Problem 75 - Prealgebra

Question

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

Step-by-Step Solution

Answer

The value is 6.

Step 1: Substitute the Value

Start by substituting the given value of \(x\), \(x = \frac{3}{4}\), into the expression \(12x - 3\). This gives us:\[12\left(\frac{3}{4}\right) - 3\]

Step 2: Multiply

Next, multiply \(12\) by \(\frac{3}{4}\). To do this, multiply \(12\) by \(3\) and then divide by \(4\).\[12 \times \frac{3}{4} = \frac{36}{4} = 9\]

Step 3: Subtracting

Subtract \(3\) from \(9\) to find the final value of the expression:\[9 - 3 = 6\]

Key Concepts

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operational symbols like plus, minus, multiply, and divide. In our exercise, we have the expression \(12x - 3\). Here, \(12x\) is a term where 12 is the coefficient of the variable \(x\), which means 12 is multiplied by \(x\). The expression \(-3\) is a constant term because it doesn’t contain any variables. Combining terms like these to form expressions is fundamental in algebra, as it allows us to describe mathematical relationships and equations.

When working with expressions, substituting a number for a variable is a common task. This means replacing the variable with a specific number, like we did by substituting \(x = \frac{3}{4}\) into the expression to solve it. Doing so converts the expression into a numerical calculation that follows algebraic rules.

Fractions

Fractions represent parts of a whole and are expressed with a numerator and a denominator. In our exercise, the expression you are working with involves a fraction: \(\frac{3}{4}\).

- **Numerator:** The top number (in this case, 3), which tells how many parts we are considering.
- **Denominator:** The bottom number (in this case, 4), which shows the total number of equal parts the whole is divided into.

Working with fractions involves operations like addition, subtraction, multiplication, and division. When substituting \(\frac{3}{4}\) for \(x\) in an expression, it's important to understand how to multiply fractions and whole numbers together, as well as simplify results when possible, as we will in our exercise.

Multiplication and Division

Multiplication and division are fundamental arithmetic operations, especially important in solving algebraic expressions.

**Multiplication:** When multiplying fractions by whole numbers, as in the expression \(12 \times \frac{3}{4}\), it's helpful to understand the steps:

• First, multiply the whole number by the numerator of the fraction. In this case, \(12 \times 3 = 36\).
• Next, divide that product by the denominator of the fraction. Here, \(\frac{36}{4} = 9\).

**Division:** Though not explicitly shown in this problem, division appears in simplifying fractions (like dividing 36 by 4). Understanding division as an operation to "split" into equal parts or to "reduce" fractions simplifies many algebraic processes.

By mastering multiplication and division within algebraic contexts, you can confidently tackle problems involving any combination of fractions and variables.