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

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

Step 1

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

Add $3 x$ to both sides of the equation.

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

$x + 3 y - 3 z = 3$

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

Add $3 z$ to both sides of the equation.

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

$x + 3 y - 3 z = 3$

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

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

$x + 3 y - 3 z = 3$

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

Step 2

Replace all occurrences of $y$ with $- 9 + 3 x + 3 z$ in each equation.

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

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

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

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

Simplify the left side.

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

Simplify each term.

Apply the distributive property.

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

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

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

Simplify.

Multiply $3$ by $- 9$ .

$x - 27 + 3 (3 x) + 3 (3 z) - 3 z = 3$

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

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

Multiply $3$ by $3$ .

$x - 27 + 9 x + 3 (3 z) - 3 z = 3$

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

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

Multiply $3$ by $3$ .

$x - 27 + 9 x + 9 z - 3 z = 3$

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

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

$x - 27 + 9 x + 9 z - 3 z = 3$

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

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

$x - 27 + 9 x + 9 z - 3 z = 3$

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

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

Simplify by adding terms.

Add $x$ and $9 x$ .

$10 x - 27 + 9 z - 3 z = 3$

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

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

Subtract $3 z$ from $9 z$ .

$10 x - 27 + 6 z = 3$

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

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

$10 x - 27 + 6 z = 3$

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

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

$10 x - 27 + 6 z = 3$

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

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

$10 x - 27 + 6 z = 3$

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

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

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

$7 x - 2 (- 9 + 3 x + 3 z) + 3 z = 3$

$10 x - 27 + 6 z = 3$

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

Simplify the left side.

Simplify $7 x - 2 (- 9 + 3 x + 3 z) + 3 z$ .

Simplify each term.

Apply the distributive property.

$7 x - 2 \cdot - 9 - 2 (3 x) - 2 (3 z) + 3 z = 3$

$10 x - 27 + 6 z = 3$

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

Simplify.

Multiply $- 2$ by $- 9$ .

$7 x + 18 - 2 (3 x) - 2 (3 z) + 3 z = 3$

$10 x - 27 + 6 z = 3$

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

Multiply $3$ by $- 2$ .

$7 x + 18 - 6 x - 2 (3 z) + 3 z = 3$

$10 x - 27 + 6 z = 3$

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

Multiply $3$ by $- 2$ .

$7 x + 18 - 6 x - 6 z + 3 z = 3$

$10 x - 27 + 6 z = 3$

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

$7 x + 18 - 6 x - 6 z + 3 z = 3$

$10 x - 27 + 6 z = 3$

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

$7 x + 18 - 6 x - 6 z + 3 z = 3$

$10 x - 27 + 6 z = 3$

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

Simplify by adding terms.

Subtract $6 x$ from $7 x$ .

$x + 18 - 6 z + 3 z = 3$

$10 x - 27 + 6 z = 3$

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

Add $- 6 z$ and $3 z$ .

$x + 18 - 3 z = 3$

$10 x - 27 + 6 z = 3$

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

$x + 18 - 3 z = 3$

$10 x - 27 + 6 z = 3$

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

$x + 18 - 3 z = 3$

$10 x - 27 + 6 z = 3$

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

$x + 18 - 3 z = 3$

$10 x - 27 + 6 z = 3$

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

$x + 18 - 3 z = 3$

$10 x - 27 + 6 z = 3$

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

Step 3

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

Subtract $18$ from both sides of the equation.

$x - 3 z = 3 - 18$

$10 x - 27 + 6 z = 3$

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

Add $3 z$ to both sides of the equation.

$x = 3 - 18 + 3 z$

$10 x - 27 + 6 z = 3$

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

Subtract $18$ from $3$ .

$x = - 15 + 3 z$

$10 x - 27 + 6 z = 3$

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

$x = - 15 + 3 z$

$10 x - 27 + 6 z = 3$

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

Step 4

Replace all occurrences of $x$ with $- 15 + 3 z$ in each equation.

Replace all occurrences of $x$ in $10 x - 27 + 6 z = 3$ with $- 15 + 3 z$ .

$10 (- 15 + 3 z) - 27 + 6 z = 3$

$x = - 15 + 3 z$

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

Simplify the left side.

Simplify $10 (- 15 + 3 z) - 27 + 6 z$ .

Simplify each term.

Apply the distributive property.

$10 \cdot - 15 + 10 (3 z) - 27 + 6 z = 3$

$x = - 15 + 3 z$

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

Multiply $10$ by $- 15$ .

$- 150 + 10 (3 z) - 27 + 6 z = 3$

$x = - 15 + 3 z$

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

Multiply $3$ by $10$ .

$- 150 + 30 z - 27 + 6 z = 3$

$x = - 15 + 3 z$

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

$- 150 + 30 z - 27 + 6 z = 3$

$x = - 15 + 3 z$

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

Simplify by adding terms.

Subtract $27$ from $- 150$ .

$30 z - 177 + 6 z = 3$

$x = - 15 + 3 z$

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

Add $30 z$ and $6 z$ .

