Solve the System of Equations 2p-q+4r=11 p+2q-6r=-11 3p-2q-10r=11

$2 p - q + 4 r = 11$ $p + 2 q - 6 r = - 11$ $3 p - 2 q - 10 r = 11$

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

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

$p - 6 r = - 11 - 2 q$

$2 p - q + 4 r = 11$

$3 p - 2 q - 10 r = 11$

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

$p = - 11 - 2 q + 6 r$

$2 p - q + 4 r = 11$

$3 p - 2 q - 10 r = 11$

$p = - 11 - 2 q + 6 r$

$2 p - q + 4 r = 11$

$3 p - 2 q - 10 r = 11$

Replace all occurrences of $p$ with $- 11 - 2 q + 6 r$ in each equation.

Replace all occurrences of $p$ in $2 p - q + 4 r = 11$ with $- 11 - 2 q + 6 r$ .

$2 (- 11 - 2 q + 6 r) - q + 4 r = 11$

$p = - 11 - 2 q + 6 r$

$3 p - 2 q - 10 r = 11$

Simplify the left side.

Simplify $2 (- 11 - 2 q + 6 r) - q + 4 r$ .

Simplify each term.

Apply the distributive property.

$2 \cdot - 11 + 2 (- 2 q) + 2 (6 r) - q + 4 r = 11$

$p = - 11 - 2 q + 6 r$

$3 p - 2 q - 10 r = 11$

Simplify.

Multiply $2$ by $- 11$ .

$- 22 + 2 (- 2 q) + 2 (6 r) - q + 4 r = 11$

$p = - 11 - 2 q + 6 r$

$3 p - 2 q - 10 r = 11$

Multiply $- 2$ by $2$ .

$- 22 - 4 q + 2 (6 r) - q + 4 r = 11$

$p = - 11 - 2 q + 6 r$

$3 p - 2 q - 10 r = 11$

Multiply $6$ by $2$ .

$- 22 - 4 q + 12 r - q + 4 r = 11$

$p = - 11 - 2 q + 6 r$

$3 p - 2 q - 10 r = 11$

$- 22 - 4 q + 12 r - q + 4 r = 11$

$p = - 11 - 2 q + 6 r$

$3 p - 2 q - 10 r = 11$

$- 22 - 4 q + 12 r - q + 4 r = 11$

$p = - 11 - 2 q + 6 r$

$3 p - 2 q - 10 r = 11$

Simplify by adding terms.

Subtract $q$ from $- 4 q$ .

$- 22 - 5 q + 12 r + 4 r = 11$

$p = - 11 - 2 q + 6 r$

$3 p - 2 q - 10 r = 11$

Add $12 r$ and $4 r$ .

$- 22 - 5 q + 16 r = 11$

$p = - 11 - 2 q + 6 r$

$3 p - 2 q - 10 r = 11$

$- 22 - 5 q + 16 r = 11$

$p = - 11 - 2 q + 6 r$

$3 p - 2 q - 10 r = 11$

$- 22 - 5 q + 16 r = 11$

$p = - 11 - 2 q + 6 r$

$3 p - 2 q - 10 r = 11$

$- 22 - 5 q + 16 r = 11$

$p = - 11 - 2 q + 6 r$

$3 p - 2 q - 10 r = 11$

Replace all occurrences of $p$ in $3 p - 2 q - 10 r = 11$ with $- 11 - 2 q + 6 r$ .

$3 (- 11 - 2 q + 6 r) - 2 q - 10 r = 11$

$- 22 - 5 q + 16 r = 11$

$p = - 11 - 2 q + 6 r$

Simplify the left side.

Simplify $3 (- 11 - 2 q + 6 r) - 2 q - 10 r$ .

Simplify each term.

Apply the distributive property.

$3 \cdot - 11 + 3 (- 2 q) + 3 (6 r) - 2 q - 10 r = 11$

$- 22 - 5 q + 16 r = 11$

$p = - 11 - 2 q + 6 r$

Simplify.

Multiply $3$ by $- 11$ .

$- 33 + 3 (- 2 q) + 3 (6 r) - 2 q - 10 r = 11$

$- 22 - 5 q + 16 r = 11$

$p = - 11 - 2 q + 6 r$

Multiply $- 2$ by $3$ .

$- 33 - 6 q + 3 (6 r) - 2 q - 10 r = 11$

$- 22 - 5 q + 16 r = 11$

$p = - 11 - 2 q + 6 r$

Multiply $6$ by $3$ .

