Problem 54 - Algebra 1

Question

Evaluate the expression for the given value(s) of the variable(s). $$\frac{3 a-b}{a} \text { when } a=\frac{1}{3} \text { and } b=-3$$

Step-by-Step Solution

Answer

The evaluated expression equals 12.

Step 1: Substitution of values into the equation

Substitute \(a = \frac{1}{3}\) and \(b = -3\) into the given equation. This will give us \(\frac{3 (\frac{1}{3}) - (-3)}{\frac{1}{3}}\).

Step 2: Simplifying the numerator

The numerator simplifies to \(3 \times \frac{1}{3} - (-3)\) = \(1 + 3\) = \(4\).

Step 3: Simplifying the denominator

The denominator simplifies to \( \frac{1}{3}\).

Step 4: Solve the equation

To find the solution, the numerator and denominator of the resulting fraction, \(\frac{4}{\frac{1}{3}}\), should be calculated. If you multiply the numerator by the reciprocal of the denominator it simplifies to \(4\times3 = 12\).

Key Concepts

Substitution

When tackling problems involving variable expressions, one of the first steps is substitution. This means replacing the variables in the expression with their given values. For instance, in the expression \( \frac{3a - b}{a} \), we have specific values for variables \( a = \frac{1}{3} \) and \( b = -3 \). By substituting these values, we transform the expression into an arithmetic form that can be easily handled.
This substitution transforms the expression from an abstract form with letters (variables) to a concrete form with numbers. As a result, the expression becomes \( \frac{3(\frac{1}{3}) - (-3)}{\frac{1}{3}} \). Each occurrence of \( a \) and \( b \) in the equation is replaced with their given values.
It's crucial to be meticulous during substitution to avoid errors, especially with operations involving negative signs and fractions.

Simplification

Once the substitution is complete, the next step in evaluating the expression is simplification. This involves performing arithmetic operations to reduce the expression to a more manageable form. In our example, the numerator of the expression is \( 3 \times \frac{1}{3} - (-3) \).
Let's break this down:

• First, perform the multiplication: \( 3 \times \frac{1}{3} = 1 \).
• Next, handle the subtraction involving a negative number: \( 1 - (-3) \) simplifies to \( 1 + 3 = 4 \).

Simplification processes both the numerator and denominator independently. In this example, the denominator \( a \) remains as \( \frac{1}{3} \), which is straightforward since it was already simplified by substitution.
Simplification clarifies the mathematical operations at play, ensuring we end up with a form like \( \frac{4}{\frac{1}{3}} \) that is ready for more operations.

Fraction Division

In mathematics, division of fractions often confuses students, but it becomes easier with the method of multiplying by the reciprocal. Our expression has reduced to the form \( \frac{4}{\frac{1}{3}} \) after substitution and simplification.
Division of fractions can be transformed into multiplication by flipping the fraction in the denominator. This process converts our division problem:

• Reciprocal of \( \frac{1}{3} \) is \( 3 \).
• Multiply the numerator by this reciprocal: \( 4 \times 3 \).

This results in \( 12 \), concluding that \( \frac{4}{\frac{1}{3}} = 12 \).
Understanding fraction division in terms of multiplication by the reciprocal can significantly ease the process, turning a potentially complex division into a simple multiplication task. This approach is not only useful in examinations but also in everyday applications of mathematics.