Finite Math Examples

$\frac{3 x}{8} - \frac{3 y}{5} = \frac{33}{80}$

Step 1

The standard form of a linear equation is $A x + B y = C$ .

Step 2

Find the LCD of $\frac{3 x}{8}, \frac{3 y}{5},$ and $\frac{33}{80}$ .

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Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$8, 5, 80$

Step 2.2

Since $8, 5, 80$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $8, 5, 80$ then find LCM for the variable part .

Step 2.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 2.4

The prime factors for $8$ are $2 \cdot 2 \cdot 2$ .

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Step 2.4.1

$8$ has factors of $2$ and $4$ .

$2 \cdot 4$

Step 2.4.2

$4$ has factors of $2$ and $2$ .

$2 \cdot 2 \cdot 2$

Step 2.5

Since $5$ has no factors besides $1$ and $5$ .

$5$ is a prime number

Step 2.6

The prime factors for $80$ are $2 \cdot 2 \cdot 2 \cdot 2 \cdot 5$ .

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Step 2.6.1

$80$ has factors of $2$ and $40$ .

$2 \cdot 40$

Step 2.6.2

$40$ has factors of $2$ and $20$ .

$2 \cdot 2 \cdot 20$

Step 2.6.3

$20$ has factors of $2$ and $10$ .

$2 \cdot 2 \cdot 2 \cdot 10$

Step 2.6.4

$10$ has factors of $2$ and $5$ .

$2 \cdot 2 \cdot 2 \cdot 2 \cdot 5$

Step 2.7

Multiply $2 \cdot 2 \cdot 2 \cdot 2 \cdot 5$ .

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Step 2.7.1

Multiply $2$ by $2$ .

$4 \cdot 2 \cdot 2 \cdot 5$

Step 2.7.2

Multiply $4$ by $2$ .

$8 \cdot 2 \cdot 5$

Step 2.7.3

Multiply $8$ by $2$ .

$16 \cdot 5$

Step 2.7.4

Multiply $16$ by $5$ .

$80$

Step 3

Multiply both sides by $80$ .

$80 (\frac{3 x}{8} - \frac{3 y}{5}) = 80 (\frac{33}{80})$

Step 4

Simplify the left side.

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Step 4.1

Simplify $80 (\frac{3 x}{8} - \frac{3 y}{5})$ .

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Step 4.1.1

Apply the distributive property.

$80 \frac{3 x}{8} + 80 (- \frac{3 y}{5}) = 80 (\frac{33}{80})$

Step 4.1.2

Cancel the common factor of $8$ .

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Step 4.1.2.1

Factor $8$ out of $80$ .

$8 (10) \frac{3 x}{8} + 80 (- \frac{3 y}{5}) = 80 (\frac{33}{80})$

Step 4.1.2.2

Cancel the common factor.

$8 \cdot 10 \frac{3 x}{8} + 80 (- \frac{3 y}{5}) = 80 (\frac{33}{80})$

Step 4.1.2.3

Rewrite the expression.

$10 (3 x) + 80 (- \frac{3 y}{5}) = 80 (\frac{33}{80})$

Step 4.1.3

Multiply $3$ by $10$ .

$30 x + 80 (- \frac{3 y}{5}) = 80 (\frac{33}{80})$

Step 4.1.4

Cancel the common factor of $5$ .

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Step 4.1.4.1

Move the leading negative in $- \frac{3 y}{5}$ into the numerator.

$30 x + 80 \frac{- 3 y}{5} = 80 (\frac{33}{80})$

Step 4.1.4.2

Factor $5$ out of $80$ .

$30 x + 5 (16) \frac{- 3 y}{5} = 80 (\frac{33}{80})$

Step 4.1.4.3

Cancel the common factor.

$30 x + 5 \cdot 16 \frac{- 3 y}{5} = 80 (\frac{33}{80})$

Step 4.1.4.4

Rewrite the expression.

$30 x + 16 (- 3 y) = 80 (\frac{33}{80})$

Step 4.1.5

Multiply $- 3$ by $16$ .

$30 x - 48 y = 80 (\frac{33}{80})$

Step 5

Simplify the right side.

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Step 5.1

Cancel the common factor of $80$ .

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Step 5.1.1

Cancel the common factor.

$30 x - 48 y = 80 (\frac{33}{80})$

Step 5.1.2

Rewrite the expression.

$30 x - 48 y = 33$

Step 6