Pre-Algebra Examples

$\frac{1}{8} x + \frac{1}{4} y = \frac{17}{8}$

Step 1

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

Step 2

Find the LCD of $\frac{1}{8}, \frac{1}{4},$ and $\frac{17}{8}$ .

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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.

$8, 4, 8$

Step 2.2

Since $8, 4, 8$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $8, 4, 8$ 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

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

$2 \cdot 2$

Step 2.6

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

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Step 2.6.1

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

$2 \cdot 4$

Step 2.6.2

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

$2 \cdot 2 \cdot 2$

Step 2.7

Multiply $2 \cdot 2 \cdot 2$ .

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Step 2.7.1

Multiply $2$ by $2$ .

$4 \cdot 2$

Step 2.7.2

Multiply $4$ by $2$ .

$8$

Step 3

Multiply both sides by $8$ .

$8 (\frac{1}{8} x + \frac{1}{4} y) = 8 (\frac{17}{8})$

Step 4

Simplify the left side.

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Step 4.1

Simplify $8 (\frac{1}{8} x + \frac{1}{4} y)$ .

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Step 4.1.1

Simplify each term.

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Step 4.1.1.1

Combine $\frac{1}{8}$ and $x$ .

$8 (\frac{x}{8} + \frac{1}{4} y) = 8 (\frac{17}{8})$

Step 4.1.1.2

Combine $\frac{1}{4}$ and $y$ .

$8 (\frac{x}{8} + \frac{y}{4}) = 8 (\frac{17}{8})$

Step 4.1.2

Simplify terms.

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Step 4.1.2.1

Apply the distributive property.

$8 \frac{x}{8} + 8 \frac{y}{4} = 8 (\frac{17}{8})$

Step 4.1.2.2

Cancel the common factor of $8$ .

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Step 4.1.2.2.1

Cancel the common factor.

$8 \frac{x}{8} + 8 \frac{y}{4} = 8 (\frac{17}{8})$

Step 4.1.2.2.2

Rewrite the expression.

$x + 8 \frac{y}{4} = 8 (\frac{17}{8})$

Step 4.1.2.3

Cancel the common factor of $4$ .

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Step 4.1.2.3.1

Factor $4$ out of $8$ .

$x + 4 (2) \frac{y}{4} = 8 (\frac{17}{8})$

Step 4.1.2.3.2

Cancel the common factor.

$x + 4 \cdot 2 \frac{y}{4} = 8 (\frac{17}{8})$

Step 4.1.2.3.3

Rewrite the expression.

$x + 2 y = 8 (\frac{17}{8})$

Step 5

Simplify the right side.

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Step 5.1

Cancel the common factor of $8$ .

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Step 5.1.1

Cancel the common factor.

$x + 2 y = 8 (\frac{17}{8})$

Step 5.1.2

Rewrite the expression.

$x + 2 y = 17$

Step 6