Finite Math Examples

$5 p + 11 q = - 7$ , $3 p - 8 q = 25$

Step 1

Solve for $p$ in $5 p + 11 q = - 7$ .

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Step 1.1

Subtract $11 q$ from both sides of the equation.

$5 p = - 7 - 11 q$

$3 p - 8 q = 25$

Step 1.2

Divide each term in $5 p = - 7 - 11 q$ by $5$ and simplify.

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Step 1.2.1

Divide each term in $5 p = - 7 - 11 q$ by $5$ .

$\frac{5 p}{5} = \frac{- 7}{5} + \frac{- 11 q}{5}$

$3 p - 8 q = 25$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $5$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{5 p}{5} = \frac{- 7}{5} + \frac{- 11 q}{5}$

$3 p - 8 q = 25$

Step 1.2.2.1.2

Divide $p$ by $1$ .

$p = \frac{- 7}{5} + \frac{- 11 q}{5}$

$3 p - 8 q = 25$

$p = \frac{- 7}{5} + \frac{- 11 q}{5}$

$3 p - 8 q = 25$

$p = \frac{- 7}{5} + \frac{- 11 q}{5}$

$3 p - 8 q = 25$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Move the negative in front of the fraction.

$p = - \frac{7}{5} + \frac{- 11 q}{5}$

$3 p - 8 q = 25$

Step 1.2.3.1.2

Move the negative in front of the fraction.

$p = - \frac{7}{5} - \frac{11 q}{5}$

$3 p - 8 q = 25$

$p = - \frac{7}{5} - \frac{11 q}{5}$

$3 p - 8 q = 25$

$p = - \frac{7}{5} - \frac{11 q}{5}$

$3 p - 8 q = 25$

$p = - \frac{7}{5} - \frac{11 q}{5}$

$3 p - 8 q = 25$

$p = - \frac{7}{5} - \frac{11 q}{5}$

$3 p - 8 q = 25$

Step 2

Replace all occurrences of $p$ with $- \frac{7}{5} - \frac{11 q}{5}$ in each equation.

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Step 2.1

Replace all occurrences of $p$ in $3 p - 8 q = 25$ with $- \frac{7}{5} - \frac{11 q}{5}$ .

$3 (- \frac{7}{5} - \frac{11 q}{5}) - 8 q = 25$

$p = - \frac{7}{5} - \frac{11 q}{5}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $3 (- \frac{7}{5} - \frac{11 q}{5}) - 8 q$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Apply the distributive property.

$3 (- \frac{7}{5}) + 3 (- \frac{11 q}{5}) - 8 q = 25$

$p = - \frac{7}{5} - \frac{11 q}{5}$

Step 2.2.1.1.2

Multiply $3 (- \frac{7}{5})$ .

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Step 2.2.1.1.2.1

Multiply $- 1$ by $3$ .

$- 3 (\frac{7}{5}) + 3 (- \frac{11 q}{5}) - 8 q = 25$

$p = - \frac{7}{5} - \frac{11 q}{5}$

Step 2.2.1.1.2.2

Combine $- 3$ and $\frac{7}{5}$ .

$\frac{- 3 \cdot 7}{5} + 3 (- \frac{11 q}{5}) - 8 q = 25$

$p = - \frac{7}{5} - \frac{11 q}{5}$

Step 2.2.1.1.2.3

Multiply $- 3$ by $7$ .

$\frac{- 21}{5} + 3 (- \frac{11 q}{5}) - 8 q = 25$

$p = - \frac{7}{5} - \frac{11 q}{5}$

$\frac{- 21}{5} + 3 (- \frac{11 q}{5}) - 8 q = 25$

$p = - \frac{7}{5} - \frac{11 q}{5}$

Step 2.2.1.1.3

Multiply $3 (- \frac{11 q}{5})$ .

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Step 2.2.1.1.3.1

Multiply $- 1$ by $3$ .

$\frac{- 21}{5} - 3 \frac{11 q}{5} - 8 q = 25$

$p = - \frac{7}{5} - \frac{11 q}{5}$

Step 2.2.1.1.3.2

Combine $- 3$ and $\frac{11 q}{5}$ .

$\frac{- 21}{5} + \frac{- 3 (11 q)}{5} - 8 q = 25$

$p = - \frac{7}{5} - \frac{11 q}{5}$

Step 2.2.1.1.3.3

Multiply $11$ by $- 3$ .

