Problem 34 - Prealgebra

Question

Simplify expression. \(2 x+3-3 x+9\)

Step-by-Step Solution

Answer

The simplified expression is \(-x + 12\).

Step 1: Identify Like Terms

In the expression \(2x + 3 - 3x + 9\), the like terms are the terms that contain the variable \(x\) and the constant terms. \(2x\) and \(-3x\) are like terms because they contain the variable \(x\). \(3\) and \(9\) are like terms as they are both constants.

Step 2: Combine Like Terms with Variables

Focus on the terms with \(x\): \(2x\) and \(-3x\). Combine them by performing the subtraction: \(2x - 3x = -1x\) or simply \(-x\).

Step 3: Combine Constant Terms

Next, add the constant terms together: \(3 + 9 = 12\).

Step 4: Write the Simplified Expression

Combine the simplified variable term with the simplified constant term. The expression becomes \(-x + 12\).

Key Concepts

Understanding Like Terms

In algebra, it is crucial to identify and work with like terms when simplifying expressions. Like terms are terms in an expression that have identical variable parts, even if their coefficients differ. This means they have the same letter or letters, raised to the same power. For example:

• In the expression \(2x + 3 - 3x + 9\), the terms \(2x\) and \(-3x\) are considered like terms because both contain the variable \(x\).
• In another instance, \(5y^2\) and \(-7y^2\) are like terms because both contain \(y^2\).

Recognizing like terms is fundamental, as it allows you to combine them to simplify algebraic expressions efficiently.

Combining Like Terms

Once like terms are identified, the next step is to combine them. This process is essential for simplifying expressions, making them easier to work with. Here's how you can do it:
When combining like terms, focus on their coefficients—numbers in front of the variables—and perform basic arithmetic operations on them.

• In our example, \(2x\) and \(-3x\) are like terms. We perform the subtraction: \(2x - 3x=(-1)x\) or simply \(-x\).
• For more examples, adding \(4a\) and \(3a\) results in \(7a\), while \(-2y\) plus \(5y\) becomes \(3y\).

This step is crucial because combining like terms reduces the number of terms in an expression, leading to a simpler, cleaner result.

Deciphering Constant Terms

Constant terms are an essential part of algebraic expressions, especially when it comes to simplifying them. They are the numbers without any variable attachment, standing alone in the expression. You can consider them the fixed numerical values.
In expressions like \(2x + 3 - 3x + 9\), the terms \(3\) and \(9\) are constants. They have no variables attached.
Combining constant terms follows the same basic arithmetic principles as combining like terms:

• Add \(3\) and \(9\) to get a combined constant of \(12\).
• If the constants were \(-4\) and \(6\), they would sum up to \(2\).

By simplifying both the variable and constant parts of the expression, you're left with an easier-to-handle form, exemplified by the final simplified expression \(-x + 12\).