$36 z - 177 = 3$

$x = - 15 + 3 z$

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

$36 z - 177 = 3$

$x = - 15 + 3 z$

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

$36 z - 177 = 3$

$x = - 15 + 3 z$

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

$36 z - 177 = 3$

$x = - 15 + 3 z$

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

Replace all occurrences of $x$ in $y = - 9 + 3 x + 3 z$ with $- 15 + 3 z$ .

$y = - 9 + 3 (- 15 + 3 z) + 3 z$

$36 z - 177 = 3$

$x = - 15 + 3 z$

Simplify the right side.

Simplify $- 9 + 3 (- 15 + 3 z) + 3 z$ .

Simplify each term.

Apply the distributive property.

$y = - 9 + 3 \cdot - 15 + 3 (3 z) + 3 z$

$36 z - 177 = 3$

$x = - 15 + 3 z$

Multiply $3$ by $- 15$ .

$y = - 9 - 45 + 3 (3 z) + 3 z$

$36 z - 177 = 3$

$x = - 15 + 3 z$

Multiply $3$ by $3$ .

$y = - 9 - 45 + 9 z + 3 z$

$36 z - 177 = 3$

$x = - 15 + 3 z$

$y = - 9 - 45 + 9 z + 3 z$

$36 z - 177 = 3$

$x = - 15 + 3 z$

Simplify by adding terms.

Subtract $45$ from $- 9$ .

$y = - 54 + 9 z + 3 z$

$36 z - 177 = 3$

$x = - 15 + 3 z$

Add $9 z$ and $3 z$ .

$y = - 54 + 12 z$

$36 z - 177 = 3$

$x = - 15 + 3 z$

$y = - 54 + 12 z$

$36 z - 177 = 3$

$x = - 15 + 3 z$

$y = - 54 + 12 z$

$36 z - 177 = 3$

$x = - 15 + 3 z$

$y = - 54 + 12 z$

$36 z - 177 = 3$

$x = - 15 + 3 z$

$y = - 54 + 12 z$

$36 z - 177 = 3$

$x = - 15 + 3 z$

Step 5

Solve for $z$ in $36 z - 177 = 3$ .

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

Add $177$ to both sides of the equation.

$36 z = 3 + 177$

$y = - 54 + 12 z$

$x = - 15 + 3 z$

Add $3$ and $177$ .

$36 z = 180$

$y = - 54 + 12 z$

$x = - 15 + 3 z$

$36 z = 180$

$y = - 54 + 12 z$

$x = - 15 + 3 z$

Divide each term in $36 z = 180$ by $36$ and simplify.

Divide each term in $36 z = 180$ by $36$ .

$\frac{36 z}{36} = \frac{180}{36}$

$y = - 54 + 12 z$

$x = - 15 + 3 z$

Simplify the left side.

Cancel the common factor of $36$ .

Cancel the common factor.

$\frac{36 z}{36} = \frac{180}{36}$

$y = - 54 + 12 z$

$x = - 15 + 3 z$

Divide $z$ by $1$ .

$z = \frac{180}{36}$

$y = - 54 + 12 z$

$x = - 15 + 3 z$

$z = \frac{180}{36}$

$y = - 54 + 12 z$

$x = - 15 + 3 z$

$z = \frac{180}{36}$

$y = - 54 + 12 z$

$x = - 15 + 3 z$

Simplify the right side.

Divide $180$ by $36$ .

$z = 5$

$y = - 54 + 12 z$

$x = - 15 + 3 z$

$z = 5$

$y = - 54 + 12 z$

$x = - 15 + 3 z$

$z = 5$

$y = - 54 + 12 z$

$x = - 15 + 3 z$

$z = 5$

$y = - 54 + 12 z$

$x = - 15 + 3 z$

Step 6

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

Replace all occurrences of $z$ in $y = - 54 + 12 z$ with $5$ .

$y = - 54 + 12 (5)$

$z = 5$

$x = - 15 + 3 z$

Simplify the right side.

Simplify $- 54 + 12 (5)$ .

Multiply $12$ by $5$ .

$y = - 54 + 60$

$z = 5$

$x = - 15 + 3 z$

Add $- 54$ and $60$ .

$y = 6$

$z = 5$

$x = - 15 + 3 z$

$y = 6$

$z = 5$

$x = - 15 + 3 z$

$y = 6$

$z = 5$

$x = - 15 + 3 z$

Replace all occurrences of $z$ in $x = - 15 + 3 z$ with $5$ .

$x = - 15 + 3 (5)$

$y = 6$

$z = 5$

Simplify the right side.

Simplify $- 15 + 3 (5)$ .

Multiply $3$ by $5$ .

$x = - 15 + 15$

$y = 6$

$z = 5$

Add $- 15$ and $15$ .

$x = 0$

$y = 6$

$z = 5$

$x = 0$

$y = 6$

$z = 5$

$x = 0$

$y = 6$

$z = 5$

$x = 0$

$y = 6$

$z = 5$

Step 7

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

$(0, 6, 5)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(0, 6, 5)$

Equation Form:

$x = 0, y = 6, z = 5$

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

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