Solve by Addition/Elimination 4a+7b=-51 , 8a+3b=-20 , 6b+3c=18

$4 a + 7 b = - 51$ , $8 a + 3 b = - 20$ , $6 b + 3 c = 18$

Step 1

Choose two equations and eliminate one variable. In this case, eliminate $a$ .

$4 a + 7 b = - 51$

$8 a + 3 b = - 20$

Step 2

Eliminate $a$ from the system.

Multiply each equation by the value that makes the coefficients of $a$ opposite.

$(- 2) \cdot (4 a + 7 b) = (- 2) (- 51)$

$8 a + 3 b = - 20$

Simplify.

Simplify the left side.

Simplify $(- 2) \cdot (4 a + 7 b)$ .

Apply the distributive property.

$- 2 (4 a) - 2 (7 b) = (- 2) (- 51)$

$8 a + 3 b = - 20$

Multiply.

Multiply $4$ by $- 2$ .

$- 8 a - 2 (7 b) = (- 2) (- 51)$

$8 a + 3 b = - 20$

Multiply $7$ by $- 2$ .

$- 8 a - 14 b = (- 2) (- 51)$

$8 a + 3 b = - 20$

$- 8 a - 14 b = (- 2) (- 51)$

$8 a + 3 b = - 20$

$- 8 a - 14 b = (- 2) (- 51)$

$8 a + 3 b = - 20$

$- 8 a - 14 b = (- 2) (- 51)$

$8 a + 3 b = - 20$

Simplify the right side.

Multiply $- 2$ by $- 51$ .

$- 8 a - 14 b = 102$

$8 a + 3 b = - 20$

$- 8 a - 14 b = 102$

$8 a + 3 b = - 20$

$- 8 a - 14 b = 102$

$8 a + 3 b = - 20$

Add the two equations together to eliminate $a$ from the system.

			$-$	$8$	$a$	$-$	$1$	$4$	$b$	$=$	$1$	$0$	$2$
$+$				$8$	$a$	$+$		$3$	$b$	$=$	$-$	$2$	$0$
						$-$	$1$	$1$	$b$	$=$		$8$	$2$

The resultant equation has $a$ eliminated.

$- 11 b = 82$

$- 11 b = 82$

Step 3

Take the resultant equation and the third original equation and eliminate another variable. In this case, eliminate $b$ .

$- 11 b = 82$

$6 b + 3 c = 18$

Step 4

Eliminate $b$ from the system.

Multiply each equation by the value that makes the coefficients of $b$ opposite.

$(6) \cdot (- 11 b) = (6) (82)$

$(11) \cdot (6 b + 3 c) = (11) (18)$

Simplify.

Simplify the left side.

Multiply $- 11$ by $6$ .

$- 66 b = (6) (82)$

$(11) \cdot (6 b + 3 c) = (11) (18)$

$- 66 b = (6) (82)$

$(11) \cdot (6 b + 3 c) = (11) (18)$

Simplify the right side.

Multiply $6$ by $82$ .

$- 66 b = 492$

$(11) \cdot (6 b + 3 c) = (11) (18)$

$- 66 b = 492$

$(11) \cdot (6 b + 3 c) = (11) (18)$

Simplify the left side.

Simplify $(11) \cdot (6 b + 3 c)$ .

Apply the distributive property.

$- 66 b = 492$

$11 (6 b) + 11 (3 c) = (11) (18)$

Multiply.

Multiply $6$ by $11$ .

$- 66 b = 492$

$66 b + 11 (3 c) = (11) (18)$

Multiply $3$ by $11$ .

$- 66 b = 492$

$66 b + 33 c = (11) (18)$

$- 66 b = 492$

$66 b + 33 c = (11) (18)$

$- 66 b = 492$

$66 b + 33 c = (11) (18)$

$- 66 b = 492$

$66 b + 33 c = (11) (18)$

Simplify the right side.

Multiply $11$ by $18$ .

$- 66 b = 492$

$66 b + 33 c = 198$

$- 66 b = 492$

$66 b + 33 c = 198$

$- 66 b = 492$

$66 b + 33 c = 198$

Add the two equations together to eliminate $b$ from the system.

					$-$	$6$	$6$	$b$	$=$	$4$	$9$	$2$
		$+$				$6$	$6$	$b$	$=$	$1$	$9$	$8$
$3$	$3$	$c$							$=$	$6$	$9$	$0$

The resultant equation has $b$ eliminated.

$33 c = 690$

Divide each term in $33 c = 690$ by $33$ and simplify.

Divide each term in $33 c = 690$ by $33$ .

$\frac{33 c}{33} = \frac{690}{33}$

Simplify the left side.

