Problem 51 - Elementary Algebra Within Reach

Question

In Exercises 51-54, determine whether each value of \(x\) is a solution of the equation. \(\frac{2}{x}-\frac{1}{x}=1\) (a) \(x=0\) (b) \(x=\frac{1}{3}\)

Step-by-Step Solution

Answer

Neither \(x = 0\) nor \(x = \frac{1}{3}\) are solutions to the equation \(\frac{2}{x}-\frac{1}{x} = 1\).

Step 1 Substitute and simplify.

First we need to check if \(x = 0\) is a solution. But we note that substituting \(x = 0\) in any equation of the form \(1 / x\) leads to an undefined mathematical expression, hence \(x = 0\) is not a solution.

Step 2 Substitute and simplify second value

Now, check if \(x = \frac{1}{3}\) is a solution by substituting \(x = \frac{1}{3}\) into the equation. The left side of the equation becomes \(\frac{2}{\frac{1}{3}}-\frac{1}{\frac{1}{3}} = 2 * 3 - 3 = 6 - 3 = 3\). Therefore, substituting \(x = \frac{1}{3}\) into the equation does not result in \(1\), so \(x = \frac{1}{3}\) is not a solution.

Key Concepts

Understanding Rational Expressions

Rational expressions are expressions that involve a ratio of two polynomials. These expressions often appear like fractions, where the numerator and the denominator are both polynomials. For instance, in the equation \( \frac{2}{x} - \frac{1}{x} = 1 \), the terms \( \frac{2}{x} \) and \( \frac{1}{x} \) are rational expressions.

• Numerators and Denominators: In rational expressions, it’s crucial to identify the numerator and the denominator. Both should allow for simplification and solving problems if applicable.
• Simplification: Rational expressions can be simplified similarly to fractions, often reducing to simpler forms by factoring and simplifying the polynomials where possible.

Understanding rational expressions is necessary for appreciating the operations involved, such as addition, subtraction, and simplification. This knowledge also helps predict the behavior of the expression when substituting different values for variables. Thus, applying algebraic operations correctly is key to solving equations involving rational expressions.

Undefined Expressions in Mathematics

The concept of undefined expressions often arises in mathematics when dealing with rational expressions. An expression becomes undefined when the denominator equals zero, as division by zero is not possible.

• Why Undefined Occurs: Mathematically, any number or term divided by zero does not yield a finite number or even an infinite one, it simply lacks any meaning in the real number system.
• Example in Exercise: For example, checking if \(x = 0\) in the expression \(\frac{2}{x} - \frac{1}{x} = 1\) results in division by zero, making the expression undefined.

Students often encounter undefined expressions when working with rational expressions and need to recognize which values may not be suitable during substitution. Identifying potential undefined expressions prevents logical errors and misinterpretations in problem-solving.

Exploring the Substitution Method

The substitution method is a powerful tool in algebra for solving equations involving variables. By substituting specific values into equations, you can verify or find solutions efficiently.

• Substitution Process: Substitute the target value into the variable of the equation; simplify the equation to determine if the initial condition holds true.
• Application in Example: In our specific exercise, substituting \(x = \frac{1}{3}\) into the equation \(\frac{2}{x} - \frac{1}{x} = 1\) does not satisfy the equation since the left side evaluates to \(3\) instead of \(1\).

Using the substitution method, helps gain insights into what values satisfy or don't satisfy an equation, reinforcing foundational algebraic concepts. This method is frequently utilized when solving both linear and non-linear equations, providing clarity and precision effectively.