$\frac{- 21}{5} + \frac{- 33 q}{5} - 8 q = 25$

$p = - \frac{7}{5} - \frac{11 q}{5}$

$\frac{- 21}{5} + \frac{- 33 q}{5} - 8 q = 25$

$p = - \frac{7}{5} - \frac{11 q}{5}$

Step 2.2.1.1.4

Simplify each term.

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Step 2.2.1.1.4.1

Move the negative in front of the fraction.

$- \frac{21}{5} + \frac{- 33 q}{5} - 8 q = 25$

$p = - \frac{7}{5} - \frac{11 q}{5}$

Step 2.2.1.1.4.2

Move the negative in front of the fraction.

$- \frac{21}{5} - \frac{33 q}{5} - 8 q = 25$

$p = - \frac{7}{5} - \frac{11 q}{5}$

$- \frac{21}{5} - \frac{33 q}{5} - 8 q = 25$

$p = - \frac{7}{5} - \frac{11 q}{5}$

$- \frac{21}{5} - \frac{33 q}{5} - 8 q = 25$

$p = - \frac{7}{5} - \frac{11 q}{5}$

Step 2.2.1.2

To write $- 8 q$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- \frac{21}{5} - \frac{33 q}{5} - 8 q \cdot \frac{5}{5} = 25$

$p = - \frac{7}{5} - \frac{11 q}{5}$

Step 2.2.1.3

Combine $- 8 q$ and $\frac{5}{5}$ .

$- \frac{21}{5} - \frac{33 q}{5} + \frac{- 8 q \cdot 5}{5} = 25$

$p = - \frac{7}{5} - \frac{11 q}{5}$

Step 2.2.1.4

Combine the numerators over the common denominator.

$- \frac{21}{5} + \frac{- 33 q - 8 q \cdot 5}{5} = 25$

$p = - \frac{7}{5} - \frac{11 q}{5}$

Step 2.2.1.5

Combine the numerators over the common denominator.

$\frac{- 21 - 33 q - 8 q \cdot 5}{5} = 25$

$p = - \frac{7}{5} - \frac{11 q}{5}$

Step 2.2.1.6

Multiply $5$ by $- 8$ .

$\frac{- 21 - 33 q - 40 q}{5} = 25$

$p = - \frac{7}{5} - \frac{11 q}{5}$

Step 2.2.1.7

Subtract $40 q$ from $- 33 q$ .

$\frac{- 21 - 73 q}{5} = 25$

$p = - \frac{7}{5} - \frac{11 q}{5}$

Step 2.2.1.8

Rewrite $- 21$ as $- 1 (21)$ .

$\frac{- 1 \cdot 21 - 73 q}{5} = 25$

$p = - \frac{7}{5} - \frac{11 q}{5}$

Step 2.2.1.9

Factor $- 1$ out of $- 73 q$ .

$\frac{- 1 \cdot 21 - (73 q)}{5} = 25$

$p = - \frac{7}{5} - \frac{11 q}{5}$

Step 2.2.1.10

Factor $- 1$ out of $- 1 (21) - (73 q)$ .

$\frac{- 1 (21 + 73 q)}{5} = 25$

$p = - \frac{7}{5} - \frac{11 q}{5}$

Step 2.2.1.11

Move the negative in front of the fraction.

$- \frac{21 + 73 q}{5} = 25$

$p = - \frac{7}{5} - \frac{11 q}{5}$

$- \frac{21 + 73 q}{5} = 25$

$p = - \frac{7}{5} - \frac{11 q}{5}$

$- \frac{21 + 73 q}{5} = 25$

$p = - \frac{7}{5} - \frac{11 q}{5}$

$- \frac{21 + 73 q}{5} = 25$

$p = - \frac{7}{5} - \frac{11 q}{5}$

Step 3

Solve for $q$ in $- \frac{21 + 73 q}{5} = 25$ .

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Step 3.1

Multiply both sides of the equation by $- 5$ .

$- 5 (- \frac{21 + 73 q}{5}) = - 5 \cdot 25$

$p = - \frac{7}{5} - \frac{11 q}{5}$

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $- 5 (- \frac{21 + 73 q}{5})$ .

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Step 3.2.1.1.1

Cancel the common factor of $5$ .

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Step 3.2.1.1.1.1

Move the leading negative in $- \frac{21 + 73 q}{5}$ into the numerator.

$- 5 \frac{- (21 + 73 q)}{5} = - 5 \cdot 25$

$p = - \frac{7}{5} - \frac{11 q}{5}$

Step 3.2.1.1.1.2

Factor $5$ out of $- 5$ .

$5 (- 1) (\frac{- (21 + 73 q)}{5}) = - 5 \cdot 25$

$p = - \frac{7}{5} - \frac{11 q}{5}$

Step 3.2.1.1.1.3

Cancel the common factor.