$- 33 - 6 q + 18 r - 2 q - 10 r = 11$

$- 22 - 5 q + 16 r = 11$

$p = - 11 - 2 q + 6 r$

$- 33 - 6 q + 18 r - 2 q - 10 r = 11$

$- 22 - 5 q + 16 r = 11$

$p = - 11 - 2 q + 6 r$

$- 33 - 6 q + 18 r - 2 q - 10 r = 11$

$- 22 - 5 q + 16 r = 11$

$p = - 11 - 2 q + 6 r$

Simplify by adding terms.

Subtract $2 q$ from $- 6 q$ .

$- 33 - 8 q + 18 r - 10 r = 11$

$- 22 - 5 q + 16 r = 11$

$p = - 11 - 2 q + 6 r$

Subtract $10 r$ from $18 r$ .

$- 33 - 8 q + 8 r = 11$

$- 22 - 5 q + 16 r = 11$

$p = - 11 - 2 q + 6 r$

$- 33 - 8 q + 8 r = 11$

$- 22 - 5 q + 16 r = 11$

$p = - 11 - 2 q + 6 r$

$- 33 - 8 q + 8 r = 11$

$- 22 - 5 q + 16 r = 11$

$p = - 11 - 2 q + 6 r$

$- 33 - 8 q + 8 r = 11$

$- 22 - 5 q + 16 r = 11$

$p = - 11 - 2 q + 6 r$

$- 33 - 8 q + 8 r = 11$

$- 22 - 5 q + 16 r = 11$

$p = - 11 - 2 q + 6 r$

Solve for $q$ in $- 33 - 8 q + 8 r = 11$ .

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

Add $33$ to both sides of the equation.

$- 8 q + 8 r = 11 + 33$

$- 22 - 5 q + 16 r = 11$

$p = - 11 - 2 q + 6 r$

Subtract $8 r$ from both sides of the equation.

$- 8 q = 11 + 33 - 8 r$

$- 22 - 5 q + 16 r = 11$

$p = - 11 - 2 q + 6 r$

Add $11$ and $33$ .

$- 8 q = 44 - 8 r$

$- 22 - 5 q + 16 r = 11$

$p = - 11 - 2 q + 6 r$

$- 8 q = 44 - 8 r$

$- 22 - 5 q + 16 r = 11$

$p = - 11 - 2 q + 6 r$

Divide each term in $- 8 q = 44 - 8 r$ by $- 8$ and simplify.

Divide each term in $- 8 q = 44 - 8 r$ by $- 8$ .

$\frac{- 8 q}{- 8} = \frac{44}{- 8} + \frac{- 8 r}{- 8}$

$- 22 - 5 q + 16 r = 11$

$p = - 11 - 2 q + 6 r$

Simplify the left side.

Cancel the common factor of $- 8$ .

Cancel the common factor.

$\frac{- 8 q}{- 8} = \frac{44}{- 8} + \frac{- 8 r}{- 8}$

$- 22 - 5 q + 16 r = 11$

$p = - 11 - 2 q + 6 r$

Divide $q$ by $1$ .

$q = \frac{44}{- 8} + \frac{- 8 r}{- 8}$

$- 22 - 5 q + 16 r = 11$

$p = - 11 - 2 q + 6 r$

$q = \frac{44}{- 8} + \frac{- 8 r}{- 8}$

$- 22 - 5 q + 16 r = 11$

$p = - 11 - 2 q + 6 r$

$q = \frac{44}{- 8} + \frac{- 8 r}{- 8}$

$- 22 - 5 q + 16 r = 11$

$p = - 11 - 2 q + 6 r$

Simplify the right side.

Simplify each term.

Cancel the common factor of $44$ and $- 8$ .

Factor $4$ out of $44$ .

$q = \frac{4 (11)}{- 8} + \frac{- 8 r}{- 8}$

$- 22 - 5 q + 16 r = 11$

$p = - 11 - 2 q + 6 r$

Cancel the common factors.

Factor $4$ out of $- 8$ .

$q = \frac{4 \cdot 11}{4 \cdot - 2} + \frac{- 8 r}{- 8}$

$- 22 - 5 q + 16 r = 11$

$p = - 11 - 2 q + 6 r$

Cancel the common factor.

$q = \frac{4 \cdot 11}{4 \cdot - 2} + \frac{- 8 r}{- 8}$

$- 22 - 5 q + 16 r = 11$

$p = - 11 - 2 q + 6 r$

Rewrite the expression.

$q = \frac{11}{- 2} + \frac{- 8 r}{- 8}$

$- 22 - 5 q + 16 r = 11$

$p = - 11 - 2 q + 6 r$

$q = \frac{11}{- 2} + \frac{- 8 r}{- 8}$

$- 22 - 5 q + 16 r = 11$

$p = - 11 - 2 q + 6 r$

$q = \frac{11}{- 2} + \frac{- 8 r}{- 8}$

$- 22 - 5 q + 16 r = 11$

$p = - 11 - 2 q + 6 r$

Move the negative in front of the fraction.

