Problem 25 - Algebra for College Students

Question

Add or subtract the rational expressions as indicated. Be sure to express your answers in simplest form. $$ \frac{3 x-1}{3}-\frac{5 x+2}{5} $$

Step-by-Step Solution

Answer

\( \frac{-11}{15} \)

Step 1: Determine the Least Common Denominator (LCD)

To combine \( \frac{3x-1}{3} \) and \( \frac{5x+2}{5} \), we must find a common denominator. The denominators are 3 and 5, with the least common denominator being \( 3 \times 5 = 15 \).

Step 2: Rewrite Each Expression

Rewrite each expression so they both have the LCD as their denominator. Multiply \( \frac{3x-1}{3} \) by \( \frac{5}{5} \) to get \( \frac{5(3x-1)}{15} \), and \( \frac{5x+2}{5} \) by \( \frac{3}{3} \) to get \( \frac{3(5x+2)}{15} \).

Step 3: Expand the Numerators

Distribute in the numerators: \( 5(3x-1) = 15x - 5 \) and \( 3(5x+2) = 15x + 6 \). The expressions become \( \frac{15x-5}{15} \) and \( \frac{15x+6}{15} \).

Step 4: Subtract the Numerators

Subtract the second numerator from the first: \( (15x-5) - (15x+6) = 15x - 5 - 15x - 6 = -11 \).

Step 5: Simplify the Resulting Expression

Replace the expression with the result: \( \frac{-11}{15} \). Since \(-11\) and \(15\) have no common factors besides 1, the expression \( \frac{-11}{15} \) is already in its simplest form.

Key Concepts

Least Common Denominator

The concept of a least common denominator (LCD) is crucial when working with rational expressions that have different denominators. Imagine you want to add or subtract fractions; you can think of it like aligning two rows of cake slices so they match up perfectly. To do this effectively, both fractions need the same denominator. This makes it easier to combine them.
For instance, in the expression \( \frac{3x-1}{3} - \frac{5x+2}{5} \), the denominators are 3 and 5. Here, the LCD is the smallest number that both denominators can divide into without leaving a remainder. Calculate this by multiplying the two different denominators:

• 3 multiplied by 5 equals 15.

So, 15 is the least common denominator in this case.

Simplification of Expressions

Once you have a least common denominator, it's time to simplify each expression to have this common denominator. Start from being a helpful middle step to finally add or subtract fractions.
For example, \( \frac{3x-1}{3} \) needs to be rewritten so that the denominator is 15 instead of 3. Multiply both the numerator and denominator by 5:

• \( \frac{3x-1}{3} \times \frac{5}{5} = \frac{5(3x-1)}{15} \)

Similarly, rewrite \( \frac{5x+2}{5} \) with the same denominator by multiplying both parts by 3:

• \( \frac{5x+2}{5} \times \frac{3}{3} = \frac{3(5x+2)}{15} \)

This approach keeps the expressions equal in value but aligns them under the same framework.

Adding and Subtracting Fractions

Finally, when adding or subtracting fractions, focus on the numerators. Thanks to having the same denominator, you only need to worry about the top part of the fractions. In our example, once the denominators of the expressions are both 15:

• Subtract the expanded numerators: \( (15x-5) - (15x+6) \)

The process is just like handling whole numbers:

• First, subtract: \(15x - 15x = 0\)
• Then, combine any constants: \(-5 - 6 = -11\)

Once calculations are completed, the result is \( \frac{-11}{15} \). Concluding the expression, double-check its simplicity. In this case, since \( -11 \) and \( 15 \) do not have common factors besides 1, \( \frac{-11}{15} \) is already simplified.