Problem 36 - Elementary Algebra

Question

Simplify. $$ -2 a+5 a-12 a $$

Step-by-Step Solution

Answer

The simplified expression is \(-9a\).

Step 1: Group Like Terms

Identify the like terms in the expression. All the terms in the expression \(-2a + 5a - 12a\) are like terms because they all contain the variable \(a\).

Step 2: Combine Coefficients

Add or subtract the coefficients of the like terms. Compute \(-2 + 5 - 12\).

Step 3: Calculate the Result

Perform the operation: \(-2 + 5 - 12 = -9\). Thus, the expression simplifies to \(-9a\).

Key Concepts

Like Terms

Understanding the concept of "like terms" is essential in algebraic simplification. Like terms are terms that have the same variables raised to the same power. In the expression \[-2a + 5a - 12a\], each term is like a family member that shares something in common: they all have the variable \(a\). When terms share the same variable and power, they can be grouped together. This makes it easier to simplify expressions. It's like organizing your closet by color or season, it makes everything clearer and more manageable.

Here’s how you recognize like terms:

• They must contain the same variable(s) and powers. For example, \(2x\) and \(-3x\) are like terms because they both contain \(x^1\).
• Numerical constants (like 3 and -5) are also considered like terms.
• Terms like \(4x^2y\) and \(x^2y\) are alike since they both include \(x^2y\).

Once you know how to spot them, you can combine these terms to make the expression simpler.

Coefficients

The next step in simplifying algebraic expressions like \[-2a + 5a - 12a\] involves understanding coefficients. A coefficient is the numerical part of a term, the number that is in front of a variable. Think of it as a label that tells you how many of that variable you have.

In the expression, the coefficients are \(-2, 5,\) and \(-12\).
When you group like terms, you focus on adding or subtracting just these numbers, while the variables remain intact. Imagine combining groups of apples where the number in each group is given by the coefficients:

• Subtract 2 apples: \(-2a\)
• Add 5 apples: \(5a\)
• Subtract 12 apples: \(-12a\)

By combining these coefficients, you determine the net effect of the operation on the variable \(a\). This concept is central to proper algebraic manipulation.

Algebraic Simplification

Algebraic simplification involves reducing expressions to their simplest form without changing their value. The goal is to make the math as straightforward as possible. Let's see how it's done using the expression \[-2a + 5a - 12a\].

Simplification is a lot like cleaning up a messy desk: you get rid of what's not needed, consolidate similar items, and make everything look tidy.
Here’s a simple process for algebraic simplification:

• Identify and group the like terms: This was where we noticed all the terms had the variable \(a\).
• Combine the coefficients: We then added and subtracted the coefficients: \(-2 + 5 - 12 = -9\)
• Rewrite the simplified expression: As a result, we derived the simplified term: \(-9a\). This is cleaner and easily understood at a glance.

Simplifying algebra not only declutters your math expressions but also makes solving more straightforward and error-free.