Algebra Examples

$7 p = 9 (p + q) + 11$ $9 q + 3 = - 4 (7 q + p)$

Step 1

Solve for $p$ in $7 p = 9 (p + q) + 11$ .

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Step 1.1

Apply the distributive property.

$7 p = 9 p + 9 q + 11$

$9 q + 3 = - 4 (7 q + p)$

Step 1.2

Move all terms containing $p$ to the left side of the equation.

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Step 1.2.1

Subtract $9 p$ from both sides of the equation.

$7 p - 9 p = 9 q + 11$

$9 q + 3 = - 4 (7 q + p)$

Step 1.2.2

Subtract $9 p$ from $7 p$ .

$- 2 p = 9 q + 11$

$9 q + 3 = - 4 (7 q + p)$

$- 2 p = 9 q + 11$

$9 q + 3 = - 4 (7 q + p)$

Step 1.3

Divide each term in $- 2 p = 9 q + 11$ by $- 2$ and simplify.

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Step 1.3.1

Divide each term in $- 2 p = 9 q + 11$ by $- 2$ .

$\frac{- 2 p}{- 2} = \frac{9 q}{- 2} + \frac{11}{- 2}$

$9 q + 3 = - 4 (7 q + p)$

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Cancel the common factor of $- 2$ .

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Step 1.3.2.1.1

Cancel the common factor.

$\frac{- 2 p}{- 2} = \frac{9 q}{- 2} + \frac{11}{- 2}$

$9 q + 3 = - 4 (7 q + p)$

Step 1.3.2.1.2

Divide $p$ by $1$ .

$p = \frac{9 q}{- 2} + \frac{11}{- 2}$

$9 q + 3 = - 4 (7 q + p)$

$p = \frac{9 q}{- 2} + \frac{11}{- 2}$

$9 q + 3 = - 4 (7 q + p)$

$p = \frac{9 q}{- 2} + \frac{11}{- 2}$

$9 q + 3 = - 4 (7 q + p)$

Step 1.3.3

Simplify the right side.

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Step 1.3.3.1

Simplify each term.

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Step 1.3.3.1.1

Move the negative in front of the fraction.

$p = - \frac{9 q}{2} + \frac{11}{- 2}$

$9 q + 3 = - 4 (7 q + p)$

Step 1.3.3.1.2

Move the negative in front of the fraction.

$p = - \frac{9 q}{2} - \frac{11}{2}$

$9 q + 3 = - 4 (7 q + p)$

$p = - \frac{9 q}{2} - \frac{11}{2}$

$9 q + 3 = - 4 (7 q + p)$

$p = - \frac{9 q}{2} - \frac{11}{2}$

$9 q + 3 = - 4 (7 q + p)$

$p = - \frac{9 q}{2} - \frac{11}{2}$

$9 q + 3 = - 4 (7 q + p)$

$p = - \frac{9 q}{2} - \frac{11}{2}$

$9 q + 3 = - 4 (7 q + p)$

Step 2

Replace all occurrences of $p$ with $- \frac{9 q}{2} - \frac{11}{2}$ in each equation.

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Step 2.1

Replace all occurrences of $p$ in $9 q + 3 = - 4 (7 q + p)$ with $- \frac{9 q}{2} - \frac{11}{2}$ .

$9 q + 3 = - 4 (7 q - \frac{9 q}{2} - \frac{11}{2})$

$p = - \frac{9 q}{2} - \frac{11}{2}$

Step 2.2

Simplify $9 q + 3 = - 4 (7 q - \frac{9 q}{2} - \frac{11}{2})$ .

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Step 2.2.1

Simplify the left side.

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Step 2.2.1.1

Remove parentheses.

$9 q + 3 = - 4 (7 q - \frac{9 q}{2} - \frac{11}{2})$

$p = - \frac{9 q}{2} - \frac{11}{2}$

$9 q + 3 = - 4 (7 q - \frac{9 q}{2} - \frac{11}{2})$

$p = - \frac{9 q}{2} - \frac{11}{2}$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Simplify $- 4 (7 q - \frac{9 q}{2} - \frac{11}{2})$ .

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Step 2.2.2.1.1

To write $7 q$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$9 q + 3 = - 4 (7 q \cdot \frac{2}{2} - \frac{9 q}{2} - \frac{11}{2})$

$p = - \frac{9 q}{2} - \frac{11}{2}$

Step 2.2.2.1.2

Combine $7 q$ and $\frac{2}{2}$ .

$9 q + 3 = - 4 (\frac{7 q \cdot 2}{2} - \frac{9 q}{2} - \frac{11}{2})$

$p = - \frac{9 q}{2} - \frac{11}{2}$

Step 2.2.2.1.3

Combine the numerators over the common denominator.

$9 q + 3 = - 4 (\frac{7 q \cdot 2 - 9 q}{2} - \frac{11}{2})$

$p = - \frac{9 q}{2} - \frac{11}{2}$

Step 2.2.2.1.4

Combine the numerators over the common denominator.

$9 q + 3 = - 4 \frac{7 q \cdot 2 - 9 q - 11}{2}$

$p = - \frac{9 q}{2} - \frac{11}{2}$

Step 2.2.2.1.5

Multiply $2$ by $7$ .

$9 q + 3 = - 4 \frac{14 q - 9 q - 11}{2}$

$p = - \frac{9 q}{2} - \frac{11}{2}$

Step 2.2.2.1.6

Subtract $9 q$ from $14 q$ .

