Problem 32 - Elementary Algebra Within Reach

Question

In Exercises \(25-32\), simplify the expression by combining like terms. $$ 2 t-4+8 t+9 $$

Step-by-Step Solution

Answer

The simplified expression is \(10t + 5\).

Step 1: Identify and Combine Likes Terms

Identify the like terms in the expression. We can see that the like terms are \(2t\) and \(8t\), and the constants -4 and 9. Rewriting the expression grouping the like terms together results in: \( (2t + 8t) + (-4 + 9)\).

Step 2: Simplify the Expression

Now simplify these groups. Adding \(2t\) and \(8t\) results in \(10t\) and adding \(-4\) and \(9\) results in \(5\). So the expression simplifies to: \(10t + 5\).

Key Concepts

Combining Like Terms

Combining like terms is an essential skill in algebra. It makes expressions simpler and easier to work with. Like terms are terms that have the same variables raised to the same power. In other words, the only difference between like terms is their coefficient. To combine them, you simply add or subtract their coefficients.
For example, consider the terms \(2t\) and \(8t\) from the expression \(2t - 4 + 8t + 9\). Both terms have the variable \(t\) raised to the first power, so they are like terms. To combine them, you add their coefficients:

• Combine like terms \(2t + 8t\).
• This results in \(10t\).

Combining like terms makes the expression more manageable, which is the goal of simplifying algebraic expressions.

Simplifying Expressions

Simplifying expressions involves reducing an expression to its simplest form. This process makes it easier to evaluate or solve equations. The key to simplifying an algebraic expression is to minimize the number of terms without changing its value.
A simplified expression facilitates easier manipulation and understanding in further calculations. In the example provided, after identifying and combining like terms, our expression is transformed from \(2t - 4 + 8t + 9\) to \(10t + 5\). Every term that can be combined is combined:

• Combine the variables: \(2t + 8t = 10t\).
• Combine the constants: \(-4 + 9 = 5\).

This simplified expression is a cleaner version that communicates the same mathematical idea.

Like Terms

Like terms in an algebraic expression are crucial for simplifying it. Identifying like terms is about finding terms that share the same variables and exponents. When we look at an expression, we group these terms together to combine them.
In our expression example, \(2t + 8t\), both terms have the same variable \(t\), making them like terms. However, terms such as \(\-4\) and \(9\) are like terms with each other due to them being constants (numbers on their own with no variable).
Recognizing like terms allows us to reduce expressions to a form that is simpler to work with, by strategically combining them.

Constants

In algebra, constants are numbers without any variables. Unlike variables, which can change values, constants remain the same throughout the problem. They are essential elements within algebraic expressions.
In the given exercise, the numbers \(-4\) and \(9\) are constants. When simplifying expressions, one of the steps is to combine these constants. It helps in reducing clutter and makes expressions less bulky. For instance, combining the constants in the example:

• \(-4 + 9 = 5\)

Removing complexity by combining constants is a part of simplifying an algebraic expression, which in turn makes solving equations more straightforward.