Problem 7 - Introductory Algebra for College Students

Question

Evaluate each expression for \(x=4\). $$\frac{28}{x}$$

Step-by-Step Solution

Answer

The result of evaluating the expression \(\frac {28}{x}\) for \(x=4\) is 7.

Step 1: Substitute the Value of \(x\)

The first step to solve this exercise is to replace the variable \(x\) in the expression with the given value, which is 4. That gives us the expression \(\frac {28}{4}\).

Step 2: Divide

The final step is to perform the division operation. The number 28 divided by 4 equals 7.

Key Concepts

Substitution Method

The substitution method is a fundamental concept used to solve problems in mathematics, particularly when dealing with expressions that contain variables. In this context, the aim is to simplify the expression by replacing the variable with a given numerical value.

Let's start by understanding how this method works in our given exercise. We have the expression \( \frac{28}{x} \) with the variable \( x \). To evaluate it, we need to substitute the variable \( x \) with its given value, which is 4 in this case. So, when we apply this method, we rewrite the expression as \( \frac{28}{4} \).

This simple act of replacing variables with numbers can simplify expressions and is widely used not only in basic problems but also in complex algebraic equations.

Division

Once the variable has been substituted, the next step is division. Division is one of the four basic arithmetic operations. In simple terms, it involves determining how many times one number fits into another.

In our exercise, after substituting \( x \) with 4, the expression becomes \( \frac{28}{4} \). This requires us to perform a division operation. Here, 28 is the dividend, and 4 is the divisor. By executing the division, we find out that 28 divided by 4 gives us the quotient, which is 7.

In this specific problem, division transforms the expression into its simplest form, which helps in finding the numerical value of the expression after the substitution.

Variable Substitution

Variable substitution is a strategic method used in mathematics to make complicated expressions easier to work with. By substituting variables with specific numerical values, it is possible to evaluate algebraic expressions.

• Identify the variable in the expression, in our example, it is \( x \).
• Determine the value to substitute, here it is 4.
• Replace the variable in the expression to simplify it, resulting in \( \frac{28}{4} \).

This approach is not only limited to simple problems but is also applicable in solving complex equations and inequalities. By simplifying expressions through variable substitution, problems become more manageable, allowing for easier execution of subsequent operations, like division.