Solve the System of Equations 2x+y=0 x+3y-z=-11 7x+5y+4z=21

$2 x + y = 0$ $x + 3 y - z = - 11$ $7 x + 5 y + 4 z = 21$

Step 1

Subtract $2 x$ from both sides of the equation.

$y = - 2 x$

$x + 3 y - z = - 11$

$7 x + 5 y + 4 z = 21$

Step 2

Replace all occurrences of $y$ with $- 2 x$ in each equation.

Replace all occurrences of $y$ in $x + 3 y - z = - 11$ with $- 2 x$ .

$x + 3 (- 2 x) - z = - 11$

$y = - 2 x$

$7 x + 5 y + 4 z = 21$

Simplify the left side.

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

Multiply $- 2$ by $3$ .

$x - 6 x - z = - 11$

$y = - 2 x$

$7 x + 5 y + 4 z = 21$

Subtract $6 x$ from $x$ .

$- 5 x - z = - 11$

$y = - 2 x$

$7 x + 5 y + 4 z = 21$

$- 5 x - z = - 11$

$y = - 2 x$

$7 x + 5 y + 4 z = 21$

$- 5 x - z = - 11$

$y = - 2 x$

$7 x + 5 y + 4 z = 21$

Replace all occurrences of $y$ in $7 x + 5 y + 4 z = 21$ with $- 2 x$ .

$7 x + 5 (- 2 x) + 4 z = 21$

$- 5 x - z = - 11$

$y = - 2 x$

Simplify the left side.

Simplify $7 x + 5 (- 2 x) + 4 z$ .

Multiply $- 2$ by $5$ .

$7 x - 10 x + 4 z = 21$

$- 5 x - z = - 11$

$y = - 2 x$

Subtract $10 x$ from $7 x$ .

$- 3 x + 4 z = 21$

$- 5 x - z = - 11$

$y = - 2 x$

$- 3 x + 4 z = 21$

$- 5 x - z = - 11$

$y = - 2 x$

$- 3 x + 4 z = 21$

$- 5 x - z = - 11$

$y = - 2 x$

$- 3 x + 4 z = 21$

$- 5 x - z = - 11$

$y = - 2 x$

Step 3

Solve for $x$ in $- 3 x + 4 z = 21$ .

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

$- 3 x = 21 - 4 z$

$- 5 x - z = - 11$

$y = - 2 x$

Divide each term in $- 3 x = 21 - 4 z$ by $- 3$ and simplify.

Divide each term in $- 3 x = 21 - 4 z$ by $- 3$ .

$\frac{- 3 x}{- 3} = \frac{21}{- 3} + \frac{- 4 z}{- 3}$

$- 5 x - z = - 11$

$y = - 2 x$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 x}{- 3} = \frac{21}{- 3} + \frac{- 4 z}{- 3}$

$- 5 x - z = - 11$

$y = - 2 x$

Divide $x$ by $1$ .

$x = \frac{21}{- 3} + \frac{- 4 z}{- 3}$

$- 5 x - z = - 11$

$y = - 2 x$

$x = \frac{21}{- 3} + \frac{- 4 z}{- 3}$

$- 5 x - z = - 11$

$y = - 2 x$

$x = \frac{21}{- 3} + \frac{- 4 z}{- 3}$

$- 5 x - z = - 11$

$y = - 2 x$

Simplify the right side.

Simplify each term.

Divide $21$ by $- 3$ .

$x = - 7 + \frac{- 4 z}{- 3}$

$- 5 x - z = - 11$

$y = - 2 x$

Dividing two negative values results in a positive value.

$x = - 7 + \frac{4 z}{3}$

$- 5 x - z = - 11$

$y = - 2 x$

$x = - 7 + \frac{4 z}{3}$

$- 5 x - z = - 11$

$y = - 2 x$

$x = - 7 + \frac{4 z}{3}$

$- 5 x - z = - 11$

$y = - 2 x$

$x = - 7 + \frac{4 z}{3}$

$- 5 x - z = - 11$

$y = - 2 x$

$x = - 7 + \frac{4 z}{3}$

$- 5 x - z = - 11$

$y = - 2 x$

Step 4

Replace all occurrences of $x$ with $- 7 + \frac{4 z}{3}$ in each equation.

