Problem 38 - Elementary Algebra Within Reach

Question

In Exercises 37-44, evaluate the algebraic expression for the given values of the variable(s). \(3 x-2\) (a) \(x=\frac{4}{3}\) (b) \(x=-1\)

Step-by-Step Solution

Answer

The evaluation of the expression \(3x - 2\) for the given values results in (a) \(2\); (b) \(-5\).

Step 1: Substitute for \(x\)

First, we replace \(x\) in the expression \(3x - 2\) with the given value. For example, for (a), if \(x = \frac{4}{3}\), the expression becomes \(3 (\frac{4}{3}) - 2\). For case (b), if \(x = -1\), the expression becomes \(3(-1) -2\)

Step 2: Simplification

In the next step, we simplify the resulting expression. For part (a), the expression simplifies to \(4 - 2 = 2\). For part (b), the expression simplifies to \(-3 - 2 = -5\).

Step 3: Conclusion

The evaluated expressions give us the solutions to the problem. Therefore, for (a), the algebraic expression \(3x - 2\) evaluates to \(2\) when \(x = \frac{4}{3}\). Similarly, for (b), the expression evaluates to \(-5\) when \(x = -1\).

Key Concepts

Substitution

Substitution is a key idea in algebra, especially when working with algebraic expressions. It involves replacing the variables in an expression with their given values. This helps simplify the problem and discover solutions.

In the example provided, we see how substitution works:

• When given the expression \(3x - 2\), and the value \(x = \frac{4}{3}\), substitution allows us to replace \(x\) with \(\frac{4}{3}\).
• The expression then becomes \(3(\frac{4}{3}) - 2\).

Following these steps allows the expression to be more manageable and ready for the next stage: simplification.

Simplification

Once substitution is complete, the next step is simplification. Simplification is the process of making an algebraic expression easier to understand and solve by performing arithmetic operations.

Consider the example of the expression \(3x - 2\) after substituting \(x = \frac{4}{3}\):

• The expression becomes \(3(\frac{4}{3}) - 2\).
• When simplified, \(3 \times \frac{4}{3} = 4\).
• Then the expression \(4 - 2\) simplifies further to \(2\).

Each simplification step helps break down the expression into a more digestible number, revealing the solution.

Variables in Algebra

Variables are an essential part of algebra, serving as placeholders for numbers, and allowing us to write expressions and equations. In algebra, variables like \(x\) provide a way to express general solutions or relations.

When solving problems, understanding variables is crucial as:

• They represent unknown quantities that can vary.
• They allow generalization of mathematical concepts.

In the expression \(3x - 2\), \(x\) is a variable. By assigning different values to this variable, we can evaluate the statement to find specific outcomes, which illustrates the power and flexibility variables offer in algebra.