Problem 71 - Prealgebra

Question

Find the value of each of \(12 x-3\) for each of the following values of \(x .\) $$\frac{1}{4}$$

Step-by-Step Solution

Answer

The value of the expression is 0 when \(x = \frac{1}{4}\).

Step 1: Substitute Value

Substitute the given value of \(x = \frac{1}{4}\) into the expression \(12x - 3\). The expression becomes \(12\left(\frac{1}{4}\right) - 3\).

Step 2: Multiply

Calculate the product \(12\times \frac{1}{4}\). Multiply the numbers to get \(12 \times \frac{1}{4} = \frac{12}{4} = 3\).

Step 3: Subtract

Subtract the result from Step 2 from 3: \(3 - 3 = 0\).

Key Concepts

Substitution Method

The substitution method is a mathematical technique used to solve expressions and equations by replacing variables with their given values or expressions. In prealgebra, substituting typically involves replacing a variable, such as \(x\), with a specific numeric value provided in a problem.
For the exercise at hand, the substitution method allows us to find the value of an expression with ease by simply plugging the value \(x = \frac{1}{4}\) into the expression \(12x - 3\)
.

• We began this method by replacing \(x\) with \(\frac{1}{4}\), which altered our initial expression to \(12\left(\frac{1}{4}\right) - 3\).
• This process makes it straightforward to later perform arithmetic operations, as the expression is now entirely numeric.

Substitution is foundational because it directly affects how we can proceed with evaluating and simplifying expressions, especially when preparing to solve complex equations involving multiple variables.

Evaluating Expressions

Evaluating expressions is the process of finding the value of an algebraic expression by performing the necessary mathematical operations once variables have been substituted. It's a crucial part of algebra that allows us to interpret and simplify mathematical statements.
In our current exercise, after substituting \(x = \frac{1}{4}\) into \(12x - 3\), our expression became \(12\left(\frac{1}{4}\right) - 3\)
This is where evaluating the expression takes place.
To efficiently evaluate:

• First, carry out any multiplication, which in this example is \(12 \times \frac{1}{4}\).
• Then, proceed with subtraction, taking the result of the multiplication and subtracting the remaining number.

Evaluating step by step ensures accuracy and improves understanding of the order of operations, which is key in all mathematical calculations.

Arithmetic Operations

Arithmetic operations refer to the basic operations of mathematics: addition, subtraction, multiplication, and division. These are essential for evaluating expressions and solving equations, and they are foundational to understanding higher-level math concepts.
In solving \(12x - 3\) with \(x = \frac{1}{4}\), we performed a series of arithmetic operations:

• First, multiplication was used to calculate \(12 \times \frac{1}{4}\), leading to \(\frac{12}{4}\) or 3. This is a straightforward operation where a whole number is multiplied by a fraction.
• Next, subtraction was performed, taking the result from the multiplication (3) and subtracting 3, which resulted in 0.

Each arithmetic operation serves a pivotal role and understanding these helps in simplifying expressions and solving them effectively. Mastery of these elementary operations strengthens a student's ability to tackle more complex algebraic problems in future studies.