Replace all occurrences of $x$ in $- 5 x - z = - 11$ with $- 7 + \frac{4 z}{3}$ .

$- 5 (- 7 + \frac{4 z}{3}) - z = - 11$

$x = - 7 + \frac{4 z}{3}$

$y = - 2 x$

Simplify the left side.

Simplify $- 5 (- 7 + \frac{4 z}{3}) - z$ .

Simplify each term.

Apply the distributive property.

$- 5 \cdot - 7 - 5 \frac{4 z}{3} - z = - 11$

$x = - 7 + \frac{4 z}{3}$

$y = - 2 x$

Multiply $- 5$ by $- 7$ .

$35 - 5 \frac{4 z}{3} - z = - 11$

$x = - 7 + \frac{4 z}{3}$

$y = - 2 x$

Multiply $- 5 \frac{4 z}{3}$ .

Combine $- 5$ and $\frac{4 z}{3}$ .

$35 + \frac{- 5 (4 z)}{3} - z = - 11$

$x = - 7 + \frac{4 z}{3}$

$y = - 2 x$

Multiply $4$ by $- 5$ .

$35 + \frac{- 20 z}{3} - z = - 11$

$x = - 7 + \frac{4 z}{3}$

$y = - 2 x$

$35 + \frac{- 20 z}{3} - z = - 11$

$x = - 7 + \frac{4 z}{3}$

$y = - 2 x$

Move the negative in front of the fraction.

$35 - \frac{20 z}{3} - z = - 11$

$x = - 7 + \frac{4 z}{3}$

$y = - 2 x$

$35 - \frac{20 z}{3} - z = - 11$

$x = - 7 + \frac{4 z}{3}$

$y = - 2 x$

To write $- z$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$35 - \frac{20 z}{3} - z \cdot \frac{3}{3} = - 11$

$x = - 7 + \frac{4 z}{3}$

$y = - 2 x$

Simplify terms.

Combine $- z$ and $\frac{3}{3}$ .

$35 - \frac{20 z}{3} + \frac{- z \cdot 3}{3} = - 11$

$x = - 7 + \frac{4 z}{3}$

$y = - 2 x$

Combine the numerators over the common denominator.

$35 + \frac{- 20 z - z \cdot 3}{3} = - 11$

$x = - 7 + \frac{4 z}{3}$

$y = - 2 x$

$35 + \frac{- 20 z - z \cdot 3}{3} = - 11$

$x = - 7 + \frac{4 z}{3}$

$y = - 2 x$

Simplify each term.

Simplify the numerator.

Factor $z$ out of $- 20 z - z \cdot 3$ .

Factor $z$ out of $- 20 z$ .

$35 + \frac{z \cdot - 20 - z \cdot 3}{3} = - 11$

$x = - 7 + \frac{4 z}{3}$

$y = - 2 x$

Factor $z$ out of $- z \cdot 3$ .

$35 + \frac{z \cdot - 20 + z (- 1 \cdot 3)}{3} = - 11$

$x = - 7 + \frac{4 z}{3}$

$y = - 2 x$

Factor $z$ out of $z \cdot - 20 + z (- 1 \cdot 3)$ .

$35 + \frac{z (- 20 - 1 \cdot 3)}{3} = - 11$

$x = - 7 + \frac{4 z}{3}$

$y = - 2 x$

$35 + \frac{z (- 20 - 1 \cdot 3)}{3} = - 11$

$x = - 7 + \frac{4 z}{3}$

$y = - 2 x$

Multiply $- 1$ by $3$ .

$35 + \frac{z (- 20 - 3)}{3} = - 11$

$x = - 7 + \frac{4 z}{3}$

$y = - 2 x$

Subtract $3$ from $- 20$ .

$35 + \frac{z \cdot - 23}{3} = - 11$

$x = - 7 + \frac{4 z}{3}$

$y = - 2 x$

$35 + \frac{z \cdot - 23}{3} = - 11$

$x = - 7 + \frac{4 z}{3}$

$y = - 2 x$

Move $- 23$ to the left of $z$ .

