Problem 8 - Introductory Algebra for College Students

Question

Evaluate each expression for \(x=4\). $$\frac{36}{x}$$

Step-by-Step Solution

Answer

The solution to the given expression for \(x=4\) is 9.

Step 1: Substitute

Replace the x in the expression with the given value. That gives us: \(\frac{36}{4}\)

Step 2: Simplify the Expression

Now that we have replaced x in the expression, we can simplify the expression by performing the division operation, i.e. 36 divided by 4.

Key Concepts

Evaluating Expressions

The process of evaluating expressions involves calculating the value of an expression when numbers are substituted for variables. This helps in understanding the behavior of functions or expressions for different values. To evaluate an expression, such as \( \frac{36}{x} \) for \( x=4 \), we seek to find out what the expression amounts to when \( x \) takes on the specified value.

The evaluation process entails a straightforward substitution, replacing the variable with its given numerical counterpart, and then executing the necessary arithmetic operations. In our example, once \( x \) is replaced with 4, the expression simplifies to \( \frac{36}{4} \) which equals 9. This operation shows us that when \( x \) is valued at 4, the expression's output is 9.

Substitution Method

The substitution method is a fundamental algebraic tool for evaluating expressions. It involves replacing variables in an equation or expression with their corresponding values. This is a necessary first step in simplifying expressions and solving equations.

To use this method effectively, it is crucial to replace the variable accurately and ensure that every instance of that variable is substituted. In the given exercise, the first step correctly substitutes the value of 4 in place of \( x \), which led us to the expression \( \frac{36}{4} \). The substitution method lays the foundation for finding the final solution by contextualizing the abstract expression with a specific number.

Algebraic Division

When it comes to algebraic division, it's essential to understand that it's simply the process of dividing terms containing variables and/or numbers. Although it can sometimes involve more complex manipulations, at its core, it is comparable to basic numerical division.

In our previous example, after substituting \( x \) with 4, we arrive at a division operation, 36 divided by 4. Crucially, in algebraic division, like any division, we must carefully consider the divisor; because division by zero is undefined, ensuring the divisor is a valid number is vital to simplifying the expression correctly. Once the division is completed, the simplified result reflects the value of the original algebraic expression for the given input.