Problem 61 - Beginning and Intermediate Algebra

Question

Simplify each expression. $$ t+(-t)+\frac{1}{2}(2) $$

Step-by-Step Solution

Answer

1

Step 1: Identify Like Terms

Look at the expression: First identify the terms, which are: The like terms in this expression are: and .

Step 2: Combine Like Terms

Combine the like terms and . Combining them: . The updated expression is now: .

Step 3: Simplify Constants

Simplify the remaining term . Multiply and . This becomes: .

Step 4: Combine Terms to Simplify

After combining and simplifying terms, the expression is now simplified. .

Key Concepts

Understanding Like Terms

To simplify expressions, we first need to understand what like terms are. Like terms are terms that have the same variable raised to the same power. For example, in the expression \(2t + 3t - 5\), the terms \(2t\) and \(3t\) are like terms because they both contain the variable \(t\) raised to the first power. However, the term \(-5\) is not a like term because it has no variable. Identifying like terms allows us to combine them simply by adding or subtracting their coefficients. This will simplify the expression and make further calculations easier.

Combining Like Terms

Once you have identified like terms, you can combine them to simplify the expression. Let's look at the example \(t + (-t) + \frac{1}{2}(2)\). First, identify the like terms. Here, \(t\) and \(-t\) are like terms. To combine these, add their coefficients: \(1t\) and \(-1t\). This gives us \(t - t = 0\). Hence, \(t + (-t)\) cancels out to zero. By combining these like terms, the expression becomes simpler. It is now left with just \(\frac{1}{2}(2)\).

Simplifying Constants

After combining like terms, we are left with constants or terms without variables. In our example, the expression is now \(\frac{1}{2}(2)\). Simplifying constants usually involves basic arithmetic. Here, we need to multiply \(\frac{1}{2}\) by \(2\). This can be done as follows: \(\frac{1}{2} \times 2 = 1\). Thus, the expression simplifies to \(1\). Now the entire original expression \(t + (-t) + \frac{1}{2}(2)\) has been simplified to the constant \(1\). By following these steps consistently, you can easily simplify any algebraic expression.