Problem 70 - Prealgebra

Question

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

Step-by-Step Solution

Answer

When \(x = \frac{1}{3}\), \(12x - 3 = 1\).

Step 1: Substitute the given value of x into the expression

We need to find the value of \(12x - 3\) with \(x = \frac{1}{3}\). Substitute \(\frac{1}{3}\) for \(x\) in the expression, so it becomes \(12 \left( \frac{1}{3} \right) - 3\).

Step 2: Simplify the multiplication inside the expression

Now, calculate \(12 \times \frac{1}{3}\). This can be done by multiplying the numerator of the fraction by 12: \(12 \times \frac{1}{3} = \frac{12}{3}\).

Step 3: Simplify the division

Next, simplify the fraction \(\frac{12}{3}\) which results in 4, because \(12\) divided by \(3\) is \(4\).

Step 4: Complete the subtraction

Finally, subtract \(3\) from \(4\). So, we have \(4 - 3 = 1\). This means for \(x = \frac{1}{3}\), the expression \(12x - 3\) equals \(1\).

Key Concepts

Substitution

Substitution is an essential concept in prealgebra. When faced with an expression where the variable's value is given, substitution is the first step to solving the problem.
In this particular exercise, we substitute the value of \(x\), which is \(\frac{1}{3}\), into the expression \(12x - 3\).

The process involves replacing the variable \(x\) with its specified value. It requires you to be precise. When substituting, carefully place the given number inside parentheses to avoid mistakes. In our example, we substitute \(x\) with \(\frac{1}{3}\) in the expression, transforming it into \(12 \left( \frac{1}{3} \right) - 3\). This sets the stage for the next steps of solving the expression.

Simplifying Expressions

Simplifying expressions is about breaking down the expression into a more manageable form. In mathematics, expressions are often reduced to their simplest form for clarity.

After substituting, the expression became \(12 \left( \frac{1}{3} \right) - 3\). The next step is to simplify the multiplication. Multiply \(12\) by \(\frac{1}{3}\), which gives us \(\frac{12}{3}\).

Simplifying this fraction involves performing division. In this case, \(\frac{12}{3}\) simplifies to \(4\), because \(12\) divided by \(3\) equals \(4\). This simplification helps to evaluate parts of the expression sequentially, making it easier to solve step by step.

Arithmetic Operations

Once we have simplified the multiplication, arithmetic operations allow us to combine numbers to find the final result. In this exercise, we perform subtraction.
After simplifying the expression, we have reduced it to \(4 - 3\).

Arithmetic operations are the basic calculations like addition, subtraction, multiplication, and division. They follow the rules of BODMAS (Brackets, Orders, Division, Multiplication, Addition, Subtraction) to ensure correct results.
In this context, subtraction is straightforward. We subtract \(3\) from \(4\), resulting in \(1\). This final calculation concludes our solution. Using arithmetic operations effectively enables us to reach the solution efficiently and accurately.