Finite Math Examples

$40 a + 60 b = 32000$ , $1.5 a + 2 b = 60000$

Step 1

Solve for $a$ in $40 a + 60 b = 32000$ .

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Step 1.1

Subtract $60 b$ from both sides of the equation.

$40 a = 32000 - 60 b$

$1.5 a + 2 b = 60000$

Step 1.2

Divide each term in $40 a = 32000 - 60 b$ by $40$ and simplify.

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Step 1.2.1

Divide each term in $40 a = 32000 - 60 b$ by $40$ .

$\frac{40 a}{40} = \frac{32000}{40} + \frac{- 60 b}{40}$

$1.5 a + 2 b = 60000$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $40$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{40 a}{40} = \frac{32000}{40} + \frac{- 60 b}{40}$

$1.5 a + 2 b = 60000$

Step 1.2.2.1.2

Divide $a$ by $1$ .

$a = \frac{32000}{40} + \frac{- 60 b}{40}$

$1.5 a + 2 b = 60000$

$a = \frac{32000}{40} + \frac{- 60 b}{40}$

$1.5 a + 2 b = 60000$

$a = \frac{32000}{40} + \frac{- 60 b}{40}$

$1.5 a + 2 b = 60000$

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

Divide $32000$ by $40$ .

$a = 800 + \frac{- 60 b}{40}$

$1.5 a + 2 b = 60000$

Step 1.2.3.1.2

Cancel the common factor of $- 60$ and $40$ .

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Step 1.2.3.1.2.1

Factor $20$ out of $- 60 b$ .

$a = 800 + \frac{20 (- 3 b)}{40}$

$1.5 a + 2 b = 60000$

Step 1.2.3.1.2.2

Cancel the common factors.

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Step 1.2.3.1.2.2.1

Factor $20$ out of $40$ .

$a = 800 + \frac{20 (- 3 b)}{20 (2)}$

$1.5 a + 2 b = 60000$

Step 1.2.3.1.2.2.2

Cancel the common factor.

$a = 800 + \frac{20 (- 3 b)}{20 \cdot 2}$

$1.5 a + 2 b = 60000$

Step 1.2.3.1.2.2.3

Rewrite the expression.

$a = 800 + \frac{- 3 b}{2}$

$1.5 a + 2 b = 60000$

$a = 800 + \frac{- 3 b}{2}$

$1.5 a + 2 b = 60000$

$a = 800 + \frac{- 3 b}{2}$

$1.5 a + 2 b = 60000$

Step 1.2.3.1.3

Move the negative in front of the fraction.

$a = 800 - \frac{3 b}{2}$

$1.5 a + 2 b = 60000$

$a = 800 - \frac{3 b}{2}$

$1.5 a + 2 b = 60000$

$a = 800 - \frac{3 b}{2}$

$1.5 a + 2 b = 60000$

$a = 800 - \frac{3 b}{2}$

$1.5 a + 2 b = 60000$

$a = 800 - \frac{3 b}{2}$

$1.5 a + 2 b = 60000$

Step 2

Replace all occurrences of $a$ with $800 - \frac{3 b}{2}$ in each equation.

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Step 2.1

Replace all occurrences of $a$ in $1.5 a + 2 b = 60000$ with $800 - \frac{3 b}{2}$ .

$1.5 (800 - \frac{3 b}{2}) + 2 b = 60000$

$a = 800 - \frac{3 b}{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $1.5 (800 - \frac{3 b}{2}) + 2 b$ .

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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.

$1.5 \cdot 800 + 1.5 (- \frac{3 b}{2}) + 2 b = 60000$

$a = 800 - \frac{3 b}{2}$

Step 2.2.1.1.2

Multiply $1.5$ by $800$ .

$1200 + 1.5 (- \frac{3 b}{2}) + 2 b = 60000$

$a = 800 - \frac{3 b}{2}$

Step 2.2.1.1.3

Multiply $1.5 (- \frac{3 b}{2})$ .

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Step 2.2.1.1.3.1

Multiply $- 1$ by $1.5$ .

$1200 - 1.5 \frac{3 b}{2} + 2 b = 60000$

$a = 800 - \frac{3 b}{2}$

Step 2.2.1.1.3.2

Combine $- 1.5$ and $\frac{3 b}{2}$ .

$1200 + \frac{- 1.5 (3 b)}{2} + 2 b = 60000$

$a = 800 - \frac{3 b}{2}$

Step 2.2.1.1.3.3

Multiply $3$ by $- 1.5$ .

$1200 + \frac{- 4.5 b}{2} + 2 b = 60000$

$a = 800 - \frac{3 b}{2}$

$1200 + \frac{- 4.5 b}{2} + 2 b = 60000$

$a = 800 - \frac{3 b}{2}$

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.

$1200 - \frac{4.5 b}{2} + 2 b = 60000$

$a = 800 - \frac{3 b}{2}$

Step 2.2.1.1.4.2

Factor $4.5$ out of $4.5 b$ .

$1200 - \frac{4.5 (b)}{2} + 2 b = 60000$

$a = 800 - \frac{3 b}{2}$

Step 2.2.1.1.4.3

Factor $2$ out of $2$ .

$1200 - \frac{4.5 (b)}{2 (1)} + 2 b = 60000$

$a = 800 - \frac{3 b}{2}$

Step 2.2.1.1.4.4

Separate fractions.

