Problem 37 - Elementary Algebra Within Reach

Question

In Exercises 37-44, evaluate the algebraic expression for the given values of the variable(s). \(2 x-1\) (a) \(x=\frac{1}{2}\) (b) \(x=-4\)

Step-by-Step Solution

Answer

The evaluated expressions are \(0\) when \(x = \frac{1}{2}\) and \(-9\) when \(x = -4\)

Step 1: Substitute \(x = \frac{1}{2}\) into the expression

To evaluate the expression \(2 x - 1\) when \(x = \frac{1}{2}\), substitute \(\frac{1}{2}\) for \(x\) in the expression: \(2(\frac{1}{2}) - 1\)

Step 2: Simplify the expression

Multiplying \(2\) by \(\frac{1}{2}\) yields \(1\), so the expression becomes:\(1 - 1\) which simplifies to \(0\)

Step 3: Substitute \(x = -4\) into the expression

To evaluate the expression \(2 x - 1\) when \(x = -4\), substitute \(-4\) for \(x\) in the expression: \(2(-4) - 1\)

Step 4: Simplify the expression

Multiplying \(2\) by \(-4\) yields \(-8\), so the expression becomes:\(-8 - 1\) which simplifies to \(-9\)

Key Concepts

Substitution Method

The substitution method is a helpful technique when dealing with algebraic expressions. Its main goal is to replace variables with actual numbers to make calculations simpler. Here’s how it works in our example:
- Begin by identifying the variable in the expression. In this case, it’s \(x\) in the expression \(2x - 1\).- Replace this variable with the value given in the exercise. This means if \(x = \frac{1}{2}\), you substitute \(\frac{1}{2}\) into \(2x - 1\), which turns it into \(2(\frac{1}{2}) - 1\).
Substitution helps to convert an abstract expression into a numerical one. This makes it possible to move on to the next step, simplification.

Simplification Process

After substituting the given value into the expression, the next step is the simplification process. Simplifying algebraic expressions means reducing them to their simplest form to easily find the answer.
- Perform the arithmetic operations as instructed by the expression’s structure. For \(2(\frac{1}{2}) - 1\): first multiply \(2\) by \(\frac{1}{2}\), to get \(1\).- Then subtract \(1\) from \(1\), resulting in \(0\).
This process is repeated for every new substituted value if there are more parts to evaluate. Simplicity in expression offers clarity and reduces room for error, making algebra manageable.

Algebraic Operations

Understanding the various algebraic operations is fundamental to successfully evaluating expressions. These operations include addition, subtraction, multiplication, and division, and each plays a role depending on the expression.
- In the expression \(2x - 1\), multiplication and subtraction are the key operations.- First, handle any multiplication. Here, multiply \(2\) by the value of \(x\) (e.g., \(-4\) in the second evaluation).- Follow up with subtraction, which involves subtracting \(1\) from the result of your previous operation. This forms a crucial part of getting to the final simplified value.
Mastering algebraic operations ensures accuracy in evaluations and enables you to solve complex expressions confidently.