Problem 69 - Prealgebra

Question

Find the value of each of \(12 x-3\) for each of the following values of \(x .\) $$\frac{1}{2}$$

Step-by-Step Solution

Answer

The value of the expression is 3.

Step 1: Plug in the given value of x

Replace the variable \(x\) in the expression \(12x - 3\) with the given value \(x = \frac{1}{2}\). This gives us: \(12\left(\frac{1}{2}\right) - 3\).

Step 2: Evaluate the expression

Calculate \(12 \times \frac{1}{2}\) first. This is equal to \(\frac{12}{2} = 6\).

Step 3: Subtract 3 from the result

Subtract 3 from 6, which gives: \[ 6 - 3 = 3 \].

Key Concepts

Prealgebra

Prealgebra is a fundamental branch of mathematics that prepares students for studying algebra. It usually involves arithmetic operations and the manipulation of mathematical expressions and equations without reliance on abstract concepts. One common area explored in prealgebra is working with basic expressions, such as linear expressions, which are equations where variables are raised to the power of one.

Prealgebra is essential because it lays the groundwork for understanding algebraic concepts later on. Students learn to combine like terms, evaluate expressions using basic operations like addition, subtraction, multiplication, and division, and solve simple equations.

For instance, if you're tasked with evaluating an expression like "12x - 3" for different values of \(x\), you're applying prealgebra techniques to plug in different numbers and simplify the result. This exercise strengthens understanding of variable manipulation and operational hierarchy.

Step-by-step solution

A step-by-step solution is a systematic approach to solving mathematical problems. It breaks down each stage of the solution into manageable parts, which makes it easier to follow and understand.

When working through a problem like "12x - 3" with \(x = \frac{1}{2}\), the solution process can be divided into clear stages:

• Start by substituting the given value into the expression. Replace all instances of the variable with the given number.
• Next, solve any arithmetic operations within the expression. Follow the order of operations: perform any multiplication or division first, then addition or subtraction.
• Finish by simplifying the expression to its simplest form.

This kind of structured approach helps prevent mistakes and reinforces a logical progression towards solving any math problem. It also enhances confidence in working with more complex equations in the future.

Substitution in expressions

Substitution is a key process in algebra and prealgebra, where you replace a variable in an equation or expression with a specific value. This technique is crucial for evaluating expressions and solving equations.

In the original exercise, the expression given is "12x - 3". To evaluate this for \(x = \frac{1}{2}\), we substitute the \(\frac{1}{2}\) for \(x\), turning the expression into "12\left(\frac{1}{2}\right) - 3". Substitution is the gateway to transforming variables into known quantities, which can then be worked with using basic arithmetic.

This practice helps develop skills needed for more advanced algebra, where multiple variables and their relationships become more complex. By mastering substitution, students pave the way for tackling intricate topics with ease.