Algebra Examples

$p - q = 8$ $2 p - q = 19$

Step 1

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

$p - q = 8$

$(- 1) \cdot (2 p - q) = (- 1) (19)$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $(- 1) \cdot (2 p - q)$ .

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Step 2.1.1.1

Apply the distributive property.

$p - q = 8$

$- 1 (2 p) - 1 (- q) = (- 1) (19)$

Step 2.1.1.2

Multiply $2$ by $- 1$ .

$p - q = 8$

$- 2 p - 1 (- q) = (- 1) (19)$

Step 2.1.1.3

Multiply $- 1 (- q)$ .

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Step 2.1.1.3.1

Multiply $- 1$ by $- 1$ .

$p - q = 8$

$- 2 p + 1 q = (- 1) (19)$

Step 2.1.1.3.2

Multiply $q$ by $1$ .

$p - q = 8$

$- 2 p + q = (- 1) (19)$

$p - q = 8$

$- 2 p + q = (- 1) (19)$

$p - q = 8$

$- 2 p + q = (- 1) (19)$

$p - q = 8$

$- 2 p + q = (- 1) (19)$

Step 2.2

Simplify the right side.

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Step 2.2.1

Multiply $- 1$ by $19$ .

$p - q = 8$

$- 2 p + q = - 19$

$p - q = 8$

$- 2 p + q = - 19$

$p - q = 8$

$- 2 p + q = - 19$

Step 3

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

			$p$	$-$	$q$	$=$			$8$
$+$	$-$	$2$	$p$	$+$	$q$	$=$	$-$	$1$	$9$
		$-$	$p$			$=$	$-$	$1$	$1$

Step 4

Divide each term in $- p = - 11$ by $- 1$ and simplify.

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Step 4.1

Divide each term in $- p = - 11$ by $- 1$ .

$\frac{- p}{- 1} = \frac{- 11}{- 1}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Dividing two negative values results in a positive value.

$\frac{p}{1} = \frac{- 11}{- 1}$

Step 4.2.2

Divide $p$ by $1$ .

$p = \frac{- 11}{- 1}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $- 11$ by $- 1$ .

$p = 11$

Step 5

Substitute the value found for $p$ into one of the original equations, then solve for $q$ .

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Step 5.1

Substitute the value found for $p$ into one of the original equations to solve for $q$ .

$(11) - q = 8$

Step 5.2

Remove parentheses.

$11 - q = 8$

Step 5.3

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

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Step 5.3.1

Subtract $11$ from both sides of the equation.

$- q = 8 - 11$

Step 5.3.2

Subtract $11$ from $8$ .

$- q = - 3$

Step 5.4

Divide each term in $- q = - 3$ by $- 1$ and simplify.

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Step 5.4.1

Divide each term in $- q = - 3$ by $- 1$ .

$\frac{- q}{- 1} = \frac{- 3}{- 1}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Dividing two negative values results in a positive value.

$\frac{q}{1} = \frac{- 3}{- 1}$

Step 5.4.2.2

Divide $q$ by $1$ .

$q = \frac{- 3}{- 1}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Divide $- 3$ by $- 1$ .

$q = 3$

Step 6

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

$p = 11$

$q = 3$

			$p$	$-$	$q$	$=$			$8$
$+$	$-$	$2$	$p$	$+$	$q$	$=$	$-$	$1$	$9$
		$-$	$p$			$=$	$-$	$1$	$1$

			$p$	$-$	$q$	$=$			$8$
$+$	$-$	$2$	$p$	$+$	$q$	$=$	$-$	$1$	$9$
		$-$	$p$			$=$	$-$	$1$	$1$

			$p$	$-$	$q$	$=$			$8$
$+$	$-$	$2$	$p$	$+$	$q$	$=$	$-$	$1$	$9$
		$-$	$p$			$=$	$-$	$1$	$1$