Solve the System of Equations 5a+b+c=11 a+6b+3c=-1 2a+b-c=-1

$5 a + b + c = 11$ $a + 6 b + 3 c = - 1$ $2 a + b - c = - 1$

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

Subtract $5 a$ from both sides of the equation.

$b + c = 11 - 5 a$

$a + 6 b + 3 c = - 1$

$2 a + b - c = - 1$

Subtract $c$ from both sides of the equation.

$b = 11 - 5 a - c$

$a + 6 b + 3 c = - 1$

$2 a + b - c = - 1$

$b = 11 - 5 a - c$

$a + 6 b + 3 c = - 1$

$2 a + b - c = - 1$

Replace all occurrences of $b$ with $11 - 5 a - c$ in each equation.

Replace all occurrences of $b$ in $a + 6 b + 3 c = - 1$ with $11 - 5 a - c$ .

$a + 6 (11 - 5 a - c) + 3 c = - 1$

$b = 11 - 5 a - c$

$2 a + b - c = - 1$

Simplify the left side.

Simplify $a + 6 (11 - 5 a - c) + 3 c$ .

Simplify each term.

Apply the distributive property.

$a + 6 \cdot 11 + 6 (- 5 a) + 6 (- c) + 3 c = - 1$

$b = 11 - 5 a - c$

$2 a + b - c = - 1$

Simplify.

Multiply $6$ by $11$ .

$a + 66 + 6 (- 5 a) + 6 (- c) + 3 c = - 1$

$b = 11 - 5 a - c$

$2 a + b - c = - 1$

Multiply $- 5$ by $6$ .

$a + 66 - 30 a + 6 (- c) + 3 c = - 1$

$b = 11 - 5 a - c$

$2 a + b - c = - 1$

Multiply $- 1$ by $6$ .

$a + 66 - 30 a - 6 c + 3 c = - 1$

$b = 11 - 5 a - c$

$2 a + b - c = - 1$

$a + 66 - 30 a - 6 c + 3 c = - 1$

$b = 11 - 5 a - c$

$2 a + b - c = - 1$

$a + 66 - 30 a - 6 c + 3 c = - 1$

$b = 11 - 5 a - c$

$2 a + b - c = - 1$

Simplify by adding terms.

Subtract $30 a$ from $a$ .

$- 29 a + 66 - 6 c + 3 c = - 1$

$b = 11 - 5 a - c$

$2 a + b - c = - 1$

Add $- 6 c$ and $3 c$ .

$- 29 a + 66 - 3 c = - 1$

$b = 11 - 5 a - c$

$2 a + b - c = - 1$

$- 29 a + 66 - 3 c = - 1$

$b = 11 - 5 a - c$

$2 a + b - c = - 1$

$- 29 a + 66 - 3 c = - 1$

$b = 11 - 5 a - c$

$2 a + b - c = - 1$

$- 29 a + 66 - 3 c = - 1$

$b = 11 - 5 a - c$

$2 a + b - c = - 1$

Replace all occurrences of $b$ in $2 a + b - c = - 1$ with $11 - 5 a - c$ .

$2 a + 11 - 5 a - c - c = - 1$

$- 29 a + 66 - 3 c = - 1$

$b = 11 - 5 a - c$

Simplify the left side.

Simplify $2 a + 11 - 5 a - c - c$ .

Remove parentheses.

$2 a + 11 - 5 a - c - c = - 1$

$- 29 a + 66 - 3 c = - 1$

$b = 11 - 5 a - c$

Subtract $5 a$ from $2 a$ .

$- 3 a + 11 - c - c = - 1$

$- 29 a + 66 - 3 c = - 1$

$b = 11 - 5 a - c$

Subtract $c$ from $- c$ .

