Finite Math Examples

$2 p + 5 q = 100$ , $p - 2 q = 20$

Step 1

Add $2 q$ to both sides of the equation.

$p = 20 + 2 q$

$2 p + 5 q = 100$

Step 2

Replace all occurrences of $p$ with $20 + 2 q$ in each equation.

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Step 2.1

Replace all occurrences of $p$ in $2 p + 5 q = 100$ with $20 + 2 q$ .

$2 (20 + 2 q) + 5 q = 100$

$p = 20 + 2 q$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $2 (20 + 2 q) + 5 q$ .

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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.

$2 \cdot 20 + 2 (2 q) + 5 q = 100$

$p = 20 + 2 q$

Step 2.2.1.1.2

Multiply $2$ by $20$ .

$40 + 2 (2 q) + 5 q = 100$

$p = 20 + 2 q$

Step 2.2.1.1.3

Multiply $2$ by $2$ .

$40 + 4 q + 5 q = 100$

$p = 20 + 2 q$

$40 + 4 q + 5 q = 100$

$p = 20 + 2 q$

Step 2.2.1.2

Add $4 q$ and $5 q$ .

$40 + 9 q = 100$

$p = 20 + 2 q$

$40 + 9 q = 100$

$p = 20 + 2 q$

$40 + 9 q = 100$

$p = 20 + 2 q$

$40 + 9 q = 100$

$p = 20 + 2 q$

Step 3

Solve for $q$ in $40 + 9 q = 100$ .

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Step 3.1

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

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Step 3.1.1

Subtract $40$ from both sides of the equation.

$9 q = 100 - 40$

$p = 20 + 2 q$

Step 3.1.2

Subtract $40$ from $100$ .

$9 q = 60$

$p = 20 + 2 q$

$9 q = 60$

$p = 20 + 2 q$

Step 3.2

Divide each term in $9 q = 60$ by $9$ and simplify.

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Step 3.2.1

Divide each term in $9 q = 60$ by $9$ .

$\frac{9 q}{9} = \frac{60}{9}$

$p = 20 + 2 q$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $9$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{9 q}{9} = \frac{60}{9}$

$p = 20 + 2 q$

Step 3.2.2.1.2

Divide $q$ by $1$ .

$q = \frac{60}{9}$

$p = 20 + 2 q$

$q = \frac{60}{9}$

$p = 20 + 2 q$

$q = \frac{60}{9}$

$p = 20 + 2 q$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Cancel the common factor of $60$ and $9$ .

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Step 3.2.3.1.1

Factor $3$ out of $60$ .

$q = \frac{3 (20)}{9}$

$p = 20 + 2 q$

Step 3.2.3.1.2

Cancel the common factors.

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Step 3.2.3.1.2.1

Factor $3$ out of $9$ .

$q = \frac{3 \cdot 20}{3 \cdot 3}$

$p = 20 + 2 q$

Step 3.2.3.1.2.2

Cancel the common factor.

$q = \frac{3 \cdot 20}{3 \cdot 3}$

$p = 20 + 2 q$

Step 3.2.3.1.2.3

Rewrite the expression.

$q = \frac{20}{3}$

$p = 20 + 2 q$

$q = \frac{20}{3}$

$p = 20 + 2 q$

$q = \frac{20}{3}$

$p = 20 + 2 q$

$q = \frac{20}{3}$

$p = 20 + 2 q$

$q = \frac{20}{3}$

$p = 20 + 2 q$

$q = \frac{20}{3}$

$p = 20 + 2 q$

Step 4

Replace all occurrences of $q$ with $\frac{20}{3}$ in each equation.

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Step 4.1

Replace all occurrences of $q$ in $p = 20 + 2 q$ with $\frac{20}{3}$ .

$p = 20 + 2 (\frac{20}{3})$

$q = \frac{20}{3}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $20 + 2 (\frac{20}{3})$ .

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Step 4.2.1.1

Multiply $2 (\frac{20}{3})$ .

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Step 4.2.1.1.1

Combine $2$ and $\frac{20}{3}$ .

$p = 20 + \frac{2 \cdot 20}{3}$

$q = \frac{20}{3}$

Step 4.2.1.1.2

Multiply $2$ by $20$ .

$p = 20 + \frac{40}{3}$

$q = \frac{20}{3}$

$p = 20 + \frac{40}{3}$

$q = \frac{20}{3}$

Step 4.2.1.2

To write $20$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$p = 20 \cdot \frac{3}{3} + \frac{40}{3}$

$q = \frac{20}{3}$

Step 4.2.1.3

Combine $20$ and $\frac{3}{3}$ .

$p = \frac{20 \cdot 3}{3} + \frac{40}{3}$

$q = \frac{20}{3}$

Step 4.2.1.4

Combine the numerators over the common denominator.

$p = \frac{20 \cdot 3 + 40}{3}$

$q = \frac{20}{3}$

Step 4.2.1.5

Simplify the numerator.

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Step 4.2.1.5.1

Multiply $20$ by $3$ .

$p = \frac{60 + 40}{3}$

$q = \frac{20}{3}$

Step 4.2.1.5.2

Add $60$ and $40$ .

$p = \frac{100}{3}$

$q = \frac{20}{3}$

$p = \frac{100}{3}$

$q = \frac{20}{3}$

$p = \frac{100}{3}$

$q = \frac{20}{3}$

$p = \frac{100}{3}$

$q = \frac{20}{3}$

$p = \frac{100}{3}$

$q = \frac{20}{3}$

Step 5

List all of the solutions.

$p = \frac{100}{3}, q = \frac{20}{3}$