Problem 14 - Introductory Algebra for College Students

Question

Evaluate each expression for \(x=4\). $$\frac{5 x+52}{3 x}$$

Step-by-Step Solution

Answer

The value of the expression \(\frac{5x+52}{3x}\) for \(x=4\) is \(6\).

Step 1: Substitute \(x\) in the expression

Firstly, replace all instances of variable \(x\) in the equation \(\frac{5x+52}{3x}\) with the value \(4\). The equation thus becomes \(\frac{5*4+52}{3*4}\).

Step 2: Simplify the numerator

After substituting \(4\) for \(x\), simplify the numerator i.e. \(5*4+52\), which equals to \(20+52=72\). The equation now becomes \(\frac{72}{3*4}\).

Step 3: Simplify the denominator

Simplify the denominator i.e. \(3*4\), which equals \(12\). Now, the equation becomes \(\frac{72}{12}\).

Step 4: Simplify the fraction

Perform the division in the fraction i.e. \(72\div 12\), which equals \(6\).

Key Concepts

Substitution

When solving algebraic expressions by evaluating them at a specific value, substitution is the first and crucial step. In this exercise, we start by substituting the given value of the variable into the expression. Here, we're given that \(x = 4\) and we need to substitute this into the expression \(\frac{5x + 52}{3x}\).
Try to imagine substitution as a simple replacement process: wherever you see an \(x\), you replace it with \(4\). So the expression transforms to \(\frac{5 \times 4 + 52}{3 \times 4}\), making it easier to handle numerically.
Substitution simplifies expressions by turning variables into numbers, hence removing any abstractness. This step usually leads directly into simplifying the parts of the fraction, beginning with the numerator.

Numerator Simplification

The numerator is the top part of a fraction, which often requires simplification to understand the fraction better. After substitution, our numerator becomes \(5 \times 4 + 52\). Numerical simplification involves performing arithmetic operations like multiplication and addition. First, multiply \(5\) by \(4\), which gives \(20\). Then, add \(52\) to \(20\) to get a total of \(72\).
So, the simplified numerator is \(72\). Simplification is crucial because it reduces complexity, allowing for easier handling of the expression in subsequent steps, such as simplifying the denominator.

Denominator Simplification

Just like with the numerator, the denominator of a fraction needs to be simplified for ease of division. In this expression, the denominator is \(3 \times x\) after substitution, which for our equation becomes \(3 \times 4\). The simplification process here involves simple multiplication: multiplying \(3\) by \(4\) gives us \(12\). Now, the fraction reduces to \(\frac{72}{12}\).
This simplification step is vital as it prepares the fraction for the final simplification. By reducing complex expressions to more straightforward terms, it sets us up for a successful simplification of the entire fraction.

Fraction Simplification

Once both the numerator and denominator have been simplified, the final step is to simplify the fraction itself. You execute this by performing the division within the fraction.
For our expression, we now have \(\frac{72}{12}\). Simply divide the numerator by the denominator: \(72 \div 12\). The result of this operation is \(6\).
Simplifying fractions enables you to arrive at the most concise and manageable form of an expression, which is often required in math problems. This step concludes the process, giving us a clear, simplified, and easily understandable result from the original algebraic expression. Choosing the simplest representation helps in clear communication and better understanding of mathematical results.