$- 3 a + 11 - 2 c = - 1$

$- 29 a + 66 - 3 c = - 1$

$b = 11 - 5 a - c$

$- 3 a + 11 - 2 c = - 1$

$- 29 a + 66 - 3 c = - 1$

$b = 11 - 5 a - c$

$- 3 a + 11 - 2 c = - 1$

$- 29 a + 66 - 3 c = - 1$

$b = 11 - 5 a - c$

$- 3 a + 11 - 2 c = - 1$

$- 29 a + 66 - 3 c = - 1$

$b = 11 - 5 a - c$

Solve for $a$ in $- 3 a + 11 - 2 c = - 1$ .

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

Subtract $11$ from both sides of the equation.

$- 3 a - 2 c = - 1 - 11$

$- 29 a + 66 - 3 c = - 1$

$b = 11 - 5 a - c$

Add $2 c$ to both sides of the equation.

$- 3 a = - 1 - 11 + 2 c$

$- 29 a + 66 - 3 c = - 1$

$b = 11 - 5 a - c$

Subtract $11$ from $- 1$ .

$- 3 a = - 12 + 2 c$

$- 29 a + 66 - 3 c = - 1$

$b = 11 - 5 a - c$

$- 3 a = - 12 + 2 c$

$- 29 a + 66 - 3 c = - 1$

$b = 11 - 5 a - c$

Divide each term in $- 3 a = - 12 + 2 c$ by $- 3$ and simplify.

Divide each term in $- 3 a = - 12 + 2 c$ by $- 3$ .

$\frac{- 3 a}{- 3} = \frac{- 12}{- 3} + \frac{2 c}{- 3}$

$- 29 a + 66 - 3 c = - 1$

$b = 11 - 5 a - c$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 a}{- 3} = \frac{- 12}{- 3} + \frac{2 c}{- 3}$

$- 29 a + 66 - 3 c = - 1$

$b = 11 - 5 a - c$

Divide $a$ by $1$ .

$a = \frac{- 12}{- 3} + \frac{2 c}{- 3}$

$- 29 a + 66 - 3 c = - 1$

$b = 11 - 5 a - c$

$a = \frac{- 12}{- 3} + \frac{2 c}{- 3}$

$- 29 a + 66 - 3 c = - 1$

$b = 11 - 5 a - c$

$a = \frac{- 12}{- 3} + \frac{2 c}{- 3}$

$- 29 a + 66 - 3 c = - 1$

$b = 11 - 5 a - c$

Simplify the right side.

Simplify each term.

Divide $- 12$ by $- 3$ .

$a = 4 + \frac{2 c}{- 3}$

$- 29 a + 66 - 3 c = - 1$

$b = 11 - 5 a - c$

Move the negative in front of the fraction.

$a = 4 - \frac{2 c}{3}$

$- 29 a + 66 - 3 c = - 1$

$b = 11 - 5 a - c$

$a = 4 - \frac{2 c}{3}$

$- 29 a + 66 - 3 c = - 1$

$b = 11 - 5 a - c$

$a = 4 - \frac{2 c}{3}$

$- 29 a + 66 - 3 c = - 1$

$b = 11 - 5 a - c$

$a = 4 - \frac{2 c}{3}$

$- 29 a + 66 - 3 c = - 1$

$b = 11 - 5 a - c$

$a = 4 - \frac{2 c}{3}$

$- 29 a + 66 - 3 c = - 1$

$b = 11 - 5 a - c$

Replace all occurrences of $a$ with $4 - \frac{2 c}{3}$ in each equation.

Replace all occurrences of $a$ in $- 29 a + 66 - 3 c = - 1$ with $4 - \frac{2 c}{3}$ .

$- 29 (4 - \frac{2 c}{3}) + 66 - 3 c = - 1$

$a = 4 - \frac{2 c}{3}$

$b = 11 - 5 a - c$

Simplify the left side.

Simplify $- 29 (4 - \frac{2 c}{3}) + 66 - 3 c$ .

Simplify each term.

Apply the distributive property.

$- 29 \cdot 4 - 29 (- \frac{2 c}{3}) + 66 - 3 c = - 1$

$a = 4 - \frac{2 c}{3}$

$b = 11 - 5 a - c$

Multiply $- 29$ by $4$ .

