Problem 29 - Elementary Algebra

Question

Simplify. $$ 1 x-3+1 x+5 $$

Step-by-Step Solution

Answer

The simplified expression is \(2x + 2\).

Step 1: Identify Like Terms

The expression given is \(1x - 3 + 1x + 5\). First, identify like terms in the expression. Like terms in our case are the ones that have the same variable part, which are \(1x\) and \(1x\). The constants are \(-3\) and \(5\).

Step 2: Combine Like Terms with Variables

Add the coefficients of the like terms involving \(x\). Here we have \(1x\) and \(1x\), which gives us \((1 + 1)x = 2x\).

Step 3: Combine Constant Terms

Next, combine the constant terms \(-3\) and \(5\). Adding them gives \(-3 + 5 = 2\).

Step 4: Write the Simplified Expression

Now, combine the results from step 2 and step 3 to form the simplified expression: \(2x + 2\).

Key Concepts

Like Terms

In algebra, not all terms can be directly combined, only those that involve the same variables raised to the same powers. These are called "Like Terms". For example, in the expression \(1x - 3 + 1x + 5\), the terms \(1x\) and \(1x\) are like terms because they both have the variable \(x\) with the same power. On the other hand, the constants \(-3\) and \(5\) are also considered like terms because they do not involve variables and can be combined on their own.
To identify like terms in any given expression:

• Look for terms with the same variables.
• Ensure these variables have the same exponents.
• Constant terms (numbers without variables) can always be combined with each other.

Recognizing like terms is crucial because it allows us to simplify expressions by combining them, making the expression more manageable.

Simplification

Simplification refers to the process of reducing an algebraic expression to its simplest form. This typically means combining like terms to make the expression cleaner and more straightforward. For our given example \(1x - 3 + 1x + 5\), simplification involves combining terms that are either similar in variable or are constants.
Steps to simplify an algebraic expression include:

• First, identify and group like terms together.
• Combine the coefficients of the variable terms.
• Add or subtract constant terms accordingly.

By simplifying expressions, we aim to make them easier to work with in further calculations. For instance, after simplification, \(1x - 3 + 1x + 5\) turns into \(2x + 2\), which is more compact and easier to interpret.

Coefficients

A coefficient in algebra is the numerical factor that multiplies a variable in an algebraic term. In our expression \(1x - 3 + 1x + 5\), the numbers \(1\) in \(1x\) are the coefficients of \(x\). Similarly, \(-3\) and \(5\) can be thought of as coefficients of the constant terms, even though they are not affecting a variable.
Understanding coefficients is essential because they determine the magnitude and direction (positive or negative) of the variable term. When simplifying expressions, knowing the coefficients allows you to:

• Easily combine like terms by adding or subtracting their coefficients.
• Potentially factor expressions if needed.
• Understand the relative size and influence of each term within the expression.

In our problem, adding the coefficients \(1\) and \(1\) in \(1x + 1x\) results in the simplified \(2x\), which succinctly represents how many times \(x\) is present in the expression after simplification.