Problem 52 - Elementary Algebra Within Reach

Question

In Exercises 51-54, determine whether each value of \(x\) is a solution of the equation. \(\frac{4}{x}+\frac{2}{x}=1\) (a) \(x=0\) (b) \(x=6\)

Step-by-Step Solution

Answer

The solution for the provided values are: (a) \(x = 0\) is not a solution to the equation. (b) \(x = 6\) is a solution to the equation.

Step 1: Substitution for x = 0

First replace \(x\) with 0 in the equation: \(\frac{4}{0} + \frac{2}{0} = 1\). Due to division by zero, this operation is undefined. So, \(x = 0\) is not a solution.

Step 2: Substitution for x = 6

Next, substitute \(x\) with 6: \(\frac{4}{6} + \frac{2}{6} = 1\). Simplifying this gives \(\frac{2}{3} + \frac{1}{3} = 1\), which is a true statement. Therefore, \(x = 6\) is a solution to the equation.

Key Concepts

Division by Zero

When solving rational equations, one of the most critical rules to remember is that division by zero is undefined. This means you cannot divide by zero in any mathematical operation because it does not result in a meaningful or definite number. Let's break down this concept further:

Attempting to divide any number by zero leads to a logical inconsistency, as there is no number that, when multiplied by zero, returns the initial number.
Understanding that division by zero produces an indeterminate form is crucial in mathematics.

In the context of rational equations, if you substitute a value of \(x\) that results in any denominator becoming zero, the solution is not valid. Always ensure to check the denominators to confirm that they are not zero before proceeding with calculations.

Solution Verification

After solving a rational equation by substituting a given value for \(x\), verifying that your solution is correct is essential. Verification ensures that all steps reflect the original equation's logic and that no undefined math operations occur.

Follow these steps to verify solutions effectively:

• Substitute the solution back into the original equation.
• Simplify both sides of the equation thoroughly, maintaining equality.
• Check that no division by zero occurs at any step during simplification.
• If both sides of the equation are equal after simplification, the substitution test is valid, confirming the solution.

In our example exercise, substituting \(x = 6\) back into the original equation resulted in a true statement \(\frac{2}{3} + \frac{1}{3} = 1\), confirming that \(x = 6\) is indeed a solution.

Substitution Method

The substitution method is a straightforward technique to determine if a proposed solution satisfies an equation. When working with rational equations, substitution involves replacing \(x\) with a specific value and verifying the resulting expression.

To use the substitution method:

• Take the value provided, such as \(x = 6\), and replace \(x\) in the equation with this number.
• Simplify the expression. For example, substituting \(x = 6\) into the equation gives \(\frac{4}{6} + \frac{2}{6}\).
• Continue simplifying until you determine whether the left-hand side equals the right-hand side of the equation.
• Identify any instances of division by zero during this process to avoid incorrect conclusions.

In our exercise, substituting \(x = 0\) was invalid due to division by zero, while \(x = 6\) confirmed a true equation. Utilizing substitution is a reliable approach to ensure values meet all criteria of the original rational equation.