$9 q + 3 = - 4 \frac{5 q - 11}{2}$

$p = - \frac{9 q}{2} - \frac{11}{2}$

Step 2.2.2.1.7

Cancel the common factor of $2$ .

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Step 2.2.2.1.7.1

Factor $2$ out of $- 4$ .

$9 q + 3 = 2 (- 2) (\frac{5 q - 11}{2})$

$p = - \frac{9 q}{2} - \frac{11}{2}$

Step 2.2.2.1.7.2

Cancel the common factor.

$9 q + 3 = 2 \cdot (- 2 \frac{5 q - 11}{2})$

$p = - \frac{9 q}{2} - \frac{11}{2}$

Step 2.2.2.1.7.3

Rewrite the expression.

$9 q + 3 = - 2 (5 q - 11)$

$p = - \frac{9 q}{2} - \frac{11}{2}$

$9 q + 3 = - 2 (5 q - 11)$

$p = - \frac{9 q}{2} - \frac{11}{2}$

Step 2.2.2.1.8

Apply the distributive property.

$9 q + 3 = - 2 (5 q) - 2 \cdot - 11$

$p = - \frac{9 q}{2} - \frac{11}{2}$

Step 2.2.2.1.9

Multiply.

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Step 2.2.2.1.9.1

Multiply $5$ by $- 2$ .

$9 q + 3 = - 10 q - 2 \cdot - 11$

$p = - \frac{9 q}{2} - \frac{11}{2}$

Step 2.2.2.1.9.2

Multiply $- 2$ by $- 11$ .

$9 q + 3 = - 10 q + 22$

$p = - \frac{9 q}{2} - \frac{11}{2}$

$9 q + 3 = - 10 q + 22$

$p = - \frac{9 q}{2} - \frac{11}{2}$

$9 q + 3 = - 10 q + 22$

$p = - \frac{9 q}{2} - \frac{11}{2}$

$9 q + 3 = - 10 q + 22$

$p = - \frac{9 q}{2} - \frac{11}{2}$

$9 q + 3 = - 10 q + 22$

$p = - \frac{9 q}{2} - \frac{11}{2}$

$9 q + 3 = - 10 q + 22$

$p = - \frac{9 q}{2} - \frac{11}{2}$

Step 3

Solve for $q$ in $9 q + 3 = - 10 q + 22$ .

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Step 3.1

Move all terms containing $q$ to the left side of the equation.

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Step 3.1.1

Add $10 q$ to both sides of the equation.

$9 q + 3 + 10 q = 22$

$p = - \frac{9 q}{2} - \frac{11}{2}$

Step 3.1.2

Add $9 q$ and $10 q$ .

$19 q + 3 = 22$

$p = - \frac{9 q}{2} - \frac{11}{2}$

$19 q + 3 = 22$

$p = - \frac{9 q}{2} - \frac{11}{2}$

Step 3.2

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

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Step 3.2.1

Subtract $3$ from both sides of the equation.

$19 q = 22 - 3$

$p = - \frac{9 q}{2} - \frac{11}{2}$

Step 3.2.2

Subtract $3$ from $22$ .

$19 q = 19$

$p = - \frac{9 q}{2} - \frac{11}{2}$

$19 q = 19$

$p = - \frac{9 q}{2} - \frac{11}{2}$

Step 3.3

Divide each term in $19 q = 19$ by $19$ and simplify.

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Step 3.3.1

Divide each term in $19 q = 19$ by $19$ .

$\frac{19 q}{19} = \frac{19}{19}$

$p = - \frac{9 q}{2} - \frac{11}{2}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $19$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{19 q}{19} = \frac{19}{19}$

$p = - \frac{9 q}{2} - \frac{11}{2}$

Step 3.3.2.1.2

Divide $q$ by $1$ .

$q = \frac{19}{19}$

$p = - \frac{9 q}{2} - \frac{11}{2}$

$q = \frac{19}{19}$

$p = - \frac{9 q}{2} - \frac{11}{2}$

$q = \frac{19}{19}$

$p = - \frac{9 q}{2} - \frac{11}{2}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Divide $19$ by $19$ .

$q = 1$

$p = - \frac{9 q}{2} - \frac{11}{2}$

$q = 1$

$p = - \frac{9 q}{2} - \frac{11}{2}$

$q = 1$

$p = - \frac{9 q}{2} - \frac{11}{2}$

$q = 1$

$p = - \frac{9 q}{2} - \frac{11}{2}$

Step 4

Replace all occurrences of $q$ with $1$ in each equation.

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Step 4.1

Replace all occurrences of $q$ in $p = - \frac{9 q}{2} - \frac{11}{2}$ with $1$ .

$p = - \frac{9 (1)}{2} - \frac{11}{2}$

$q = 1$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- \frac{9 (1)}{2} - \frac{11}{2}$ .

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Step 4.2.1.1

Combine the numerators over the common denominator.

$p = \frac{- 9 \cdot 1 - 11}{2}$

$q = 1$

Step 4.2.1.2

Simplify the expression.

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Step 4.2.1.2.1

Multiply $- 9$ by $1$ .

$p = \frac{- 9 - 11}{2}$

$q = 1$

Step 4.2.1.2.2

Subtract $11$ from $- 9$ .

$p = \frac{- 20}{2}$

$q = 1$

Step 4.2.1.2.3

Divide $- 20$ by $2$ .

$p = - 10$

$q = 1$

$p = - 10$

$q = 1$

$p = - 10$

$q = 1$

$p = - 10$

$q = 1$

$p = - 10$

$q = 1$

Step 5

List all of the solutions.

$p = - 10, q = 1$