$- 116 - 29 (- \frac{2 c}{3}) + 66 - 3 c = - 1$

$a = 4 - \frac{2 c}{3}$

$b = 11 - 5 a - c$

Multiply $- 29 (- \frac{2 c}{3})$ .

Multiply $- 1$ by $- 29$ .

$- 116 + 29 (\frac{2 c}{3}) + 66 - 3 c = - 1$

$a = 4 - \frac{2 c}{3}$

$b = 11 - 5 a - c$

Combine $29$ and $\frac{2 c}{3}$ .

$- 116 + \frac{29 (2 c)}{3} + 66 - 3 c = - 1$

$a = 4 - \frac{2 c}{3}$

$b = 11 - 5 a - c$

Multiply $2$ by $29$ .

$- 116 + \frac{58 c}{3} + 66 - 3 c = - 1$

$a = 4 - \frac{2 c}{3}$

$b = 11 - 5 a - c$

$- 116 + \frac{58 c}{3} + 66 - 3 c = - 1$

$a = 4 - \frac{2 c}{3}$

$b = 11 - 5 a - c$

$- 116 + \frac{58 c}{3} + 66 - 3 c = - 1$

$a = 4 - \frac{2 c}{3}$

$b = 11 - 5 a - c$

Add $- 116$ and $66$ .

$\frac{58 c}{3} - 50 - 3 c = - 1$

$a = 4 - \frac{2 c}{3}$

$b = 11 - 5 a - c$

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

$\frac{58 c}{3} - 3 c \cdot \frac{3}{3} - 50 = - 1$

$a = 4 - \frac{2 c}{3}$

$b = 11 - 5 a - c$

Simplify terms.

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

$\frac{58 c}{3} + \frac{- 3 c \cdot 3}{3} - 50 = - 1$

$a = 4 - \frac{2 c}{3}$

$b = 11 - 5 a - c$

Combine the numerators over the common denominator.

$\frac{58 c - 3 c \cdot 3}{3} - 50 = - 1$

$a = 4 - \frac{2 c}{3}$

$b = 11 - 5 a - c$

$\frac{58 c - 3 c \cdot 3}{3} - 50 = - 1$

$a = 4 - \frac{2 c}{3}$

$b = 11 - 5 a - c$

Simplify each term.

Simplify the numerator.

Factor $c$ out of $58 c - 3 c \cdot 3$ .

Factor $c$ out of $58 c$ .

$\frac{c \cdot 58 - 3 c \cdot 3}{3} - 50 = - 1$

$a = 4 - \frac{2 c}{3}$

$b = 11 - 5 a - c$

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

$\frac{c \cdot 58 + c (- 3 \cdot 3)}{3} - 50 = - 1$

$a = 4 - \frac{2 c}{3}$

$b = 11 - 5 a - c$

Factor $c$ out of $c \cdot 58 + c (- 3 \cdot 3)$ .

$\frac{c (58 - 3 \cdot 3)}{3} - 50 = - 1$

$a = 4 - \frac{2 c}{3}$

$b = 11 - 5 a - c$

$\frac{c (58 - 3 \cdot 3)}{3} - 50 = - 1$

$a = 4 - \frac{2 c}{3}$

$b = 11 - 5 a - c$

Multiply $- 3$ by $3$ .

$\frac{c (58 - 9)}{3} - 50 = - 1$

$a = 4 - \frac{2 c}{3}$

$b = 11 - 5 a - c$

Subtract $9$ from $58$ .

$\frac{c \cdot 49}{3} - 50 = - 1$

$a = 4 - \frac{2 c}{3}$

$b = 11 - 5 a - c$

$\frac{c \cdot 49}{3} - 50 = - 1$

$a = 4 - \frac{2 c}{3}$

$b = 11 - 5 a - c$

Move $49$ to the left of $c$ .

