Precalculus Examples

$5 a + 4 y + 6 = 0$ , $2 y = - 9 a - 4$

Step 1

Move all terms containing variables to the left.

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Step 1.1

Subtract $6$ from both sides of the equation.

$5 a + 4 y = - 6, 2 y = - 9 a - 4$

Step 1.2

Add $9 a$ to both sides of the equation.

$5 a + 4 y = - 6, 2 y + 9 a = - 4$

Step 2

Reorder the polynomial.

$9 a + 2 y = - 4$

$5 a + 4 y = - 6$

Step 3

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

$5 a + 4 y = - 6$

$(- 2) \cdot (9 a + 2 y) = (- 2) (- 4)$

Step 4

Simplify.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Simplify $(- 2) \cdot (9 a + 2 y)$ .

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Step 4.1.1.1

Apply the distributive property.

$5 a + 4 y = - 6$

$- 2 (9 a) - 2 (2 y) = (- 2) (- 4)$

Step 4.1.1.2

Multiply.

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Step 4.1.1.2.1

Multiply $9$ by $- 2$ .

$5 a + 4 y = - 6$

$- 18 a - 2 (2 y) = (- 2) (- 4)$

Step 4.1.1.2.2

Multiply $2$ by $- 2$ .

$5 a + 4 y = - 6$

$- 18 a - 4 y = (- 2) (- 4)$

$5 a + 4 y = - 6$

$- 18 a - 4 y = (- 2) (- 4)$

$5 a + 4 y = - 6$

$- 18 a - 4 y = (- 2) (- 4)$

$5 a + 4 y = - 6$

$- 18 a - 4 y = (- 2) (- 4)$

Step 4.2

Simplify the right side.

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Step 4.2.1

Multiply $- 2$ by $- 4$ .

$5 a + 4 y = - 6$

$- 18 a - 4 y = 8$

$5 a + 4 y = - 6$

$- 18 a - 4 y = 8$

$5 a + 4 y = - 6$

$- 18 a - 4 y = 8$

Step 5

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

			$5$	$a$	$+$	$4$	$y$	$=$	$-$	$6$
$+$	$-$	$1$	$8$	$a$	$-$	$4$	$y$	$=$		$8$
	$-$	$1$	$3$	$a$				$=$		$2$

Step 6

Divide each term in $- 13 a = 2$ by $- 13$ and simplify.

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Step 6.1

Divide each term in $- 13 a = 2$ by $- 13$ .

$\frac{- 13 a}{- 13} = \frac{2}{- 13}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $- 13$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{- 13 a}{- 13} = \frac{2}{- 13}$

Step 6.2.1.2

Divide $a$ by $1$ .

$a = \frac{2}{- 13}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Move the negative in front of the fraction.

$a = - \frac{2}{13}$

Step 7

Substitute the value found for $a$ into one of the original equations, then solve for $y$ .

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Step 7.1

Substitute the value found for $a$ into one of the original equations to solve for $y$ .

$5 (- \frac{2}{13}) + 4 y = - 6$

Step 7.2

Simplify each term.

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Step 7.2.1

Multiply $5 (- \frac{2}{13})$ .

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Step 7.2.1.1

Multiply $- 1$ by $5$ .

$- 5 (\frac{2}{13}) + 4 y = - 6$

Step 7.2.1.2

Combine $- 5$ and $\frac{2}{13}$ .

$\frac{- 5 \cdot 2}{13} + 4 y = - 6$

Step 7.2.1.3

Multiply $- 5$ by $2$ .

$\frac{- 10}{13} + 4 y = - 6$

Step 7.2.2

Move the negative in front of the fraction.

$- \frac{10}{13} + 4 y = - 6$

Step 7.3

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

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Step 7.3.1

Add $\frac{10}{13}$ to both sides of the equation.

$4 y = - 6 + \frac{10}{13}$

Step 7.3.2

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{13}{13}$ .

$4 y = - 6 \cdot \frac{13}{13} + \frac{10}{13}$

Step 7.3.3

Combine $- 6$ and $\frac{13}{13}$ .

$4 y = \frac{- 6 \cdot 13}{13} + \frac{10}{13}$

Step 7.3.4

Combine the numerators over the common denominator.

$4 y = \frac{- 6 \cdot 13 + 10}{13}$

Step 7.3.5

Simplify the numerator.

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Step 7.3.5.1

Multiply $- 6$ by $13$ .

$4 y = \frac{- 78 + 10}{13}$

Step 7.3.5.2

Add $- 78$ and $10$ .

$4 y = \frac{- 68}{13}$

Step 7.3.6

Move the negative in front of the fraction.

$4 y = - \frac{68}{13}$

Step 7.4

Divide each term in $4 y = - \frac{68}{13}$ by $4$ and simplify.

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Step 7.4.1

Divide each term in $4 y = - \frac{68}{13}$ by $4$ .

$\frac{4 y}{4} = \frac{- \frac{68}{13}}{4}$

Step 7.4.2

Simplify the left side.

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Step 7.4.2.1

Cancel the common factor of $4$ .

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Step 7.4.2.1.1

Cancel the common factor.

$\frac{4 y}{4} = \frac{- \frac{68}{13}}{4}$

Step 7.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{- \frac{68}{13}}{4}$

Step 7.4.3

Simplify the right side.

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Step 7.4.3.1

Multiply the numerator by the reciprocal of the denominator.

$y = - \frac{68}{13} \cdot \frac{1}{4}$

Step 7.4.3.2

Cancel the common factor of $4$ .

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Step 7.4.3.2.1

Move the leading negative in $- \frac{68}{13}$ into the numerator.

$y = \frac{- 68}{13} \cdot \frac{1}{4}$

Step 7.4.3.2.2

Factor $4$ out of $- 68$ .

$y = \frac{4 (- 17)}{13} \cdot \frac{1}{4}$

Step 7.4.3.2.3

Cancel the common factor.

$y = \frac{4 \cdot - 17}{13} \cdot \frac{1}{4}$

Step 7.4.3.2.4

Rewrite the expression.

$y = \frac{- 17}{13}$

Step 7.4.3.3

Move the negative in front of the fraction.

$y = - \frac{17}{13}$

Step 8

This is the final solution to the independent system of equations.

$a = - \frac{2}{13}$

$y = - \frac{17}{13}$

			$5$	$a$	$+$	$4$	$y$	$=$	$-$	$6$
$+$	$-$	$1$	$8$	$a$	$-$	$4$	$y$	$=$		$8$
	$-$	$1$	$3$	$a$				$=$		$2$

			$5$	$a$	$+$	$4$	$y$	$=$	$-$	$6$
$+$	$-$	$1$	$8$	$a$	$-$	$4$	$y$	$=$		$8$
	$-$	$1$	$3$	$a$				$=$		$2$

			$5$	$a$	$+$	$4$	$y$	$=$	$-$	$6$
$+$	$-$	$1$	$8$	$a$	$-$	$4$	$y$	$=$		$8$
	$-$	$1$	$3$	$a$				$=$		$2$