$q = - \frac{11}{2} + \frac{- 8 r}{- 8}$

$- 22 - 5 q + 16 r = 11$

$p = - 11 - 2 q + 6 r$

Cancel the common factor of $- 8$ .

Cancel the common factor.

$q = - \frac{11}{2} + \frac{- 8 r}{- 8}$

$- 22 - 5 q + 16 r = 11$

$p = - 11 - 2 q + 6 r$

Divide $r$ by $1$ .

$q = - \frac{11}{2} + r$

$- 22 - 5 q + 16 r = 11$

$p = - 11 - 2 q + 6 r$

$q = - \frac{11}{2} + r$

$- 22 - 5 q + 16 r = 11$

$p = - 11 - 2 q + 6 r$

$q = - \frac{11}{2} + r$

$- 22 - 5 q + 16 r = 11$

$p = - 11 - 2 q + 6 r$

$q = - \frac{11}{2} + r$

$- 22 - 5 q + 16 r = 11$

$p = - 11 - 2 q + 6 r$

$q = - \frac{11}{2} + r$

$- 22 - 5 q + 16 r = 11$

$p = - 11 - 2 q + 6 r$

$q = - \frac{11}{2} + r$

$- 22 - 5 q + 16 r = 11$

$p = - 11 - 2 q + 6 r$

Replace all occurrences of $q$ with $- \frac{11}{2} + r$ in each equation.

Replace all occurrences of $q$ in $- 22 - 5 q + 16 r = 11$ with $- \frac{11}{2} + r$ .

$- 22 - 5 (- \frac{11}{2} + r) + 16 r = 11$

$q = - \frac{11}{2} + r$

$p = - 11 - 2 q + 6 r$

Simplify the left side.

Simplify $- 22 - 5 (- \frac{11}{2} + r) + 16 r$ .

Simplify each term.

Apply the distributive property.

$- 22 - 5 (- \frac{11}{2}) - 5 r + 16 r = 11$

$q = - \frac{11}{2} + r$

$p = - 11 - 2 q + 6 r$

Multiply $- 5 (- \frac{11}{2})$ .

Multiply $- 1$ by $- 5$ .

$- 22 + 5 (\frac{11}{2}) - 5 r + 16 r = 11$

$q = - \frac{11}{2} + r$

$p = - 11 - 2 q + 6 r$

Combine $5$ and $\frac{11}{2}$ .

$- 22 + \frac{5 \cdot 11}{2} - 5 r + 16 r = 11$

$q = - \frac{11}{2} + r$

$p = - 11 - 2 q + 6 r$

Multiply $5$ by $11$ .

$- 22 + \frac{55}{2} - 5 r + 16 r = 11$

$q = - \frac{11}{2} + r$

$p = - 11 - 2 q + 6 r$

$- 22 + \frac{55}{2} - 5 r + 16 r = 11$

$q = - \frac{11}{2} + r$

$p = - 11 - 2 q + 6 r$

$- 22 + \frac{55}{2} - 5 r + 16 r = 11$

$q = - \frac{11}{2} + r$

$p = - 11 - 2 q + 6 r$

To write $- 22$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- 22 \cdot \frac{2}{2} + \frac{55}{2} - 5 r + 16 r = 11$

$q = - \frac{11}{2} + r$

$p = - 11 - 2 q + 6 r$

Combine $- 22$ and $\frac{2}{2}$ .

$\frac{- 22 \cdot 2}{2} + \frac{55}{2} - 5 r + 16 r = 11$

$q = - \frac{11}{2} + r$

$p = - 11 - 2 q + 6 r$

Combine the numerators over the common denominator.

$\frac{- 22 \cdot 2 + 55}{2} - 5 r + 16 r = 11$

$q = - \frac{11}{2} + r$

$p = - 11 - 2 q + 6 r$

Simplify the numerator.

Multiply $- 22$ by $2$ .

$\frac{- 44 + 55}{2} - 5 r + 16 r = 11$

$q = - \frac{11}{2} + r$

$p = - 11 - 2 q + 6 r$

Add $- 44$ and $55$ .

$\frac{11}{2} - 5 r + 16 r = 11$

$q = - \frac{11}{2} + r$

$p = - 11 - 2 q + 6 r$

$\frac{11}{2} - 5 r + 16 r = 11$

$q = - \frac{11}{2} + r$

$p = - 11 - 2 q + 6 r$

Add $- 5 r$ and $16 r$ .