$\frac{49 c}{3} - 50 = - 1$

$a = 4 - \frac{2 c}{3}$

$b = 11 - 5 a - c$

$\frac{49 c}{3} - 50 = - 1$

$a = 4 - \frac{2 c}{3}$

$b = 11 - 5 a - c$

$\frac{49 c}{3} - 50 = - 1$

$a = 4 - \frac{2 c}{3}$

$b = 11 - 5 a - c$

$\frac{49 c}{3} - 50 = - 1$

$a = 4 - \frac{2 c}{3}$

$b = 11 - 5 a - c$

Replace all occurrences of $a$ in $b = 11 - 5 a - c$ with $4 - \frac{2 c}{3}$ .

$b = 11 - 5 (4 - \frac{2 c}{3}) - c$

$\frac{49 c}{3} - 50 = - 1$

$a = 4 - \frac{2 c}{3}$

Simplify the right side.

Simplify $11 - 5 (4 - \frac{2 c}{3}) - c$ .

Simplify each term.

Apply the distributive property.

$b = 11 - 5 \cdot 4 - 5 (- \frac{2 c}{3}) - c$

$\frac{49 c}{3} - 50 = - 1$

$a = 4 - \frac{2 c}{3}$

Multiply $- 5$ by $4$ .

$b = 11 - 20 - 5 (- \frac{2 c}{3}) - c$

$\frac{49 c}{3} - 50 = - 1$

$a = 4 - \frac{2 c}{3}$

Multiply $- 5 (- \frac{2 c}{3})$ .

Multiply $- 1$ by $- 5$ .

$b = 11 - 20 + 5 (\frac{2 c}{3}) - c$

$\frac{49 c}{3} - 50 = - 1$

$a = 4 - \frac{2 c}{3}$

Combine $5$ and $\frac{2 c}{3}$ .

$b = 11 - 20 + \frac{5 (2 c)}{3} - c$

$\frac{49 c}{3} - 50 = - 1$

$a = 4 - \frac{2 c}{3}$

Multiply $2$ by $5$ .

$b = 11 - 20 + \frac{10 c}{3} - c$

$\frac{49 c}{3} - 50 = - 1$

$a = 4 - \frac{2 c}{3}$

$b = 11 - 20 + \frac{10 c}{3} - c$

$\frac{49 c}{3} - 50 = - 1$

$a = 4 - \frac{2 c}{3}$

$b = 11 - 20 + \frac{10 c}{3} - c$

$\frac{49 c}{3} - 50 = - 1$

$a = 4 - \frac{2 c}{3}$

Subtract $20$ from $11$ .

$b = - 9 + \frac{10 c}{3} - c$

$\frac{49 c}{3} - 50 = - 1$

$a = 4 - \frac{2 c}{3}$

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

$b = - 9 + \frac{10 c}{3} - c \cdot \frac{3}{3}$

$\frac{49 c}{3} - 50 = - 1$

$a = 4 - \frac{2 c}{3}$

Simplify terms.

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

$b = - 9 + \frac{10 c}{3} + \frac{- c \cdot 3}{3}$

$\frac{49 c}{3} - 50 = - 1$

$a = 4 - \frac{2 c}{3}$

Combine the numerators over the common denominator.

$b = - 9 + \frac{10 c - c \cdot 3}{3}$

$\frac{49 c}{3} - 50 = - 1$

$a = 4 - \frac{2 c}{3}$

$b = - 9 + \frac{10 c - c \cdot 3}{3}$

$\frac{49 c}{3} - 50 = - 1$

$a = 4 - \frac{2 c}{3}$

Simplify each term.

Simplify the numerator.

Factor $c$ out of $10 c - c \cdot 3$ .

Factor $c$ out of $10 c$ .

