Solving Linear Equations that Contain Fractions

Flashcards covering key techniques for solving linear equations with single or multiple fractions, including clearing denominators, using reciprocals, and isolating the variable.

What is the quickest way to eliminate a single fractional denominator such as 2⁄3x = 8?

Multiply both sides of the equation by the denominator (3) to clear the fraction.

Why might you multiply both sides of 2⁄3x = 8 by the reciprocal 3⁄2 instead of just 3?

Multiplying by the reciprocal clears the fraction and immediately isolates x in one step.

When using the reciprocal method on 8 · (3⁄2), which operation is usually more efficient with large numbers?

Divide first, then multiply (e.g., 8 ÷ 2 = 4, then 4 · 3 = 12).

Before clearing fractions in 5⁄8x + 4 = 14, why is it helpful to subtract 4 first?

Removing the constant avoids creating a large product (8 · 14) after multiplying by the denominator.

After isolating 5⁄8x = 10, what operation clears the remaining denominator?

Multiply both sides by 8 to obtain 5x = 80.

How is x isolated once 5x = 80 is reached?

Divide both sides by 5, giving x = 16.

When an equation contains two different denominators, what common strategy removes all fractions at once?

Multiply every term by a common denominator (preferably the least common denominator).

If the denominators are 3 and 5, what common denominator can always be used, even if not the least?

Their product, 15, which will still eliminate both fractions.

After multiplying (1⁄3 + 2x) = 2⁄5 by 15, what is 15 · 1⁄3 and 15 · 2⁄5?

15 · 1⁄3 = 5 and 15 · 2⁄5 = 6.

In the equation x⁄2 + 5 = 1⁄4, why does multiplying the entire equation by 4 work?

Because 4 is a common multiple of the denominators 2 and 4, clearing both fractions in one step.

After clearing fractions in 2x + 20 = 1, what final operations give x?

Subtract 20 to get 2x = –19, then divide by 2 to get x = –19⁄2.