Problem 71 - Intermediate Algebra

Question

Simplify by combining like terms. See Example 5 . $$\frac{2}{5} a b-\left(-\frac{1}{2} a b\right)$$

Step-by-Step Solution

Answer

\[\frac{9}{10} a b\]

Step 1: Distribute the Negative Sign

The expression is \(\frac{2}{5} a b - \left(-\frac{1}{2} a b\right)\). Start by distributing the negative sign inside the parentheses. This changes \(-\left(-\frac{1}{2} a b\right)\) to \(+\frac{1}{2} a b\), due to the rule that subtracting a negative is equivalent to addition.

Step 2: Combine Like Terms

Now we have \(\frac{2}{5} a b + \frac{1}{2} a b\). Both terms are like terms since they both include \(a b\). To combine them, find a common denominator for the fractions, which is 10 in this case. Convert \(\frac{2}{5}\) to \(\frac{4}{10}\) and \(\frac{1}{2}\) to \(\frac{5}{10}\).

Step 3: Add the Fractions

Now add the two fractions: \(\frac{4}{10}\) and \(\frac{5}{10}\). When added, \(\frac{4}{10} + \frac{5}{10} = \frac{9}{10}\). Now, the expression simplifies to \(\frac{9}{10} a b\).

Key Concepts

Distributive Property

The distributive property is a fundamental rule in algebra that allows you to simplify expressions and solve equations more easily. It involves distributing a single term across terms inside parentheses. When you see a negative sign followed by parentheses, like in

• \( -(-\frac{1}{2} ab) \),

you are effectively multiplying. The negative sign changes the sign of the term inside the parentheses. So,

• \( -(-\frac{1}{2} ab) \) becomes \( +\frac{1}{2} ab \) because two negatives make a positive.

This rule is key in simplifying expressions because it helps remove parentheses and combine similar terms. Understanding and applying the distributive property correctly will make working with algebraic expressions much easier. Remember, subtracting a negative is always like adding a positive!

Common Denominator

When adding or subtracting fractions, finding a common denominator is a must. This is because you can't directly add or subtract fractions that have different denominators, just like you can't add apples to oranges.

• In our example, we have the fractions \( \frac{2}{5} \) and \( \frac{1}{2} \).
• Their denominators are 5 and 2, which are different.
• To combine these fractions, determine the smallest number that both denominators can divide into evenly — this is the common denominator.

For 5 and 2, the common denominator is 10.

• To convert \( \frac{2}{5} \) to a denominator of 10, multiply both the numerator and denominator by 2, turning it into \( \frac{4}{10} \).
• Similarly, convert \( \frac{1}{2} \) by multiplying both by 5, resulting in \( \frac{5}{10} \).

Now, both fractions have the same denominator, making them easy to add or subtract.

Fraction Addition

Adding fractions may seem daunting at first, but once you get the hang of it, you'll see it's as simple as lining up their pieces neatly. When adding fractions:

• Ensure they have a common denominator, as discussed previously.
• Add only the numerators and keep the denominator the same.

In our example, the fractions \( \frac{4}{10} \) and \( \frac{5}{10} \) can be added directly:

• \( \frac{4}{10} + \frac{5}{10} = \frac{9}{10} \).

There's nothing tricky about it. You're simply adding the parts above the line (the numerators) while keeping the parts below the line (the denominator) constant. This keeps the size of each fractional piece the same. By mastering these steps, you'll find working with fractions to be much easier and far less intimidating.