Problem 14 - Introductory and Intermediate Algebra for College Students 4th

Question

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

Step-by-Step Solution

Answer

The calculated value of the expression for \(x=4\) is 6

Step 1: Substitution

Substitute the value of \(x = 4\) into the equation. It will look like this:\n\(\frac{5*4 + 52}{3*4}\)

Step 2: Simplify the numerator and the denominator

Let's first simplify the numerator and denominator separately. Now, we get \( \frac{20 + 52}{12}\) which simplifies to \( \frac{72}{12}\).

Step 3: Simplify the fraction

Divide the numerator by the denominator to get the final result. So \( \frac{72}{12} = 6\)

Key Concepts

Substitution Method

The substitution method plays a vital role in evaluating algebraic expressions. It involves replacing the variables in an expression with their given numerical values. This process allows you to transform an algebraic statement into a purely numerical one, which can then be evaluated using the standard rules of arithmetic.

For example, with the expression \(\frac{5x+52}{3x}\), if we're given that \(x=4\), we apply the substitution method by replacing every \(x\) in the expression with 4. It's crucial to maintain the integrity of the expression by substituting the variable consistently. Applying this method systematically allows for a clear progression towards finding the value of the expression.

Simplifying Expressions

Simplifying expressions is a foundational skill in algebra. It involves reducing expressions to their simplest form by performing operations like addition, subtraction, multiplication, and division. When simplifying, it's important to follow the order of operations, which is remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right).

In the example \(\frac{5*4 + 52}{3*4}\), once you have substituted the variable with its value, the next step involves simplifying the expression. You start by simplifying the numerator (\(20+52\)) and the denominator (\(3*4\)) separately. Once the numerator is simplified to \(72\) and the denominator to \(12\), you are left with a much simpler fraction, \(\frac{72}{12}\), which can be easily evaluated.

Algebraic Fractions

Algebraic fractions are similar to numerical fractions, but they contain variables in the numerator, the denominator, or both. The principles of simplifying algebraic fractions are the same as those for numerical fractions: you need to reduce the fraction to its simplest form. This typically involves simplifying the numerator and denominator separately and then dividing them if possible.

Continuing with our example, after having simplified the numerator and denominator separately, we are left with the fraction \(\frac{72}{12}\). To simplify, we divide 72 by 12, which equals 6. Simplifying algebraic fractions often involves factoring, finding common denominators, and canceling out terms, but in this case, since both the numerator and the denominator are numerical after the substitution, the fraction simplifies just like a regular numerical fraction.