Problem 81 - Elementary and Intermediate Algebra

Question

Simplify by combining like terms. $$ 3 x+4-5 x+1 $$

Step-by-Step Solution

Answer

The simplified expression is \(-2x + 5\).

Step 1: Identify Like Terms

Examine the expression \(3x + 4 - 5x + 1\) and identify the terms with the variable \(x\) and the constant terms. The like terms involving \(x\) are \(3x\) and \(-5x\). The constant terms are \(+4\) and \(+1\).

Step 2: Combine Like Terms with \(x\)

Combine the terms with \(x\): \(3x - 5x\). Subtract the coefficients: \(3 - 5 = -2\). So, \(3x - 5x = -2x\).

Step 3: Combine Constant Terms

Add together the constant terms: \(4 + 1 = 5\).

Step 4: Write the Simplified Expression

Combine the results from the previous steps to form the simplified expression. Combine \(-2x\) and \(+5\) to get \(-2x + 5\).

Key Concepts

Understanding Like Terms

In algebra, a key part of simplifying expressions is identifying what's called "like terms." Like terms are terms within an algebraic expression that have the same variables raised to the same power. For example, in the expression \(3x + 4 - 5x + 1\), the like terms are the ones that contain the variable \(x\), which are \(3x\) and \(-5x\).
These terms can be combined because they share the same variable component. When you're asked to simplify an algebraic expression, your first step is to look for and group these like terms. This makes it much easier to simplify the expression by focusing on each unique term type separately.

Combining Coefficients of Like Terms

Once you've identified like terms, the next step is combining their coefficients. A coefficient is the number that is multiplied by the variable in a term. For instance, in \(3x\), the coefficient is 3. Let's take a closer look at how to combine these coefficients in our expression:

• Identify the coefficients of the like terms: Here, the coefficients are 3 (from \(3x\)) and -5 (from \(-5x\)).
• Combine these coefficients by performing the indicated operations: Combine them like you would regular numbers; here, you will subtract \(5\) from \(3\), resulting in \(-2\).
• The simplified form of the variable terms is then \(-2x\).

This process of combining coefficients reduces the expression and makes it much more manageable, clearly showing how the variables interact.

Working with Constant Terms

Aside from the terms with variables, algebraic expressions often include constant terms. Constant terms are numbers without variables, and in simplifying expressions, they are handled separately from variable terms.
Let's look at our example again: After dealing with the like terms in \(3x + 4 - 5x + 1\), the remaining terms \(4\) and \(1\) are constants. Constant terms are combined through simple addition or subtraction:

• Add \(4\) and \(1\) to get \(5\).

Once you have simplified both the variable and constant parts of the expression, combine them to give the fully simplified expression: \(-2x + 5\). Simplifying constant terms separately ensures clarity and accuracy in your final result.