Problem 22 - Algebra 1: Concepts and Skills

Question

Evaluate the expression for the given value of the variable. $$ \frac{1}{2}+t \text { when } t=2 $$

Step-by-Step Solution

Answer

The evaluated expression when \( t = 2 \) is \( \frac{5}{2} \) or 2.5 in decimal form.

Step 1: Substitute the given value into the expression

Plug the given value of \( t = 2 \) into the expression: \( \frac{1}{2} + t \). It becomes \( \frac{1}{2} + 2 \).

Step 2: Perform the Addition

Add \(\frac{1}{2}\) and 2 together. Since 2 is equal to \( \frac{4}{2} \), the expression becomes \( \frac{1}{2} + \frac{4}{2} = \frac{5}{2} \).

Key Concepts

Substitution

Substitution is a crucial technique in algebra that replaces a variable with a specific value. In the given problem, we substitute the variable \(t\) with the number 2, as specified. This step simplifies an algebraic expression so that it can be directly evaluated.

• Identify the variable and its assigned value.
• Replace every occurrence of the variable in the expression with the given value.

This process helps to transform a general equation into a numerical one, making it easier to handle other operations like addition or subtraction. For example, replacing \(t\) in our expression \(\frac{1}{2} + t\) results directly in \(\frac{1}{2} + 2\). This is the first pivotal step in any evaluation process, paving the way for further calculations.

Addition of Fractions

Adding fractions involves combining fractions to form a single value. In our exercise, this means adding a regular number represented as a fraction. To do this, you need a common denominator. For the problem:

• Convert the whole number 2 into a fraction. Here, 2 can be written as \(\frac{4}{2}\).
• With both denominators now being 2, addition becomes straightforward: \(\frac{1}{2} + \frac{4}{2} = \frac{5}{2}\).

Using a common denominator ensures that the fractions are in a proper format to be accurately added. This technique of managing denominators is essential when dealing with any fractions, whether they appear in pure numbers or algebraic expressions. Having this skill sharpens your ability to process a variety of mathematical expressions fluidly.

Variable Substitution

Variable substitution lets us transform an abstract expression into something concrete by giving specific values to variables. In many algebraic problems, like this one, it is the key step that sets off the computation. When dealing with variable substitution:

• Clearly understand the relationship between variables and their values given in the problem.
• Always reassess the expression post-substitution to ensure you substituted the correct value.

For instance, in this problem, we see \(t\) as a place-holder. By assigning it the value 2, we transition from an undefined algebraic term to a number we can calculate with. After substituting \(t = 2\) in \(\frac{1}{2} + t\), it becomes \(\frac{1}{2} + 2\). Thus, substitution allows you to move from the abstract to something solvable straightforwardly, focusing on other arithmetic steps.