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Chapter 2: Problem 65

Simplify by combining like terms whenever possible. $$2 x+5 x$$

Short Answer

Expert verified
7x

Step by step solution

01

Identify Like Terms

Find terms that have the same variable raised to the same power. In this case, both terms are like terms since they both contain the variable \(x\).
02

Combine Like Terms

Add the coefficients of the like terms. The expression is \(2x + 5x\), where \(2x\) and \(5x\) are like terms. So, add the coefficients: \(2 + 5 = 7\).
03

Write the Simplified Expression

After combining the coefficients, the simplified expression is \(7x\).

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Key Concepts

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

simplifying algebraic expressions
Simplifying algebraic expressions is a fundamental part of algebra. It makes complex expressions easier to work with. To simplify an algebraic expression, we use a few basic techniques:

First, we need to identify like terms. These are terms in the expression that contain the same variables raised to the same power.

Next, we combine the like terms by adding or subtracting their coefficients. This process is often referred to as combining like terms.

Let's break this down with an example. Consider the expression given in the exercise: \[2x + 5x\]

Identifying like terms is the first step in this process.
identifying like terms
Identifying like terms is crucial for simplifying expressions. Like terms are terms that have identical variables and exponents. This means the variables must be the same and must be raised to the same power.

For instance:
  • \(2x\) and \(5x\) are like terms because they both have the variable \(x\)
  • \(3y^2\) and \(4y^2\) are also like terms because they both have the variable \(y\) raised to the 2nd power.
Look at the given expression, \[2x + 5x\].
The terms \(2x\) and \(5x\) are like terms because they both have the variable \(x\) and no exponent or same exponent (1).

Once we have identified the like terms, we can combine them, which involves the addition of coefficients.
addition of coefficients
After identifying the like terms, the next step is to combine them through the addition of coefficients. Coefficients are the numerical part of the terms in an expression. In the term \(2x\), the coefficient is 2.

Let's take the expression from the exercise again: \[2x + 5x\]
Both terms have \(x\), so we simply add the coefficients which are 2 and 5.

Performing the addition: \[2 + 5 = 7\]
So the simplified expression is \[7x\]

Combining coefficients is a simple yet essential step in simplifying algebraic expressions, allowing us to reduce the complexity of our calculations and focus on solving the problem at hand.

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Most popular questions from this chapter

Simplify by combining like terms whenever possible. $$3(x+y)+4 x(-y)$$

Discuss what is wrong (if anything) with each of the following examples: (a) \(5-3(x-4) \stackrel{?}{=} 2(x-4) \stackrel{?}{=} 2 x-8\) (b) \(5-3(x-4) \stackrel{2}{=} 5-3 x-4 \stackrel{2}{=} 1-3 x\) (c) \(5-3(x-4) \geq 5-3 x-12 \frac{2}{2}-7-3 x\) (d) \(5-3(x-4) \geq 5-3 x+12 \leq 17-3 x\)

Evaluate the given expression for \(x=2, y=-3,\) and \(z=-4.\) $$(x+y)^{2}$$

Compare and contrast the expressions \(2^{4}\) and \(4^{2} .\) Does the fact that both of these expressions have a numerical value of 16 affect your answer?

Simplify by combining like terms whenever possible. $$-3 x^{2}+5 x y-x y-x^{2}-2 x^{2}$$
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