Problem 8 - Algebra for College Students

Question

Perform the indicated operations. Reduce answers to their lowest terms. See Example \(I\) $$ \frac{5 x^{2}}{3}+\frac{4 x^{2}}{3} $$

Step-by-Step Solution

Answer

3x^{2}

Step 1: Identify Like Terms

The given expression is \[\frac{5 x^{2}}{3} + \frac{4 x^{2}}{3}\]. Both terms have the same denominator and the same variable part (\(x^2\)), so they are like terms.

Step 2: Combine Like Terms

Add the numerators of the like terms while keeping the common denominator: \[\frac{5 x^{2} + 4 x^{2}}{3} = \frac{9x^{2}}{3}\].

Step 3: Simplify the Expression

Simplify the fraction by dividing the numerator by the denominator: \[\frac{9 x^{2}}{3} = 3x^{2}\].

Key Concepts

Like Terms

In algebra, 'like terms' are terms that have the same variable raised to the same power. For example, in the expression \(\frac{5 x^{2}}{3} + \frac{4 x^{2}}{3}\), both terms have the variable \(x^{2}\). The coefficients (numerical parts) of the terms can be different, but the variable part has to be exactly the same for terms to be considered 'like'. This allows us to combine the terms by adding or subtracting their coefficients. Identifying like terms is an important first step in simplifying expressions, as it sets the stage for combining and simplifying these terms efficiently.

Combining Fractions

Once you have identified like terms, the next step involves combining them. In the expression \(\frac{5 x^{2}}{3} + \frac{4 x^{2}}{3}\), the terms already have a common denominator, which makes it easier to add them. When combining fractions with the same denominator, you simply add or subtract the numerators while keeping the denominator unchanged.
In our example, we add the numerators \(5 x^{2}\) and \(4 x^{2}\), resulting in \(9 x^{2}\). The combined fraction now looks like this: \(\frac{9 x^{2}}{3}\). Keeping the denominator constant during the combination process ensures we haven't altered the original meaning of the expression.

Simplifying Expressions

The final step in solving the given exercise is simplifying the expression. After combining the like terms, you'll often end up with a fraction that can be reduced. In our example, we have the fraction \(\frac{9 x^{2}}{3}\).
To simplify this, divide the numerator by the denominator. Here, dividing 9 by 3 gives us the simplified result of \(3 x^{2}\). Simplification ensures that your final answer is in the lowest and simplest form, making it much easier to understand and work with in further calculations. Always check if the terms can be simplified further to ensure you've reached the simplest form of your expression.