

Multiply $2$ by $0$ .

$y = 0 + 6$

$x = 0$

$z = - 3 x - 1$

Add $0$ and $6$ .

$y = 6$

$x = 0$

$z = - 3 x - 1$

$y = 6$

$x = 0$

$z = - 3 x - 1$

$y = 6$

$x = 0$

$z = - 3 x - 1$

Replace all occurrences of $x$ in $z = - 3 x - 1$ with $0$ .

$z = - 3 \cdot 0 - 1$

$y = 6$

$x = 0$

Simplify the right side.

Simplify $- 3 \cdot 0 - 1$ .

Multiply $- 3$ by $0$ .

$z = 0 - 1$

$y = 6$

$x = 0$

Subtract $1$ from $0$ .

$z = - 1$

$y = 6$

$x = 0$

$z = - 1$

$y = 6$

$x = 0$

$z = - 1$

$y = 6$

$x = 0$

$z = - 1$

$y = 6$

$x = 0$

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

$(0, 6, - 1)$

The result can be shown in multiple forms.

Point Form:

$(0, 6, - 1)$

Equation Form:

$x = 0, y = 6, z = - 1$

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

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