$35 + \frac{- 23 \cdot z}{3} = - 11$

$x = - 7 + \frac{4 z}{3}$

$y = - 2 x$

Move the negative in front of the fraction.

$35 - \frac{23 z}{3} = - 11$

$x = - 7 + \frac{4 z}{3}$

$y = - 2 x$

$35 - \frac{23 z}{3} = - 11$

$x = - 7 + \frac{4 z}{3}$

$y = - 2 x$

$35 - \frac{23 z}{3} = - 11$

$x = - 7 + \frac{4 z}{3}$

$y = - 2 x$

$35 - \frac{23 z}{3} = - 11$

$x = - 7 + \frac{4 z}{3}$

$y = - 2 x$

Replace all occurrences of $x$ in $y = - 2 x$ with $- 7 + \frac{4 z}{3}$ .

$y = - 2 (- 7 + \frac{4 z}{3})$

$35 - \frac{23 z}{3} = - 11$

$x = - 7 + \frac{4 z}{3}$

Simplify the right side.

Simplify $- 2 (- 7 + \frac{4 z}{3})$ .

Apply the distributive property.

$y = - 2 \cdot - 7 - 2 \frac{4 z}{3}$

$35 - \frac{23 z}{3} = - 11$

$x = - 7 + \frac{4 z}{3}$

Multiply $- 2$ by $- 7$ .

$y = 14 - 2 \frac{4 z}{3}$

$35 - \frac{23 z}{3} = - 11$

$x = - 7 + \frac{4 z}{3}$

Multiply $- 2 \frac{4 z}{3}$ .

Combine $- 2$ and $\frac{4 z}{3}$ .

$y = 14 + \frac{- 2 (4 z)}{3}$

$35 - \frac{23 z}{3} = - 11$

$x = - 7 + \frac{4 z}{3}$

Multiply $4$ by $- 2$ .

$y = 14 + \frac{- 8 z}{3}$

$35 - \frac{23 z}{3} = - 11$

$x = - 7 + \frac{4 z}{3}$

$y = 14 + \frac{- 8 z}{3}$

$35 - \frac{23 z}{3} = - 11$

$x = - 7 + \frac{4 z}{3}$

Move the negative in front of the fraction.

$y = 14 - \frac{8 z}{3}$

$35 - \frac{23 z}{3} = - 11$

$x = - 7 + \frac{4 z}{3}$

$y = 14 - \frac{8 z}{3}$

$35 - \frac{23 z}{3} = - 11$

$x = - 7 + \frac{4 z}{3}$

$y = 14 - \frac{8 z}{3}$

$35 - \frac{23 z}{3} = - 11$

$x = - 7 + \frac{4 z}{3}$

$y = 14 - \frac{8 z}{3}$

$35 - \frac{23 z}{3} = - 11$

$x = - 7 + \frac{4 z}{3}$

Step 5

Solve for $z$ in $35 - \frac{23 z}{3} = - 11$ .

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

Subtract $35$ from both sides of the equation.

$- \frac{23 z}{3} = - 11 - 35$

$y = 14 - \frac{8 z}{3}$

$x = - 7 + \frac{4 z}{3}$

Subtract $35$ from $- 11$ .

$- \frac{23 z}{3} = - 46$

$y = 14 - \frac{8 z}{3}$

$x = - 7 + \frac{4 z}{3}$

$- \frac{23 z}{3} = - 46$

$y = 14 - \frac{8 z}{3}$

$x = - 7 + \frac{4 z}{3}$

Multiply both sides of the equation by $- \frac{3}{23}$ .

$- \frac{3}{23} \cdot (- \frac{23 z}{3}) = - \frac{3}{23} \cdot - 46$

$y = 14 - \frac{8 z}{3}$

$x = - 7 + \frac{4 z}{3}$

Simplify both sides of the equation.

Simplify the left side.

Simplify $- \frac{3}{23} (- \frac{23 z}{3})$ .

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{3}{23}$ into the numerator.

$\frac{- 3}{23} \cdot (- \frac{23 z}{3}) = - \frac{3}{23} \cdot - 46$

$y = 14 - \frac{8 z}{3}$

$x = - 7 + \frac{4 z}{3}$

Move the leading negative in $- \frac{23 z}{3}$ into the numerator.

