Problem 72 - Prealgebra

Question

Find the value of each of \(12 x-3\) for each of the following values of \(x .\) $$\frac{1}{6}$$

Step-by-Step Solution

Answer

The value of the expression is \(-1\).

Step 1: Substitute the Value of x

To find the value of the expression for a specific value of \( x \), we substitute \( x = \frac{1}{6} \) into the expression \( 12x - 3 \).

Step 2: Perform the Multiplication

Multiply \( 12 \) by \( \frac{1}{6} \). This simplifies to \( 2 \), since \( 12 \times \frac{1}{6} = 2 \).

Step 3: Subtract From the Result

Subtract \( 3 \) from the result obtained in Step 2. So, we have \( 2 - 3 = -1 \).

Key Concepts

Substitution

Substitution is a fundamental concept in algebra where we replace a variable with a specific value to evaluate an expression or solve an equation. This method is crucial when dealing with algebraic expressions, as it allows us to find precise numerical outputs based on given inputs.

In simple terms, whenever we have a variable, like \( x \), and we know its value, we can substitute it into an equation to work through to a solution. For instance, if we're given the expression \( 12x - 3 \) and need to evaluate it for \( x = \frac{1}{6} \), the first step would be to replace every instance of \( x \) in that expression with \( \frac{1}{6} \).

This process enables us to move from a general expression involving variables to a specific expression involving only numbers, which is much easier to manage. Remember that correctly substituting the values is key to achieving the accurate result.

Simplification in Algebra

Simplification in algebra involves reducing expressions and equations to their simplest form to make them easier to work with. This process often includes combining like terms, reducing fractions, and performing arithmetic operations logically and carefully.

In the problem at hand, after substituting \( x = \frac{1}{6} \), we had the expression \( 12 \times \frac{1}{6} - 3 \). Simplification here means performing the multiplication first to simplify the expression to \( 2 \). The expression was \( \frac{12}{6} \) due to multiplication, which reduces to 2.

Simplifying expressions is an essential skill. It aids in presenting problems in a form that is easy to understand and solve. Always look out for any factors or terms that can be reduced or combined to simplify your work effectively.

Arithmetic Operations

Arithmetic operations are the basics of mathematics and include addition, subtraction, multiplication, and division. These operations form the foundation of solving algebraic expressions.

In this exercise, you dealt with two main arithmetic operations: multiplication and subtraction. First, the multiplication of \( 12 \times \frac{1}{6} \) needed to be calculated, resulting in 2. Understanding how to multiply a whole number by a fraction is crucial, as it often comes up in algebra.

The next operation was subtraction. After finding the product, you subtract 3 to get the final result of \( -1 \). When performing arithmetic operations, follow the correct order of operations, sometimes remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)). Doing this ensures that your calculations are exact and reliable.