$1200 - (\frac{4.5}{2} \cdot \frac{b}{1}) + 2 b = 60000$

$a = 800 - \frac{3 b}{2}$

Step 2.2.1.1.4.5

Divide $4.5$ by $2$ .

$1200 - (2.25 (\frac{b}{1})) + 2 b = 60000$

$a = 800 - \frac{3 b}{2}$

Step 2.2.1.1.4.6

Divide $b$ by $1$ .

$1200 - (2.25 b) + 2 b = 60000$

$a = 800 - \frac{3 b}{2}$

Step 2.2.1.1.4.7

Multiply $2.25$ by $- 1$ .

$1200 - 2.25 b + 2 b = 60000$

$a = 800 - \frac{3 b}{2}$

$1200 - 2.25 b + 2 b = 60000$

$a = 800 - \frac{3 b}{2}$

$1200 - 2.25 b + 2 b = 60000$

$a = 800 - \frac{3 b}{2}$

Step 2.2.1.2

Add $- 2.25 b$ and $2 b$ .

$1200 - 0.25 b = 60000$

$a = 800 - \frac{3 b}{2}$

$1200 - 0.25 b = 60000$

$a = 800 - \frac{3 b}{2}$

$1200 - 0.25 b = 60000$

$a = 800 - \frac{3 b}{2}$

$1200 - 0.25 b = 60000$

$a = 800 - \frac{3 b}{2}$

Step 3

Solve for $b$ in $1200 - 0.25 b = 60000$ .

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Step 3.1

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

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Step 3.1.1

Subtract $1200$ from both sides of the equation.

$- 0.25 b = 60000 - 1200$

$a = 800 - \frac{3 b}{2}$

Step 3.1.2

Subtract $1200$ from $60000$ .

$- 0.25 b = 58800$

$a = 800 - \frac{3 b}{2}$

$- 0.25 b = 58800$

$a = 800 - \frac{3 b}{2}$

Step 3.2

Divide each term in $- 0.25 b = 58800$ by $- 0.25$ and simplify.

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Step 3.2.1

Divide each term in $- 0.25 b = 58800$ by $- 0.25$ .

$\frac{- 0.25 b}{- 0.25} = \frac{58800}{- 0.25}$

$a = 800 - \frac{3 b}{2}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $- 0.25$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{- 0.25 b}{- 0.25} = \frac{58800}{- 0.25}$

$a = 800 - \frac{3 b}{2}$

Step 3.2.2.1.2

Divide $b$ by $1$ .

$b = \frac{58800}{- 0.25}$

$a = 800 - \frac{3 b}{2}$

$b = \frac{58800}{- 0.25}$

$a = 800 - \frac{3 b}{2}$

$b = \frac{58800}{- 0.25}$

$a = 800 - \frac{3 b}{2}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $58800$ by $- 0.25$ .

$b = - 235200$

$a = 800 - \frac{3 b}{2}$

$b = - 235200$

$a = 800 - \frac{3 b}{2}$

$b = - 235200$

$a = 800 - \frac{3 b}{2}$

$b = - 235200$

$a = 800 - \frac{3 b}{2}$

Step 4

Replace all occurrences of $b$ with $- 235200$ in each equation.

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Step 4.1

Replace all occurrences of $b$ in $a = 800 - \frac{3 b}{2}$ with $- 235200$ .

$a = 800 - \frac{3 (- 235200)}{2}$

$b = - 235200$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $800 - \frac{3 (- 235200)}{2}$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Cancel the common factor of $- 235200$ and $2$ .

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Step 4.2.1.1.1.1

Factor $2$ out of $3 (- 235200)$ .

$a = 800 - \frac{2 (3 \cdot (- 117600))}{2}$

$b = - 235200$

Step 4.2.1.1.1.2

Cancel the common factors.

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Step 4.2.1.1.1.2.1

Factor $2$ out of $2$ .

$a = 800 - \frac{2 (3 \cdot (- 117600))}{2 (1)}$

$b = - 235200$

Step 4.2.1.1.1.2.2

Cancel the common factor.

$a = 800 - \frac{2 (3 \cdot (- 117600))}{2 \cdot 1}$

$b = - 235200$

Step 4.2.1.1.1.2.3

Rewrite the expression.

$a = 800 - \frac{3 \cdot (- 117600)}{1}$

$b = - 235200$

Step 4.2.1.1.1.2.4

Divide $3 \cdot (- 117600)$ by $1$ .

$a = 800 - (3 \cdot (- 117600))$

$b = - 235200$

$a = 800 - (3 \cdot (- 117600))$

$b = - 235200$

$a = 800 - (3 \cdot (- 117600))$

$b = - 235200$

Step 4.2.1.1.2

Multiply $- (3 \cdot (- 117600))$ .

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Step 4.2.1.1.2.1

Multiply $3$ by $- 117600$ .

$a = 800 + 352800$

$b = - 235200$

Step 4.2.1.1.2.2

Multiply $- 1$ by $- 352800$ .

$a = 800 + 352800$

$b = - 235200$

$a = 800 + 352800$

$b = - 235200$

$a = 800 + 352800$

$b = - 235200$

Step 4.2.1.2

Add $800$ and $352800$ .

$a = 353600$

$b = - 235200$

$a = 353600$

$b = - 235200$

$a = 353600$

$b = - 235200$

$a = 353600$

$b = - 235200$

Step 5

List all of the solutions.

$a = 353600, b = - 235200$