$\frac{11}{2} + 11 r = 11$

$q = - \frac{11}{2} + r$

$p = - 11 - 2 q + 6 r$

$\frac{11}{2} + 11 r = 11$

$q = - \frac{11}{2} + r$

$p = - 11 - 2 q + 6 r$

$\frac{11}{2} + 11 r = 11$

$q = - \frac{11}{2} + r$

$p = - 11 - 2 q + 6 r$

Replace all occurrences of $q$ in $p = - 11 - 2 q + 6 r$ with $- \frac{11}{2} + r$ .

$p = - 11 - 2 (- \frac{11}{2} + r) + 6 r$

$\frac{11}{2} + 11 r = 11$

$q = - \frac{11}{2} + r$

Simplify the right side.

Simplify $- 11 - 2 (- \frac{11}{2} + r) + 6 r$ .

Simplify each term.

Apply the distributive property.

$p = - 11 - 2 (- \frac{11}{2}) - 2 r + 6 r$

$\frac{11}{2} + 11 r = 11$

$q = - \frac{11}{2} + r$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{11}{2}$ into the numerator.

$p = - 11 - 2 (\frac{- 11}{2}) - 2 r + 6 r$

$\frac{11}{2} + 11 r = 11$

$q = - \frac{11}{2} + r$

Factor $2$ out of $- 2$ .

$p = - 11 + 2 (- 1) (\frac{- 11}{2}) - 2 r + 6 r$

$\frac{11}{2} + 11 r = 11$

$q = - \frac{11}{2} + r$

Cancel the common factor.

$p = - 11 + 2 \cdot (- 1 (\frac{- 11}{2})) - 2 r + 6 r$

$\frac{11}{2} + 11 r = 11$

$q = - \frac{11}{2} + r$

Rewrite the expression.

$p = - 11 - 1 \cdot - 11 - 2 r + 6 r$

$\frac{11}{2} + 11 r = 11$

$q = - \frac{11}{2} + r$

$p = - 11 - 1 \cdot - 11 - 2 r + 6 r$

$\frac{11}{2} + 11 r = 11$

$q = - \frac{11}{2} + r$

Multiply $- 1$ by $- 11$ .

$p = - 11 + 11 - 2 r + 6 r$

$\frac{11}{2} + 11 r = 11$

$q = - \frac{11}{2} + r$

$p = - 11 + 11 - 2 r + 6 r$

$\frac{11}{2} + 11 r = 11$

$q = - \frac{11}{2} + r$

Simplify by adding terms.

Add $- 11$ and $11$ .

$p = 0 - 2 r + 6 r$

$\frac{11}{2} + 11 r = 11$

$q = - \frac{11}{2} + r$

Subtract $2 r$ from $0$ .

$p = - 2 r + 6 r$

$\frac{11}{2} + 11 r = 11$

$q = - \frac{11}{2} + r$

Add $- 2 r$ and $6 r$ .

$p = 4 r$

$\frac{11}{2} + 11 r = 11$

$q = - \frac{11}{2} + r$

$p = 4 r$

$\frac{11}{2} + 11 r = 11$

$q = - \frac{11}{2} + r$

$p = 4 r$

$\frac{11}{2} + 11 r = 11$

$q = - \frac{11}{2} + r$

$p = 4 r$

$\frac{11}{2} + 11 r = 11$

$q = - \frac{11}{2} + r$

$p = 4 r$

$\frac{11}{2} + 11 r = 11$

$q = - \frac{11}{2} + r$

Solve for $r$ in $\frac{11}{2} + 11 r = 11$ .

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

Subtract $\frac{11}{2}$ from both sides of the equation.

$11 r = 11 - \frac{11}{2}$

$p = 4 r$

$q = - \frac{11}{2} + r$

To write $11$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$11 r = 11 \cdot \frac{2}{2} - \frac{11}{2}$

$p = 4 r$

$q = - \frac{11}{2} + r$

Combine $11$ and $\frac{2}{2}$ .

$11 r = \frac{11 \cdot 2}{2} - \frac{11}{2}$

$p = 4 r$

$q = - \frac{11}{2} + r$

Combine the numerators over the common denominator.

$11 r = \frac{11 \cdot 2 - 11}{2}$

$p = 4 r$

$q = - \frac{11}{2} + r$

Simplify the numerator.

Multiply $11$ by $2$ .

$11 r = \frac{22 - 11}{2}$

$p = 4 r$

$q = - \frac{11}{2} + r$

Subtract $11$ from $22$ .

