Problem 73 - Prealgebra

Question

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

Step-by-Step Solution

Answer

The value is 15.

Step 1: Substitute the Value of x

We are given the expression \(12x - 3\) and need to find its value when \(x = \frac{3}{2}\). Start by substituting \(x\) with \(\frac{3}{2}\) in the expression.

Step 2: Simplify Multiplication

After substitution, we have \(12 \times \frac{3}{2} - 3\). Simplify the multiplication by calculating \(12 \times \frac{3}{2}\). This gives \(12 \times \frac{3}{2} = 6 \times 3 = 18\).

Step 3: Subtract

Now, subtract 3 from 18 as per the expression \(18 - 3\). Simplify this to get \(18 - 3 = 15\).

Key Concepts

Expression Evaluation

Expressions in math are combinations of numbers, variables, and operators that represent a certain value. Evaluating an expression means determining this value based on given variables. For example, in the expression \(12x - 3\), the variable \(x\) can have different values which changes the overall result.To evaluate such an expression, follow these steps:

• Identify the expression, which is \(12x - 3\) in this case.
• Determine the value of any variables in the expression.
• Substitute the variable with its given value and perform the calculations.

Understanding expression evaluation is crucial in algebra because it sets the foundation for solving equations and inequalities. It requires substituting values and performing arithmetic operations like addition, subtraction, multiplication, and division.

Substitution

Substitution involves replacing variables in an expression with given numbers. This step is necessary to transform an algebraic expression into a numerical one that can be calculated.For example, if you have \(12x - 3\) and \(x = \frac{3}{2}\):

• Replace \(x\) with \(\frac{3}{2}\), resulting in the new expression \(12 \times \frac{3}{2} - 3\).

This new expression allows us to directly calculate its numerical value. The key to mastering substitution is paying careful attention to signs and correctly inputting the given values. It's an essential skill in prealgebra and beyond, laying the groundwork for solving more complex algebraic problems.

Simplification

Simplification refers to reducing expressions to their simplest form by performing arithmetic operations. After you've substituted a value into an expression, the next step is simplification.For instance, with the expression \(12 \times \frac{3}{2} - 3\):

• Calculate the multiplication first (\(12 \times \frac{3}{2}\)). Here, it's simplified to \(18\).
• Subtract \(3\) from the product to simplify further, leaving you with \(15\).

The goal of simplification is to make expressions as straightforward as possible, which often simplifies subsequent calculations. This concept is frequently used in algebra when dealing with expressions and equations.

Multiplication and Subtraction

Multiplication and subtraction are fundamental arithmetic operations crucial in simplifying algebraic expressions.Once substitution is complete, tackle multiplication first:

• Multiply the coefficients or constants as per the expression. For example, \(12 \times \frac{3}{2}\) equals \(18\).

Next, proceed to subtraction:

• Subtract the specified number from the result of the multiplication. In the simplified expression \(18 - 3\), the result is \(15\).

Understanding these operations is important because they frequently appear in many algebraic manipulations. Mastery ensures accuracy in evaluating and solving expressions and equations, especially in prealgebra.

Algebraic Expressions

Algebraic expressions are mathematical phrases containing numbers, variables, and operations. They are the building blocks of algebra and serve various purposes in mathematical problem-solving.For example, the expression \(12x - 3\) includes:

• The coefficient \(12\), which is multiplied by the variable \(x\).
• A constant \(-3\), which is subtracted from the product.
• The variable \(x\), which can be replaced by different numbers to find different values of the expression.

Understanding algebraic expressions is crucial for progressing in math, as they appear in equations and inequalities. They enable abstract thinking and problem-solving across various scenarios and disciplines. Recognizing how to construct and deconstruct them is a key skill in prealgebra.