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

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

Step 1

Replace all occurrences of $y$ with $x + z + 5$ in each equation.

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

$z = - 3 (x + z + 5) - 3$

$y = x + z + 5$

$2 x - y = - 4$

Simplify the right side.

Simplify $- 3 (x + z + 5) - 3$ .

Simplify each term.

Apply the distributive property.

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

$y = x + z + 5$

$2 x - y = - 4$

Multiply $- 3$ by $5$ .

$z = - 3 x - 3 z - 15 - 3$

$y = x + z + 5$

$2 x - y = - 4$

$z = - 3 x - 3 z - 15 - 3$

$y = x + z + 5$

$2 x - y = - 4$

Subtract $3$ from $- 15$ .

$z = - 3 x - 3 z - 18$

$y = x + z + 5$

$2 x - y = - 4$

$z = - 3 x - 3 z - 18$

$y = x + z + 5$

$2 x - y = - 4$

$z = - 3 x - 3 z - 18$

$y = x + z + 5$

$2 x - y = - 4$

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

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

$z = - 3 x - 3 z - 18$

$y = x + z + 5$

Simplify the left side.

Simplify $2 x - (x + z + 5)$ .

Simplify each term.

Apply the distributive property.

$2 x - x - z - 1 \cdot 5 = - 4$

$z = - 3 x - 3 z - 18$

$y = x + z + 5$

Multiply $- 1$ by $5$ .

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

$z = - 3 x - 3 z - 18$

$y = x + z + 5$

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

$z = - 3 x - 3 z - 18$

$y = x + z + 5$

Subtract $x$ from $2 x$ .

$x - z - 5 = - 4$

$z = - 3 x - 3 z - 18$

$y = x + z + 5$

$x - z - 5 = - 4$

$z = - 3 x - 3 z - 18$

$y = x + z + 5$

$x - z - 5 = - 4$

$z = - 3 x - 3 z - 18$

$y = x + z + 5$

$x - z - 5 = - 4$

$z = - 3 x - 3 z - 18$

$y = x + z + 5$

Step 2

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

Add $z$ to both sides of the equation.

$x - 5 = - 4 + z$

$z = - 3 x - 3 z - 18$

$y = x + z + 5$

Add $5$ to both sides of the equation.

$x = - 4 + z + 5$

$z = - 3 x - 3 z - 18$

$y = x + z + 5$

Add $- 4$ and $5$ .

$x = z + 1$

$z = - 3 x - 3 z - 18$

$y = x + z + 5$

$x = z + 1$

$z = - 3 x - 3 z - 18$

$y = x + z + 5$

Step 3

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

Replace all occurrences of $x$ in $z = - 3 x - 3 z - 18$ with $z + 1$ .

$z = - 3 (z + 1) - 3 z - 18$

$x = z + 1$

$y = x + z + 5$

Simplify the right side.

Simplify $- 3 (z + 1) - 3 z - 18$ .

Simplify each term.

Apply the distributive property.

$z = - 3 z - 3 \cdot 1 - 3 z - 18$

$x = z + 1$

$y = x + z + 5$

Multiply $- 3$ by $1$ .

$z = - 3 z - 3 - 3 z - 18$

$x = z + 1$

$y = x + z + 5$

$z = - 3 z - 3 - 3 z - 18$

$x = z + 1$

$y = x + z + 5$

Simplify by adding terms.

Subtract $3 z$ from $- 3 z$ .

$z = - 6 z - 3 - 18$

$x = z + 1$

$y = x + z + 5$

Subtract $18$ from $- 3$ .

$z = - 6 z - 21$

$x = z + 1$

$y = x + z + 5$

$z = - 6 z - 21$

$x = z + 1$

$y = x + z + 5$

$z = - 6 z - 21$

$x = z + 1$

$y = x + z + 5$

$z = - 6 z - 21$

$x = z + 1$

$y = x + z + 5$

Replace all occurrences of $x$ in $y = x + z + 5$ with $z + 1$ .

$y = (z + 1) + z + 5$

$z = - 6 z - 21$

$x = z + 1$

Simplify the right side.

Simplify $(z + 1) + z + 5$ .

Add $z$ and $z$ .

$y = 2 z + 1 + 5$

$z = - 6 z - 21$

$x = z + 1$

Add $1$ and $5$ .

$y = 2 z + 6$

$z = - 6 z - 21$

$x = z + 1$

$y = 2 z + 6$

$z = - 6 z - 21$

$x = z + 1$

$y = 2 z + 6$

$z = - 6 z - 21$

$x = z + 1$

$y = 2 z + 6$

$z = - 6 z - 21$

$x = z + 1$

Step 4

Solve for $z$ in $z = - 6 z - 21$ .

Move all terms containing $z$ to the left side of the equation.

Add $6 z$ to both sides of the equation.

$z + 6 z = - 21$

$y = 2 z + 6$

$x = z + 1$

Add $z$ and $6 z$ .

$7 z = - 21$

$y = 2 z + 6$

$x = z + 1$

$7 z = - 21$

$y = 2 z + 6$

$x = z + 1$

Divide each term in $7 z = - 21$ by $7$ and simplify.

Divide each term in $7 z = - 21$ by $7$ .

$\frac{7 z}{7} = \frac{- 21}{7}$

$y = 2 z + 6$

$x = z + 1$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 z}{7} = \frac{- 21}{7}$

$y = 2 z + 6$

$x = z + 1$

Divide $z$ by $1$ .

$z = \frac{- 21}{7}$

$y = 2 z + 6$

$x = z + 1$

$z = \frac{- 21}{7}$

$y = 2 z + 6$

$x = z + 1$

$z = \frac{- 21}{7}$

$y = 2 z + 6$

$x = z + 1$

Simplify the right side.

Divide $- 21$ by $7$ .

$z = - 3$

$y = 2 z + 6$

$x = z + 1$

$z = - 3$

$y = 2 z + 6$

$x = z + 1$

$z = - 3$

$y = 2 z + 6$

$x = z + 1$

$z = - 3$

$y = 2 z + 6$

$x = z + 1$

Step 5

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

Replace all occurrences of $z$ in $y = 2 z + 6$ with $- 3$ .

$y = 2 (- 3) + 6$

$z = - 3$

$x = z + 1$

Simplify the right side.

Simplify $2 (- 3) + 6$ .

Multiply $2$ by $- 3$ .

$y = - 6 + 6$

$z = - 3$

$x = z + 1$

Add $- 6$ and $6$ .

$y = 0$

$z = - 3$

$x = z + 1$

$y = 0$

$z = - 3$

$x = z + 1$

$y = 0$

$z = - 3$

$x = z + 1$

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

$x = (- 3) + 1$

$y = 0$

$z = - 3$

Simplify the right side.

Add $- 3$ and $1$ .

$x = - 2$

$y = 0$

$z = - 3$

$x = - 2$

$y = 0$

$z = - 3$

$x = - 2$

$y = 0$

$z = - 3$

Step 6

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

$(- 2, 0, - 3)$

Step 7

The result can be shown in multiple forms.

Point Form:

$(- 2, 0, - 3)$

Equation Form:

$x = - 2, y = 0, z = - 3$

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

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