$b = - 9 + \frac{c \cdot 10 - c \cdot 3}{3}$

$\frac{49 c}{3} - 50 = - 1$

$a = 4 - \frac{2 c}{3}$

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

$b = - 9 + \frac{c \cdot 10 + c (- 1 \cdot 3)}{3}$

$\frac{49 c}{3} - 50 = - 1$

$a = 4 - \frac{2 c}{3}$

Factor $c$ out of $c \cdot 10 + c (- 1 \cdot 3)$ .

$b = - 9 + \frac{c (10 - 1 \cdot 3)}{3}$

$\frac{49 c}{3} - 50 = - 1$

$a = 4 - \frac{2 c}{3}$

$b = - 9 + \frac{c (10 - 1 \cdot 3)}{3}$

$\frac{49 c}{3} - 50 = - 1$

$a = 4 - \frac{2 c}{3}$

Multiply $- 1$ by $3$ .

$b = - 9 + \frac{c (10 - 3)}{3}$

$\frac{49 c}{3} - 50 = - 1$

$a = 4 - \frac{2 c}{3}$

Subtract $3$ from $10$ .

$b = - 9 + \frac{c \cdot 7}{3}$

$\frac{49 c}{3} - 50 = - 1$

$a = 4 - \frac{2 c}{3}$

$b = - 9 + \frac{c \cdot 7}{3}$

$\frac{49 c}{3} - 50 = - 1$

$a = 4 - \frac{2 c}{3}$

Move $7$ to the left of $c$ .

$b = - 9 + \frac{7 c}{3}$

$\frac{49 c}{3} - 50 = - 1$

$a = 4 - \frac{2 c}{3}$

$b = - 9 + \frac{7 c}{3}$

$\frac{49 c}{3} - 50 = - 1$

$a = 4 - \frac{2 c}{3}$

$b = - 9 + \frac{7 c}{3}$

$\frac{49 c}{3} - 50 = - 1$

$a = 4 - \frac{2 c}{3}$

$b = - 9 + \frac{7 c}{3}$

$\frac{49 c}{3} - 50 = - 1$

$a = 4 - \frac{2 c}{3}$

$b = - 9 + \frac{7 c}{3}$

$\frac{49 c}{3} - 50 = - 1$

$a = 4 - \frac{2 c}{3}$

Solve for $c$ in $\frac{49 c}{3} - 50 = - 1$ .

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

Add $50$ to both sides of the equation.

$\frac{49 c}{3} = - 1 + 50$

$b = - 9 + \frac{7 c}{3}$

$a = 4 - \frac{2 c}{3}$

Add $- 1$ and $50$ .

$\frac{49 c}{3} = 49$

$b = - 9 + \frac{7 c}{3}$

$a = 4 - \frac{2 c}{3}$

$\frac{49 c}{3} = 49$

$b = - 9 + \frac{7 c}{3}$

$a = 4 - \frac{2 c}{3}$

Multiply both sides of the equation by $\frac{3}{49}$ .

$\frac{3}{49} \cdot \frac{49 c}{3} = \frac{3}{49} \cdot 49$

$b = - 9 + \frac{7 c}{3}$

$a = 4 - \frac{2 c}{3}$

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{3}{49} \cdot \frac{49 c}{3}$ .

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3}{49} \cdot \frac{49 c}{3} = \frac{3}{49} \cdot 49$

$b = - 9 + \frac{7 c}{3}$

$a = 4 - \frac{2 c}{3}$

Rewrite the expression.

$\frac{1}{49} \cdot (49 c) = \frac{3}{49} \cdot 49$

$b = - 9 + \frac{7 c}{3}$

$a = 4 - \frac{2 c}{3}$

$\frac{1}{49} \cdot (49 c) = \frac{3}{49} \cdot 49$

$b = - 9 + \frac{7 c}{3}$

$a = 4 - \frac{2 c}{3}$

Cancel the common factor of $49$ .

Factor $49$ out of $49 c$ .

$\frac{1}{49} \cdot (49 (c)) = \frac{3}{49} \cdot 49$

$b = - 9 + \frac{7 c}{3}$

$a = 4 - \frac{2 c}{3}$

Cancel the common factor.

