Problem 74 - Prealgebra

Question

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

Step-by-Step Solution

Answer

The value is 5.

Step 1: Substituting the Value of x

Start by substituting the given value of \(x\), which is \(\frac{2}{3}\), into the expression \(12x - 3\). This changes the expression to \(12\left(\frac{2}{3}\right) - 3\).

Step 2: Performing Multiplication

Multiply \(12\) by \(\frac{2}{3}\). To do this, you multiply the numerator by \(12\) and you keep the denominator as \(3\). Thus, \(12 \times \frac{2}{3} = \frac{24}{3}\).

Step 3: Simplifying the Fraction

Simplify the fraction \(\frac{24}{3}\) by dividing the numerator by the denominator. \(\frac{24}{3} = 8\).

Step 4: Subtraction Operation

Now, subtract \(3\) from \(8\), which simplifies to \(8 - 3 = 5\).

Key Concepts

Expression Evaluation

Expression evaluation is the process of finding the value of an algebraic expression by substituting the values of the variables and simplifying the resulting expression. When you have an expression like \(12x - 3\) and you're given a specific value for \(x\), you simply replace \(x\) with the given value. In this exercise, the specific value provided for \(x\) is the fraction \(\frac{2}{3}\). After substitution, the expression becomes \(12\left(\frac{2}{3}\right) - 3\).

The primary goal is to carefully substitute and then simplify the expression. This is crucial in many areas of algebra and helps develop a deeper understanding of how different mathematical operations interact. Ensure each substitution step is done clearly to avoid confusion in subsequent calculations. Evaluating expressions correctly is foundational in solving equations and modeling real-world problems.

Fraction Multiplication

Fraction multiplication involves multiplying the numerators together and the denominators together. It's a straightforward process and becomes even simpler when one of the numbers is a whole number. In our expression \(12\left(\frac{2}{3}\right)\), we treat \(12\) as \(\frac{12}{1}\). Thus, the multiplication of the fractions \(\frac{12}{1} \times \frac{2}{3}\) happens as follows:

• Multiply the numerators: \(12 \times 2 = 24\)
• Multiply the denominators: \(1 \times 3 = 3\)
• Form the new fraction: \(\frac{24}{3}\)

Next, the fraction we obtained, \(\frac{24}{3}\), needs to be simplified, which is an important check-in fraction multiplication to ensure results are as straightforward as possible.

Simplifying Fractions

Simplifying fractions is reducing them to their simplest form, where the numerator and denominator are the smallest possible integers. To simplify a fraction like \(\frac{24}{3}\), you need to find the greatest common divisor (GCD) of the numerator and denominator. Here, both 24 and 3 are divisible by 3.In this case:

• Divide the numerator by 3: \(24 \div 3 = 8\)
• Divide the denominator by 3: \(3 \div 3 = 1\)

Therefore, \(\frac{24}{3}\) simplifies to \(8\), because any number divided by 1 remains the same. Simplifying fractions is essential in ensuring calculations remain easy to work with, and results are as concise as possible, thus maintaining clarity and understanding in further mathematical operations.