Pre-Algebra Examples

$0.08 x - 0.01 y = 4.6$

Step 1

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

Step 2

Change $0.08$ into a fraction.

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Step 2.1

Multiply by $\frac{100}{100}$ to remove the decimal.

$\frac{100 \cdot 0.08}{100} x - 0.01 y = 4.6$

Step 2.2

Multiply $100$ by $0.08$ .

$\frac{8}{100} x - 0.01 y = 4.6$

Step 2.3

Cancel the common factor of $8$ and $100$ .

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Step 2.3.1

Factor $4$ out of $8$ .

$\frac{4 (2)}{100} x - 0.01 y = 4.6$

Step 2.3.2

Cancel the common factors.

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Step 2.3.2.1

Factor $4$ out of $100$ .

$\frac{4 \cdot 2}{4 \cdot 25} x - 0.01 y = 4.6$

Step 2.3.2.2

Cancel the common factor.

$\frac{4 \cdot 2}{4 \cdot 25} x - 0.01 y = 4.6$

Step 2.3.2.3

Rewrite the expression.

$\frac{2}{25} x - 0.01 y = 4.6$

Step 3

Change $- 0.01$ into a fraction.

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Step 3.1

Multiply by $\frac{100}{100}$ to remove the decimal.

$\frac{2}{25} x + \frac{100 \cdot - 0.01}{100} y = 4.6$

Step 3.2

Multiply $100$ by $- 0.01$ .

$\frac{2}{25} x + \frac{- 1}{100} y = 4.6$

Step 3.3

Move the negative in front of the fraction.

$\frac{2}{25} x - \frac{1}{100} y = 4.6$

Step 4

Change $4.6$ into a fraction.

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Step 4.1

Multiply by $\frac{10}{10}$ to remove the decimal.

$\frac{2}{25} x - \frac{1}{100} y = \frac{10 \cdot 4.6}{10}$

Step 4.2

Multiply $10$ by $4.6$ .

$\frac{2}{25} x - \frac{1}{100} y = \frac{46}{10}$

Step 4.3

Cancel the common factor of $46$ and $10$ .

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Step 4.3.1

Factor $2$ out of $46$ .

$\frac{2}{25} x - \frac{1}{100} y = \frac{2 (23)}{10}$

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

Factor $2$ out of $10$ .

$\frac{2}{25} x - \frac{1}{100} y = \frac{2 \cdot 23}{2 \cdot 5}$

Step 4.3.2.2

Cancel the common factor.

$\frac{2}{25} x - \frac{1}{100} y = \frac{2 \cdot 23}{2 \cdot 5}$

Step 4.3.2.3

Rewrite the expression.

$\frac{2}{25} x - \frac{1}{100} y = \frac{23}{5}$

Step 5

Find the LCD of $\frac{2}{25}, \frac{1}{100},$ and $\frac{23}{5}$ .

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Step 5.1

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

$25, 100, 5$

Step 5.2

Since $25, 100, 5$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $25, 100, 5$ then find LCM for the variable part .

Step 5.3

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 5.4

$25$ has factors of $5$ and $5$ .

$5 \cdot 5$

Step 5.5

The prime factors for $100$ are $2 \cdot 2 \cdot 5 \cdot 5$ .

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Step 5.5.1

$100$ has factors of $2$ and $50$ .

$2 \cdot 50$

Step 5.5.2

$50$ has factors of $2$ and $25$ .

$2 \cdot 2 \cdot 25$

Step 5.5.3

$25$ has factors of $5$ and $5$ .

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

Step 5.6

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

$5$ is a prime number

Step 5.7

The LCM of $25, 100, 5$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

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

Step 5.8

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

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Step 5.8.1

Multiply $2$ by $2$ .

$4 \cdot 5 \cdot 5$

Step 5.8.2

Multiply $4$ by $5$ .

$20 \cdot 5$

Step 5.8.3

Multiply $20$ by $5$ .

$100$

Step 6

Multiply both sides by $100$ .

$100 (\frac{2}{25} x - \frac{1}{100} y) = 100 (\frac{23}{5})$

Step 7

Simplify the left side.

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Step 7.1

Simplify $100 (\frac{2}{25} x - \frac{1}{100} y)$ .

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Step 7.1.1

Simplify each term.

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Step 7.1.1.1

Combine $\frac{2}{25}$ and $x$ .

$100 (\frac{2 x}{25} - \frac{1}{100} y) = 100 (\frac{23}{5})$

Step 7.1.1.2

Combine $y$ and $\frac{1}{100}$ .

$100 (\frac{2 x}{25} - \frac{y}{100}) = 100 (\frac{23}{5})$

Step 7.1.2

Simplify terms.

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Step 7.1.2.1

Apply the distributive property.

$100 \frac{2 x}{25} + 100 (- \frac{y}{100}) = 100 (\frac{23}{5})$

Step 7.1.2.2

Cancel the common factor of $25$ .

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Step 7.1.2.2.1

Factor $25$ out of $100$ .

$25 (4) \frac{2 x}{25} + 100 (- \frac{y}{100}) = 100 (\frac{23}{5})$

Step 7.1.2.2.2

Cancel the common factor.

$25 \cdot 4 \frac{2 x}{25} + 100 (- \frac{y}{100}) = 100 (\frac{23}{5})$

Step 7.1.2.2.3

Rewrite the expression.

$4 (2 x) + 100 (- \frac{y}{100}) = 100 (\frac{23}{5})$

Step 7.1.2.3

Multiply $2$ by $4$ .

$8 x + 100 (- \frac{y}{100}) = 100 (\frac{23}{5})$

Step 7.1.2.4

Cancel the common factor of $100$ .

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Step 7.1.2.4.1

Move the leading negative in $- \frac{y}{100}$ into the numerator.

$8 x + 100 \frac{- y}{100} = 100 (\frac{23}{5})$

Step 7.1.2.4.2

Cancel the common factor.

$8 x + 100 \frac{- y}{100} = 100 (\frac{23}{5})$

Step 7.1.2.4.3

Rewrite the expression.

$8 x - y = 100 (\frac{23}{5})$

Step 8

Simplify the right side.

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Step 8.1

Simplify $100 (\frac{23}{5})$ .

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Step 8.1.1

Cancel the common factor of $5$ .

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Step 8.1.1.1

Factor $5$ out of $100$ .

$8 x - y = 5 (20) \frac{23}{5}$

Step 8.1.1.2

Cancel the common factor.

$8 x - y = 5 \cdot 20 \frac{23}{5}$

Step 8.1.1.3

Rewrite the expression.

$8 x - y = 20 \cdot 23$

Step 8.1.2

Multiply $20$ by $23$ .

$8 x - y = 460$

Step 9