$11 r = \frac{11}{2}$

$p = 4 r$

$q = - \frac{11}{2} + r$

$11 r = \frac{11}{2}$

$p = 4 r$

$q = - \frac{11}{2} + r$

$11 r = \frac{11}{2}$

$p = 4 r$

$q = - \frac{11}{2} + r$

Divide each term in $11 r = \frac{11}{2}$ by $11$ and simplify.

Divide each term in $11 r = \frac{11}{2}$ by $11$ .

$\frac{11 r}{11} = \frac{\frac{11}{2}}{11}$

$p = 4 r$

$q = - \frac{11}{2} + r$

Simplify the left side.

Cancel the common factor of $11$ .

Cancel the common factor.

$\frac{11 r}{11} = \frac{\frac{11}{2}}{11}$

$p = 4 r$

$q = - \frac{11}{2} + r$

Divide $r$ by $1$ .

$r = \frac{\frac{11}{2}}{11}$

$p = 4 r$

$q = - \frac{11}{2} + r$

$r = \frac{\frac{11}{2}}{11}$

$p = 4 r$

$q = - \frac{11}{2} + r$

$r = \frac{\frac{11}{2}}{11}$

$p = 4 r$

$q = - \frac{11}{2} + r$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$r = \frac{11}{2} \cdot \frac{1}{11}$

$p = 4 r$

$q = - \frac{11}{2} + r$

Cancel the common factor of $11$ .

Cancel the common factor.

$r = \frac{11}{2} \cdot \frac{1}{11}$

$p = 4 r$

$q = - \frac{11}{2} + r$

Rewrite the expression.

$r = \frac{1}{2}$

$p = 4 r$

$q = - \frac{11}{2} + r$

$r = \frac{1}{2}$

$p = 4 r$

$q = - \frac{11}{2} + r$

$r = \frac{1}{2}$

$p = 4 r$

$q = - \frac{11}{2} + r$

$r = \frac{1}{2}$

$p = 4 r$

$q = - \frac{11}{2} + r$

$r = \frac{1}{2}$

$p = 4 r$

$q = - \frac{11}{2} + r$

Replace all occurrences of $r$ with $\frac{1}{2}$ in each equation.

Replace all occurrences of $r$ in $p = 4 r$ with $\frac{1}{2}$ .

$p = 4 (\frac{1}{2})$

$r = \frac{1}{2}$

$q = - \frac{11}{2} + r$

Simplify the right side.

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

$p = 2 (2) (\frac{1}{2})$

$r = \frac{1}{2}$

$q = - \frac{11}{2} + r$

Cancel the common factor.

$p = 2 \cdot (2 (\frac{1}{2}))$

$r = \frac{1}{2}$

$q = - \frac{11}{2} + r$

Rewrite the expression.

$p = 2$

$r = \frac{1}{2}$

$q = - \frac{11}{2} + r$

$p = 2$

$r = \frac{1}{2}$

$q = - \frac{11}{2} + r$

$p = 2$

$r = \frac{1}{2}$

$q = - \frac{11}{2} + r$

Replace all occurrences of $r$ in $q = - \frac{11}{2} + r$ with $\frac{1}{2}$ .

$q = - \frac{11}{2} + \frac{1}{2}$

$p = 2$

$r = \frac{1}{2}$

Simplify $q = - \frac{11}{2} + \frac{1}{2}$ .

Simplify the left side.

Remove parentheses.

$q = - \frac{11}{2} + \frac{1}{2}$

$p = 2$

$r = \frac{1}{2}$

$q = - \frac{11}{2} + \frac{1}{2}$

$p = 2$

$r = \frac{1}{2}$

Simplify the right side.

Simplify $- \frac{11}{2} + \frac{1}{2}$ .

Combine the numerators over the common denominator.

$q = \frac{- 11 + 1}{2}$

$p = 2$

$r = \frac{1}{2}$

Simplify the expression.

Add $- 11$ and $1$ .

$q = \frac{- 10}{2}$

$p = 2$

$r = \frac{1}{2}$

Divide $- 10$ by $2$ .

$q = - 5$

$p = 2$

$r = \frac{1}{2}$

$q = - 5$

$p = 2$

$r = \frac{1}{2}$

$q = - 5$

$p = 2$

$r = \frac{1}{2}$

$q = - 5$

$p = 2$

$r = \frac{1}{2}$

$q = - 5$

$p = 2$

$r = \frac{1}{2}$

$q = - 5$

$p = 2$

$r = \frac{1}{2}$

List all of the solutions.

$q = - 5, p = 2, r = \frac{1}{2}$

Solve the System of Equations 2p-q+4r=11 p+2q-6r=-11 3p-2q-10r=11

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