Problem 9 - Algebra for College Students

Question

Simplify the algebraic expressions in Problems \(1-14\) by combining similar terms. $$-3 a^{2}+7 b^{2}+9 a^{2}-2 b^{2}$$

Step-by-Step Solution

Answer

The simplified expression is \(6a^{2} + 5b^{2}\).

Step 1: Identify Like Terms

Identify terms with the same variables and exponents. In the expression \(-3a^{2} + 7b^{2} + 9a^{2} - 2b^{2}\), \(-3a^{2}\) and \(9a^{2}\) are like terms, and \(7b^{2}\) and \(-2b^{2}\) are also like terms.

Step 2: Combine Like Terms for \(a^2\)

Add the coefficients of the like terms for \(a^2\).\(-3 + 9 = 6\).This simplifies \(-3a^{2} + 9a^{2}\) to \(6a^{2}\).

Step 3: Combine Like Terms for \(b^2\)

Add the coefficients of the like terms for \(b^2\).\(7 - 2 = 5\).This simplifies \(7b^{2} - 2b^{2}\) to \(5b^{2}\).

Step 4: Write the Simplified Expression

Combine the simplified terms for \(a^2\) and \(b^2\) into a single expression.The expression becomes \(6a^{2} + 5b^{2}\).

Key Concepts

Like Terms

In algebra, **like terms** are terms that have the same variable part. This means the variables themselves and any exponents must match; however, the coefficients, which are the numbers in front, can differ. For example, in the expression \(-3a^{2} + 7b^{2} + 9a^{2} - 2b^{2}\), the like terms are \(-3a^{2}\) and \(9a^{2}\), as well as \(7b^{2}\) and \(-2b^{2}\). Recognizing like terms is crucial because only these terms can be combined to simplify the expression.

• Like terms must have identical variable components.
• Coefficients do not affect whether terms are alike or not.
• Combining like terms reduces the complexity of algebraic expressions.

To easily identify like terms, look for terms that include the same letters with exactly the same exponents. Spotting these terms will help you in the process of simplification by grouping them together.

Combining Coefficients

Once you've identified like terms, the next step is **combining coefficients**. Coefficients are the numerical factors in the terms, and combining them involves simple arithmetic operations, like addition or subtraction. In our expression, watch closely how this works:

• For \(a^2\): Combine \(-3\) and \(9\) to get \(6\) for the total coefficient. Thus, \(-3a^{2} + 9a^{2}\) becomes \(6a^{2}\).
• For \(b^2\): Combine \(7\) and \(-2\) to result in \(5\), transforming \(7b^{2} - 2b^{2}\) to \(5b^{2}\).

Combining coefficients is about performing the same arithmetic operation across the identified like terms. It simplifies an expression, making it more concise and easier to handle in further calculations or equations.

Algebraic Manipulation

**Algebraic Manipulation** features the strategic rearranging and simplification of expressions to achieve a simpler form or a solution. In our example problem, manipulation starts from recognizing and combining like terms all the way to writing the final simplified expression:

1. Identify any potential like terms using variable and exponent matching.
2. Combine coefficients of the like terms to simplify.
3. Rewrite the expression into a simpler, condensed form.

This exercise culminates with the expression \(6a^{2} + 5b^{2}\), derived by performing these manipulations effectively. Ultimately, mastering algebraic manipulation means gaining the skills to modify equations and expressions seamlessly, whether for simplification, solving, or further analysis.