Problem 97 - Elementary and Intermediate Algebra

Question

Simplify each expression, if possible. $$ -4 x+4 x $$

Step-by-Step Solution

Answer

The simplified expression is 0.

Step 1: Identify Like Terms

Look at the expression \(-4x + 4x\). Notice that both terms are like terms because they both contain the variable "\(x\)" with the same power.

Step 2: Combine Like Terms

Since \(-4x\) and \(+4x\) are like terms, you can combine them by adding their coefficients together. The coefficient of \(-4x\) is \(-4\) and the coefficient of \(+4x\) is \(4\).

Step 3: Simplify the Expression

Add the coefficients together: \(-4 + 4 = 0\). So the expression simplifies to \(0x\,=\,0\).

Key Concepts

Simplifying expressions

Simplifying algebraic expressions is like tidying up a messy room. You want to make the expression as clean and simple as possible. When we talk about simplifying expressions, we mean reducing them to their simplest form.

• This involves combining like terms and performing basic arithmetic operations.
• By making an expression simpler, you're essentially making it easier to work with or understand in future steps.

To simplify an expression like \(-4x + 4x\), the goal is to perform operations that help condense the expression. The result is often a numeric value or a cleaner form of the original expression. In our example, simplifying yielded the clean and straightforward result of zero.

Like terms

In algebra, like terms refer to terms in an expression that have the same variable part raised to the same power. Recognizing like terms is crucial when simplifying expressions because only these can be combined.

• For instance, terms like \(-4x\) and \(+4x\) are considered like terms.
• This is because they both contain the variable \(x\), and each term involves \(x\) raised to the first power.

Combining like terms typically involves adding or subtracting their coefficients. In our example, since both terms are like terms, we could easily add their coefficients, leading to the simplification step where \(-4 + 4 = 0\). Identifying like terms ensures that algebraic expressions are simplified correctly and efficiently.

Coefficients

Coefficients are the numbers found in front of the variables in algebraic expressions. They tell you how many of a particular variable you're working with. Understanding coefficients is vital because they are what you adjust or combine when working with like terms.

• In \(-4x\), \(-4\) is the coefficient, meaning you're dealing with \(-4\) lots of \(x\).
• Similarly, \(+4x\) has a coefficient of \(+4\).

When simplifying expressions, you focus on combining these coefficients for like terms. Here, adding \(-4\) and \(+4\) gave us zero, which significantly simplified the expression. Coefficients guide you in determining how many terms you have and what happens when terms are added or subtracted.