$\frac{1}{49} \cdot (49 c) = \frac{3}{49} \cdot 49$

$b = - 9 + \frac{7 c}{3}$

$a = 4 - \frac{2 c}{3}$

Rewrite the expression.

$c = \frac{3}{49} \cdot 49$

$b = - 9 + \frac{7 c}{3}$

$a = 4 - \frac{2 c}{3}$

$c = \frac{3}{49} \cdot 49$

$b = - 9 + \frac{7 c}{3}$

$a = 4 - \frac{2 c}{3}$

$c = \frac{3}{49} \cdot 49$

$b = - 9 + \frac{7 c}{3}$

$a = 4 - \frac{2 c}{3}$

$c = \frac{3}{49} \cdot 49$

$b = - 9 + \frac{7 c}{3}$

$a = 4 - \frac{2 c}{3}$

Simplify the right side.

Cancel the common factor of $49$ .

Cancel the common factor.

$c = \frac{3}{49} \cdot 49$

$b = - 9 + \frac{7 c}{3}$

$a = 4 - \frac{2 c}{3}$

Rewrite the expression.

$c = 3$

$b = - 9 + \frac{7 c}{3}$

$a = 4 - \frac{2 c}{3}$

$c = 3$

$b = - 9 + \frac{7 c}{3}$

$a = 4 - \frac{2 c}{3}$

$c = 3$

$b = - 9 + \frac{7 c}{3}$

$a = 4 - \frac{2 c}{3}$

$c = 3$

$b = - 9 + \frac{7 c}{3}$

$a = 4 - \frac{2 c}{3}$

$c = 3$

$b = - 9 + \frac{7 c}{3}$

$a = 4 - \frac{2 c}{3}$

Replace all occurrences of $c$ with $3$ in each equation.

Replace all occurrences of $c$ in $b = - 9 + \frac{7 c}{3}$ with $3$ .

$b = - 9 + \frac{7 (3)}{3}$

$c = 3$

$a = 4 - \frac{2 c}{3}$

Simplify the right side.

Simplify $- 9 + \frac{7 (3)}{3}$ .

Multiply $7$ by $3$ .

$b = - 9 + \frac{21}{3}$

$c = 3$

$a = 4 - \frac{2 c}{3}$

Divide $21$ by $3$ .

$b = - 9 + 7$

$c = 3$

$a = 4 - \frac{2 c}{3}$

Add $- 9$ and $7$ .

$b = - 2$

$c = 3$

$a = 4 - \frac{2 c}{3}$

$b = - 2$

$c = 3$

$a = 4 - \frac{2 c}{3}$

$b = - 2$

$c = 3$

$a = 4 - \frac{2 c}{3}$

Replace all occurrences of $c$ in $a = 4 - \frac{2 c}{3}$ with $3$ .

$a = 4 - \frac{2 (3)}{3}$

$b = - 2$

$c = 3$

Simplify the right side.

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

Simplify each term.

Cancel the common factor of $3$ .

Cancel the common factor.

$a = 4 - \frac{2 \cdot 3}{3}$

$b = - 2$

$c = 3$

Divide $2$ by $1$ .

$a = 4 - 1 \cdot 2$

$b = - 2$

$c = 3$

$a = 4 - 1 \cdot 2$

$b = - 2$

$c = 3$

Multiply $- 1$ by $2$ .

$a = 4 - 2$

$b = - 2$

$c = 3$

$a = 4 - 2$

$b = - 2$

$c = 3$

Subtract $2$ from $4$ .

$a = 2$

$b = - 2$

$c = 3$

$a = 2$

$b = - 2$

$c = 3$

$a = 2$

$b = - 2$

$c = 3$

$a = 2$

$b = - 2$

$c = 3$

List all of the solutions.

$a = 2, b = - 2, c = 3$

Solve the System of Equations 5a+b+c=11 a+6b+3c=-1 2a+b-c=-1

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