$\frac{- 3}{23} \cdot \frac{- 23 z}{3} = - \frac{3}{23} \cdot - 46$

$y = 14 - \frac{8 z}{3}$

$x = - 7 + \frac{4 z}{3}$

Factor $3$ out of $- 3$ .

$\frac{3 (- 1)}{23} \cdot \frac{- 23 z}{3} = - \frac{3}{23} \cdot - 46$

$y = 14 - \frac{8 z}{3}$

$x = - 7 + \frac{4 z}{3}$

Cancel the common factor.

$\frac{3 \cdot - 1}{23} \cdot \frac{- 23 z}{3} = - \frac{3}{23} \cdot - 46$

$y = 14 - \frac{8 z}{3}$

$x = - 7 + \frac{4 z}{3}$

Rewrite the expression.

$\frac{- 1}{23} \cdot (- 23 z) = - \frac{3}{23} \cdot - 46$

$y = 14 - \frac{8 z}{3}$

$x = - 7 + \frac{4 z}{3}$

$\frac{- 1}{23} \cdot (- 23 z) = - \frac{3}{23} \cdot - 46$

$y = 14 - \frac{8 z}{3}$

$x = - 7 + \frac{4 z}{3}$

Cancel the common factor of $23$ .

Factor $23$ out of $- 23 z$ .

$\frac{- 1}{23} \cdot (23 (- z)) = - \frac{3}{23} \cdot - 46$

$y = 14 - \frac{8 z}{3}$

$x = - 7 + \frac{4 z}{3}$

Cancel the common factor.

$\frac{- 1}{23} \cdot (23 (- z)) = - \frac{3}{23} \cdot - 46$

$y = 14 - \frac{8 z}{3}$

$x = - 7 + \frac{4 z}{3}$

Rewrite the expression.

$z = - \frac{3}{23} \cdot - 46$

$y = 14 - \frac{8 z}{3}$

$x = - 7 + \frac{4 z}{3}$

$z = - \frac{3}{23} \cdot - 46$

$y = 14 - \frac{8 z}{3}$

$x = - 7 + \frac{4 z}{3}$

Multiply.

Multiply $- 1$ by $- 1$ .

$1 z = - \frac{3}{23} \cdot - 46$

$y = 14 - \frac{8 z}{3}$

$x = - 7 + \frac{4 z}{3}$

Multiply $z$ by $1$ .

$z = - \frac{3}{23} \cdot - 46$

$y = 14 - \frac{8 z}{3}$

$x = - 7 + \frac{4 z}{3}$

$z = - \frac{3}{23} \cdot - 46$

$y = 14 - \frac{8 z}{3}$

$x = - 7 + \frac{4 z}{3}$

$z = - \frac{3}{23} \cdot - 46$

$y = 14 - \frac{8 z}{3}$

$x = - 7 + \frac{4 z}{3}$

$z = - \frac{3}{23} \cdot - 46$

$y = 14 - \frac{8 z}{3}$

$x = - 7 + \frac{4 z}{3}$

Simplify the right side.

Simplify $- \frac{3}{23} \cdot - 46$ .

Cancel the common factor of $23$ .

Move the leading negative in $- \frac{3}{23}$ into the numerator.

$z = \frac{- 3}{23} \cdot - 46$

$y = 14 - \frac{8 z}{3}$

$x = - 7 + \frac{4 z}{3}$

Factor $23$ out of $- 46$ .

$z = \frac{- 3}{23} \cdot (23 (- 2))$

$y = 14 - \frac{8 z}{3}$

$x = - 7 + \frac{4 z}{3}$

Cancel the common factor.

$z = \frac{- 3}{23} \cdot (23 \cdot - 2)$

$y = 14 - \frac{8 z}{3}$

$x = - 7 + \frac{4 z}{3}$

Rewrite the expression.

$z = - 3 \cdot - 2$

$y = 14 - \frac{8 z}{3}$

$x = - 7 + \frac{4 z}{3}$

$z = - 3 \cdot - 2$

$y = 14 - \frac{8 z}{3}$

$x = - 7 + \frac{4 z}{3}$

Multiply $- 3$ by $- 2$ .

