Problem 316 - Intermediate Algebra

Question

In the following exercises, simplify. $$ \frac{5}{6} a+\frac{3}{10} b+\frac{1}{6} a+\frac{9}{10} b $$

Step-by-Step Solution

Answer

a + \( \frac{6}{5} \)b

Step 1 - Identify Like Terms

Group the terms involving 'a' together and the terms involving 'b' together: \( \frac{5}{6} a + \frac{1}{6} a + \frac{3}{10} b + \frac{9}{10} b \)

Step 2 - Combine Like Terms for 'a'

Combine the coefficients of 'a': \( \left(\frac{5}{6} + \frac{1}{6}\right) a = \frac{6}{6} a = a \)

Step 3 - Combine Like Terms for 'b'

Combine the coefficients of 'b': \( \left(\frac{3}{10} + \frac{9}{10}\right) b = \frac{12}{10} b = \frac{6}{5} b \)

Step 4 - Write Final Expression

Combine the simplified terms to get the final simplified expression: \( a + \frac{6}{5} b \)

Key Concepts

Combining Like Terms

When simplifying algebraic expressions, combining like terms is a crucial step. Like terms are terms that have the same variable raised to the same power. In simpler terms, you can combine them because they

Coefficients

Coefficients are the numerical parts of terms that include variables. For instance, in the term \(3x\), the number 3 is the coefficient. Coefficients tell you how many of each term you have. Properly managing these numbers is key to simplifying expressions correctly.

Simplification Process

The simplification process involves identifying like terms, combining them, and then re-writing the expression in a simpler form. Breaking it down into steps: First, identify the like terms. Next, combine these by adding or subtracting the coefficients. Finally, write the new expression in a simplified form.