Cancel the common factor of $33$ .

Cancel the common factor.

$\frac{33 c}{33} = \frac{690}{33}$

Divide $c$ by $1$ .

$c = \frac{690}{33}$

$c = \frac{690}{33}$

$c = \frac{690}{33}$

Simplify the right side.

Cancel the common factor of $690$ and $33$ .

Factor $3$ out of $690$ .

$c = \frac{3 (230)}{33}$

Cancel the common factors.

Factor $3$ out of $33$ .

$c = \frac{3 \cdot 230}{3 \cdot 11}$

Cancel the common factor.

$c = \frac{3 \cdot 230}{3 \cdot 11}$

Rewrite the expression.

$c = \frac{230}{11}$

$c = \frac{230}{11}$

$c = \frac{230}{11}$

$c = \frac{230}{11}$

$c = \frac{230}{11}$

$c = \frac{230}{11}$

Step 5

Substitute the value of $c$ into an equation with $a$ eliminated already and solve for the remaining variable.

Substitute the value of $c$ into an equation with $a$ eliminated already.

$- 11 b = 82$

Divide each term in $- 11 b = 82$ by $- 11$ and simplify.

Divide each term in $- 11 b = 82$ by $- 11$ .

$\frac{- 11 b}{- 11} = \frac{82}{- 11}$

Simplify the left side.

Cancel the common factor of $- 11$ .

Cancel the common factor.

$\frac{- 11 b}{- 11} = \frac{82}{- 11}$

Divide $b$ by $1$ .

$b = \frac{82}{- 11}$

$b = \frac{82}{- 11}$

$b = \frac{82}{- 11}$

Simplify the right side.

Move the negative in front of the fraction.

$b = - \frac{82}{11}$

$b = - \frac{82}{11}$

$b = - \frac{82}{11}$

$b = - \frac{82}{11}$

Step 6

Substitute the value of each known variable into one of the initial equations and solve for the last variable.

Substitute the value of each known variable into one of the initial equations.

$4 a + 7 (- \frac{82}{11}) = - 51$

Solve for $a$ .

Simplify each term.

Multiply $7 (- \frac{82}{11})$ .

Multiply $- 1$ by $7$ .

$4 a - 7 (\frac{82}{11}) = - 51$

Combine $- 7$ and $\frac{82}{11}$ .

$4 a + \frac{- 7 \cdot 82}{11} = - 51$

Multiply $- 7$ by $82$ .

$4 a + \frac{- 574}{11} = - 51$

$4 a + \frac{- 574}{11} = - 51$

Move the negative in front of the fraction.

$4 a - \frac{574}{11} = - 51$

$4 a - \frac{574}{11} = - 51$

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

Add $\frac{574}{11}$ to both sides of the equation.

$4 a = - 51 + \frac{574}{11}$

To write $- 51$ as a fraction with a common denominator, multiply by $\frac{11}{11}$ .

$4 a = - 51 \cdot \frac{11}{11} + \frac{574}{11}$

Combine $- 51$ and $\frac{11}{11}$ .

$4 a = \frac{- 51 \cdot 11}{11} + \frac{574}{11}$

Combine the numerators over the common denominator.

$4 a = \frac{- 51 \cdot 11 + 574}{11}$

Simplify the numerator.

Multiply $- 51$ by $11$ .

$4 a = \frac{- 561 + 574}{11}$

Add $- 561$ and $574$ .

$4 a = \frac{13}{11}$

$4 a = \frac{13}{11}$

$4 a = \frac{13}{11}$

Divide each term in $4 a = \frac{13}{11}$ by $4$ and simplify.

Divide each term in $4 a = \frac{13}{11}$ by $4$ .

$\frac{4 a}{4} = \frac{\frac{13}{11}}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 a}{4} = \frac{\frac{13}{11}}{4}$

Divide $a$ by $1$ .

$a = \frac{\frac{13}{11}}{4}$

$a = \frac{\frac{13}{11}}{4}$

$a = \frac{\frac{13}{11}}{4}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$a = \frac{13}{11} \cdot \frac{1}{4}$

Multiply $\frac{13}{11} \cdot \frac{1}{4}$ .

Multiply $\frac{13}{11}$ by $\frac{1}{4}$ .

$a = \frac{13}{11 \cdot 4}$

Multiply $11$ by $4$ .

$a = \frac{13}{44}$

$a = \frac{13}{44}$

$a = \frac{13}{44}$

$a = \frac{13}{44}$

$a = \frac{13}{44}$

$a = \frac{13}{44}$

Step 7

This is the final solution to the system of equations.

$c = \frac{230}{11}$

$b = - \frac{82}{11}$

$a = \frac{13}{44}$

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