$z = 6$

$y = 14 - \frac{8 z}{3}$

$x = - 7 + \frac{4 z}{3}$

$z = 6$

$y = 14 - \frac{8 z}{3}$

$x = - 7 + \frac{4 z}{3}$

$z = 6$

$y = 14 - \frac{8 z}{3}$

$x = - 7 + \frac{4 z}{3}$

$z = 6$

$y = 14 - \frac{8 z}{3}$

$x = - 7 + \frac{4 z}{3}$

$z = 6$

$y = 14 - \frac{8 z}{3}$

$x = - 7 + \frac{4 z}{3}$

Step 6

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

Replace all occurrences of $z$ in $y = 14 - \frac{8 z}{3}$ with $6$ .

$y = 14 - \frac{8 (6)}{3}$

$z = 6$

$x = - 7 + \frac{4 z}{3}$

Simplify the right side.

Simplify $14 - \frac{8 (6)}{3}$ .

Simplify each term.

Cancel the common factor of $6$ and $3$ .

Factor $3$ out of $8 (6)$ .

$y = 14 - \frac{3 (8 \cdot (2))}{3}$

$z = 6$

$x = - 7 + \frac{4 z}{3}$

Cancel the common factors.

Factor $3$ out of $3$ .

$y = 14 - \frac{3 (8 \cdot (2))}{3 (1)}$

$z = 6$

$x = - 7 + \frac{4 z}{3}$

Cancel the common factor.

$y = 14 - \frac{3 (8 \cdot (2))}{3 \cdot 1}$

$z = 6$

$x = - 7 + \frac{4 z}{3}$

Rewrite the expression.

$y = 14 - \frac{8 \cdot (2)}{1}$

$z = 6$

$x = - 7 + \frac{4 z}{3}$

Divide $8 \cdot (2)$ by $1$ .

$y = 14 - (8 \cdot (2))$

$z = 6$

$x = - 7 + \frac{4 z}{3}$

$y = 14 - (8 \cdot (2))$

$z = 6$

$x = - 7 + \frac{4 z}{3}$

$y = 14 - (8 \cdot (2))$

$z = 6$

$x = - 7 + \frac{4 z}{3}$

Multiply $- (8 \cdot (2))$ .

Multiply $8$ by $2$ .

$y = 14 - 1 \cdot 16$

$z = 6$

$x = - 7 + \frac{4 z}{3}$

Multiply $- 1$ by $16$ .

$y = 14 - 16$

$z = 6$

$x = - 7 + \frac{4 z}{3}$

$y = 14 - 16$

$z = 6$

$x = - 7 + \frac{4 z}{3}$

$y = 14 - 16$

$z = 6$

$x = - 7 + \frac{4 z}{3}$

Subtract $16$ from $14$ .

$y = - 2$

$z = 6$

$x = - 7 + \frac{4 z}{3}$

$y = - 2$

$z = 6$

$x = - 7 + \frac{4 z}{3}$

$y = - 2$

$z = 6$

$x = - 7 + \frac{4 z}{3}$

Replace all occurrences of $z$ in $x = - 7 + \frac{4 z}{3}$ with $6$ .

$x = - 7 + \frac{4 (6)}{3}$

$y = - 2$

$z = 6$

Simplify the right side.

Simplify $- 7 + \frac{4 (6)}{3}$ .

Multiply $4$ by $6$ .

$x = - 7 + \frac{24}{3}$

$y = - 2$

$z = 6$

Divide $24$ by $3$ .

$x = - 7 + 8$

$y = - 2$

$z = 6$

Add $- 7$ and $8$ .

$x = 1$

$y = - 2$

$z = 6$

$x = 1$

$y = - 2$

$z = 6$

$x = 1$

$y = - 2$

$z = 6$

$x = 1$

$y = - 2$

$z = 6$

Step 7

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

$(1, - 2, 6)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(1, - 2, 6)$

Equation Form:

$x = 1, y = - 2, z = 6$

Solve the System of Equations 2x+y=0 x+3y-z=-11 7x+5y+4z=21

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