Pre-Algebra Examples

$0.1 x + 0.4 y = 0.25 (100)$

Step 1

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

Step 2

Change $0.1$ into a fraction.

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Step 2.1

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

$\frac{10 \cdot 0.1}{10} x + 0.4 y = 0.25 (100)$

Step 2.2

Multiply $10$ by $0.1$ .

$\frac{1}{10} x + 0.4 y = 0.25 (100)$

Step 2.3

Cancel the common factor of $1$ and $10$ .

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Step 2.3.1

Rewrite $1$ as $1 (1)$ .

$\frac{1 (1)}{10} x + 0.4 y = 0.25 (100)$

Step 2.3.2

Cancel the common factors.

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Step 2.3.2.1

Rewrite $10$ as $1 (10)$ .

$\frac{1 \cdot 1}{1 \cdot 10} x + 0.4 y = 0.25 (100)$

Step 2.3.2.2

Cancel the common factor.

$\frac{1 \cdot 1}{1 \cdot 10} x + 0.4 y = 0.25 (100)$

Step 2.3.2.3

Rewrite the expression.

$\frac{1}{10} x + 0.4 y = 0.25 (100)$

Step 3

Change $0.4$ into a fraction.

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Step 3.1

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

$\frac{1}{10} x + \frac{10 \cdot 0.4}{10} y = 0.25 (100)$

Step 3.2

Multiply $10$ by $0.4$ .

$\frac{1}{10} x + \frac{4}{10} y = 0.25 (100)$

Step 3.3

Cancel the common factor of $4$ and $10$ .

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Step 3.3.1

Factor $2$ out of $4$ .

$\frac{1}{10} x + \frac{2 (2)}{10} y = 0.25 (100)$

Step 3.3.2

Cancel the common factors.

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Step 3.3.2.1

Factor $2$ out of $10$ .

$\frac{1}{10} x + \frac{2 \cdot 2}{2 \cdot 5} y = 0.25 (100)$

Step 3.3.2.2

Cancel the common factor.

$\frac{1}{10} x + \frac{2 \cdot 2}{2 \cdot 5} y = 0.25 (100)$

Step 3.3.2.3

Rewrite the expression.

$\frac{1}{10} x + \frac{2}{5} y = 0.25 (100)$

Step 4

Change $0.25$ into a fraction.

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Step 4.1

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

$\frac{1}{10} x + \frac{2}{5} y = \frac{100 \cdot 0.25}{100} \cdot 100$

Step 4.2

Multiply $100$ by $0.25$ .

$\frac{1}{10} x + \frac{2}{5} y = \frac{25}{100} \cdot 100$

Step 4.3

Cancel the common factor of $25$ and $100$ .

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Step 4.3.1

Factor $25$ out of $25$ .

$\frac{1}{10} x + \frac{2}{5} y = \frac{25 (1)}{100} \cdot 100$

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

Factor $25$ out of $100$ .

$\frac{1}{10} x + \frac{2}{5} y = \frac{25 \cdot 1}{25 \cdot 4} \cdot 100$

Step 4.3.2.2

Cancel the common factor.

$\frac{1}{10} x + \frac{2}{5} y = \frac{25 \cdot 1}{25 \cdot 4} \cdot 100$

Step 4.3.2.3

Rewrite the expression.

$\frac{1}{10} x + \frac{2}{5} y = \frac{1}{4} \cdot 100$

Step 5

Find the LCD of $\frac{1}{10}, \frac{2}{5},$ and $\frac{1}{4}$ .

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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.

$10, 5, 4$

Step 5.2

Since $10, 5, 4$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $10, 5, 4$ 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

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

$2 \cdot 5$

Step 5.5

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

$5$ is a prime number

Step 5.6

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

$2 \cdot 2$

Step 5.7

The LCM of $10, 5, 4$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$2 \cdot 2 \cdot 5$

Step 5.8

Multiply $2 \cdot 2 \cdot 5$ .

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Step 5.8.1

Multiply $2$ by $2$ .

$4 \cdot 5$

Step 5.8.2

Multiply $4$ by $5$ .

$20$

Step 6

Multiply both sides by $20$ .

$20 (\frac{1}{10} x + \frac{2}{5} y) = 20 (\frac{1}{4} \cdot 100)$

Step 7

Simplify the left side.

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Step 7.1

Simplify $20 (\frac{1}{10} x + \frac{2}{5} y)$ .

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Step 7.1.1

Simplify each term.

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Step 7.1.1.1

Combine $\frac{1}{10}$ and $x$ .

$20 (\frac{x}{10} + \frac{2}{5} y) = 20 (\frac{1}{4} \cdot 100)$

Step 7.1.1.2

Combine $\frac{2}{5}$ and $y$ .

$20 (\frac{x}{10} + \frac{2 y}{5}) = 20 (\frac{1}{4} \cdot 100)$

Step 7.1.2

Simplify terms.

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Step 7.1.2.1

Apply the distributive property.

$20 \frac{x}{10} + 20 \frac{2 y}{5} = 20 (\frac{1}{4} \cdot 100)$

Step 7.1.2.2

Cancel the common factor of $10$ .

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Step 7.1.2.2.1

Factor $10$ out of $20$ .

$10 (2) \frac{x}{10} + 20 \frac{2 y}{5} = 20 (\frac{1}{4} \cdot 100)$

Step 7.1.2.2.2

Cancel the common factor.

$10 \cdot 2 \frac{x}{10} + 20 \frac{2 y}{5} = 20 (\frac{1}{4} \cdot 100)$

Step 7.1.2.2.3

Rewrite the expression.

$2 x + 20 \frac{2 y}{5} = 20 (\frac{1}{4} \cdot 100)$

Step 7.1.2.3

Cancel the common factor of $5$ .

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Step 7.1.2.3.1

Factor $5$ out of $20$ .

$2 x + 5 (4) \frac{2 y}{5} = 20 (\frac{1}{4} \cdot 100)$

Step 7.1.2.3.2

Cancel the common factor.

$2 x + 5 \cdot 4 \frac{2 y}{5} = 20 (\frac{1}{4} \cdot 100)$

Step 7.1.2.3.3

Rewrite the expression.

$2 x + 4 (2 y) = 20 (\frac{1}{4} \cdot 100)$

Step 7.1.2.4

Multiply $2$ by $4$ .

$2 x + 8 y = 20 (\frac{1}{4} \cdot 100)$

Step 8

Simplify the right side.

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Step 8.1

Simplify $20 (\frac{1}{4} \cdot 100)$ .

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Step 8.1.1

Cancel the common factor of $4$ .

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Step 8.1.1.1

Factor $4$ out of $100$ .

$2 x + 8 y = 20 (\frac{1}{4} \cdot (4 (25)))$

Step 8.1.1.2

Cancel the common factor.

$2 x + 8 y = 20 (\frac{1}{4} \cdot (4 \cdot 25))$

Step 8.1.1.3

Rewrite the expression.

$2 x + 8 y = 20 \cdot 25$

Step 8.1.2

Multiply $20$ by $25$ .

$2 x + 8 y = 500$

Step 9