Algebra Examples

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

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.

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Step 2.1

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

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

$8 a + 3 b = - 20$

Step 2.2

Simplify.

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Step 2.2.1

Simplify the left side.

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Step 2.2.1.1

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

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Step 2.2.1.1.1

Apply the distributive property.

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

$8 a + 3 b = - 20$

Step 2.2.1.1.2

Multiply.

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Step 2.2.1.1.2.1

Multiply $4$ by $- 2$ .

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

$8 a + 3 b = - 20$

Step 2.2.1.1.2.2

Multiply $7$ by $- 2$ .

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

$8 a + 3 b = - 20$

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

$8 a + 3 b = - 20$

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

$8 a + 3 b = - 20$

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

$8 a + 3 b = - 20$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Multiply $- 2$ by $- 51$ .

$- 8 a - 14 b = 102$

$8 a + 3 b = - 20$

$- 8 a - 14 b = 102$

$8 a + 3 b = - 20$

$- 8 a - 14 b = 102$

$8 a + 3 b = - 20$

Step 2.3

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

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

Step 2.4

The resultant equation has $a$ eliminated.

$- 11 b = 82$

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.

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Step 4.1

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

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

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

Step 4.2

Simplify.

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

Multiply $- 11$ by $6$ .

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

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

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

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

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Multiply $6$ by $82$ .

$- 66 b = 492$

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

$- 66 b = 492$

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

Step 4.2.3

Simplify the left side.

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Step 4.2.3.1

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

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Step 4.2.3.1.1

Apply the distributive property.

$- 66 b = 492$

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

Step 4.2.3.1.2

Multiply.

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Step 4.2.3.1.2.1

Multiply $6$ by $11$ .

$- 66 b = 492$

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

Step 4.2.3.1.2.2

Multiply $3$ by $11$ .

$- 66 b = 492$

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

$- 66 b = 492$

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

$- 66 b = 492$

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

$- 66 b = 492$

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

Step 4.2.4

Simplify the right side.

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Step 4.2.4.1

Multiply $11$ by $18$ .

$- 66 b = 492$

$66 b + 33 c = 198$

$- 66 b = 492$

$66 b + 33 c = 198$

$- 66 b = 492$

$66 b + 33 c = 198$

Step 4.3

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

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

Step 4.4

The resultant equation has $b$ eliminated.

$33 c = 690$

Step 4.5

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

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Step 4.5.1

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

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

Step 4.5.2

Simplify the left side.

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Step 4.5.2.1

Cancel the common factor of $33$ .

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Step 4.5.2.1.1

Cancel the common factor.

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

Step 4.5.2.1.2

Divide $c$ by $1$ .

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

Step 4.5.3

Simplify the right side.

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Step 4.5.3.1

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

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Step 4.5.3.1.1

Factor $3$ out of $690$ .

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

Step 4.5.3.1.2

Cancel the common factors.

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Step 4.5.3.1.2.1

Factor $3$ out of $33$ .

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

Step 4.5.3.1.2.2

Cancel the common factor.

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

Step 4.5.3.1.2.3

Rewrite the expression.

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

Step 5

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

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Step 5.1

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

$- 11 b = 82$

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $- 11$ .

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Step 5.2.2.1.1

Cancel the common factor.

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

Step 5.2.2.1.2

Divide $b$ by $1$ .

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

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Move the negative in front of the fraction.

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

Step 6

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

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Step 6.1

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

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

Step 6.2

Solve for $a$ .

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

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

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Step 6.2.1.1.1

Multiply $- 1$ by $7$ .

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

Step 6.2.1.1.2

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

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

Step 6.2.1.1.3

Multiply $- 7$ by $82$ .

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

Step 6.2.1.2

Move the negative in front of the fraction.

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

Step 6.2.2

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

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Step 6.2.2.1

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

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

Step 6.2.2.2

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

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

Step 6.2.2.3

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

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

Step 6.2.2.4

Combine the numerators over the common denominator.

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

Step 6.2.2.5

Simplify the numerator.

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Step 6.2.2.5.1

Multiply $- 51$ by $11$ .

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

Step 6.2.2.5.2

Add $- 561$ and $574$ .

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

Step 6.2.3

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

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Step 6.2.3.1

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

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

Step 6.2.3.2

Simplify the left side.

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Step 6.2.3.2.1

Cancel the common factor of $4$ .

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Step 6.2.3.2.1.1

Cancel the common factor.

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

Step 6.2.3.2.1.2

Divide $a$ by $1$ .

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

Step 6.2.3.3

Simplify the right side.

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Step 6.2.3.3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 6.2.3.3.2

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

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Step 6.2.3.3.2.1

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

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

Step 6.2.3.3.2.2

Multiply $11$ by $4$ .

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

Step 7

This is the final solution to the system of equations.

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

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

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

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

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

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

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

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

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