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Chapter 5: Problem 13

In the following exercises, add or subtract the monomials. \(7 x^{2}+5 x^{2}+4 a-9 a\)

Short Answer

Expert verified
12x^2 - 5a

Step by step solution

01

Identify Like Terms

First, look at the given expression: \(7x^2 + 5x^2 + 4a - 9a\). Identify the like terms. Like terms have the same variables raised to the same powers. Here, \(7x^2\) and \(5x^2\) are like terms, and \(4a\) and \(-9a\) are like terms.
02

Combine Like Terms Involving \(x^2\)

Add the coefficients of the \(x^2\) terms together: \(7x^2 + 5x^2 = (7 + 5)x^2 = 12x^2\).
03

Combine Like Terms Involving \(a\)

Subtract the coefficients of the \(a\) terms: \(4a - 9a = (4 - 9)a = -5a\).
04

Write the Final Expression

Combine the results of the previous steps: \(12x^2 - 5a\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Monomials
Monomials are algebraic expressions that consist of only one term. They can include numbers, variables, or both. For example, in the expression given in the exercise, terms like \(7x^2\) and \(4a\) are monomials.
To be classified as monomials, the terms must not have plus or minus signs separating different parts within them.
When working with monomials, we often need to combine them, especially in expressions involving addition and subtraction.
Addition and Subtraction of Algebraic Expressions
In algebra, adding and subtracting expressions involves combining like terms. This means we group together terms that have the same variable raised to the same power.
For the given exercise, the focus is on adding and subtracting the monomials: \(7x^2 + 5x^2 + 4a - 9a\).
To do this, we first need to identify which terms are 'like terms'. Once identified, we combine their coefficients, which are the numerical factors in front of the variables.
Addition and subtraction of algebraic expressions can simplify problems and reveal common patterns, making it easier to solve algebra problems.
Like Terms
Like terms are terms that have the same variables raised to the same powers. This is crucial in algebra because only like terms can be combined together. For example, in the exercise provided, \(7x^2\) and \(5x^2\) are like terms because they both contain the variable \(x\) raised to the second power.
Similarly, \(4a\) and \(-9a\) are like terms because they both involve the variable \(a\).
When combining like terms, we focus on the coefficients. For instance, adding \(7x^2\) and \(5x^2\) gives us \(12x^2\), while subtracting \(9a\) from \(4a\) results in \(-5a\). Remember, it's essential to handle the signs appropriately to get accurate results.

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