$5 \cdot (- 1 \frac{- (21 + 73 q)}{5}) = - 5 \cdot 25$

$p = - \frac{7}{5} - \frac{11 q}{5}$

Step 3.2.1.1.1.4

Rewrite the expression.

$21 + 73 q = - 5 \cdot 25$

$p = - \frac{7}{5} - \frac{11 q}{5}$

$21 + 73 q = - 5 \cdot 25$

$p = - \frac{7}{5} - \frac{11 q}{5}$

Step 3.2.1.1.2

Multiply.

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Step 3.2.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 (21 + 73 q) = - 5 \cdot 25$

$p = - \frac{7}{5} - \frac{11 q}{5}$

Step 3.2.1.1.2.2

Multiply $21 + 73 q$ by $1$ .

$21 + 73 q = - 5 \cdot 25$

$p = - \frac{7}{5} - \frac{11 q}{5}$

$21 + 73 q = - 5 \cdot 25$

$p = - \frac{7}{5} - \frac{11 q}{5}$

$21 + 73 q = - 5 \cdot 25$

$p = - \frac{7}{5} - \frac{11 q}{5}$

$21 + 73 q = - 5 \cdot 25$

$p = - \frac{7}{5} - \frac{11 q}{5}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Multiply $- 5$ by $25$ .

$21 + 73 q = - 125$

$p = - \frac{7}{5} - \frac{11 q}{5}$

$21 + 73 q = - 125$

$p = - \frac{7}{5} - \frac{11 q}{5}$

$21 + 73 q = - 125$

$p = - \frac{7}{5} - \frac{11 q}{5}$

Step 3.3

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

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Step 3.3.1

Subtract $21$ from both sides of the equation.

$73 q = - 125 - 21$

$p = - \frac{7}{5} - \frac{11 q}{5}$

Step 3.3.2

Subtract $21$ from $- 125$ .

$73 q = - 146$

$p = - \frac{7}{5} - \frac{11 q}{5}$

$73 q = - 146$

$p = - \frac{7}{5} - \frac{11 q}{5}$

Step 3.4

Divide each term in $73 q = - 146$ by $73$ and simplify.

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Step 3.4.1

Divide each term in $73 q = - 146$ by $73$ .

$\frac{73 q}{73} = \frac{- 146}{73}$

$p = - \frac{7}{5} - \frac{11 q}{5}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $73$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{73 q}{73} = \frac{- 146}{73}$

$p = - \frac{7}{5} - \frac{11 q}{5}$

Step 3.4.2.1.2

Divide $q$ by $1$ .

$q = \frac{- 146}{73}$

$p = - \frac{7}{5} - \frac{11 q}{5}$

$q = \frac{- 146}{73}$

$p = - \frac{7}{5} - \frac{11 q}{5}$

$q = \frac{- 146}{73}$

$p = - \frac{7}{5} - \frac{11 q}{5}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Divide $- 146$ by $73$ .

$q = - 2$

$p = - \frac{7}{5} - \frac{11 q}{5}$

$q = - 2$

$p = - \frac{7}{5} - \frac{11 q}{5}$

$q = - 2$

$p = - \frac{7}{5} - \frac{11 q}{5}$

$q = - 2$

$p = - \frac{7}{5} - \frac{11 q}{5}$

Step 4

Replace all occurrences of $q$ with $- 2$ in each equation.

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Step 4.1

Replace all occurrences of $q$ in $p = - \frac{7}{5} - \frac{11 q}{5}$ with $- 2$ .

$p = - \frac{7}{5} - \frac{11 (- 2)}{5}$

$q = - 2$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- \frac{7}{5} - \frac{11 (- 2)}{5}$ .

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Step 4.2.1.1

Combine the numerators over the common denominator.

$p = \frac{- 7 - 11 \cdot - 2}{5}$

$q = - 2$

Step 4.2.1.2

Simplify the expression.

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Step 4.2.1.2.1

Multiply $- 11$ by $- 2$ .

$p = \frac{- 7 + 22}{5}$

$q = - 2$

Step 4.2.1.2.2

Add $- 7$ and $22$ .

$p = \frac{15}{5}$

$q = - 2$

Step 4.2.1.2.3

Divide $15$ by $5$ .

$p = 3$

$q = - 2$

$p = 3$

$q = - 2$

$p = 3$

$q = - 2$

$p = 3$

$q = - 2$

$p = 3$

$q = - 2$

Step 5

List all of the solutions.

$p = 3, q = - 2$