Problem 8 - Algebra and Trigonometry

Question

Determine whether the given value is a solution of the equation. \(\frac{1}{x}-\frac{1}{x-4}=1\) (a) \(x=2 \quad\) (b) \(x=4\)

Step-by-Step Solution

Answer

(a) x = 2 is a solution, (b) x = 4 is not a solution.

Step 1: Substitute x = 2 into the equation

First, substitute \(x = 2\) into the equation \(\frac{1}{x} - \frac{1}{x-4} = 1\). This gives us \(\frac{1}{2} - \frac{1}{2-4} = 1\). Simplify the terms to get \(\frac{1}{2} + \frac{1}{2} = 1\) which results in \(1 = 1\). Since both sides of the equation are equal, \(x = 2\) is a solution.

Step 2: Substitute x = 4 into the equation

Now, substitute \(x = 4\) into the equation \(\frac{1}{x} - \frac{1}{x-4} = 1\). This results in \(\frac{1}{4} - \frac{1}{4-4} = 1\). At this point, we encounter \(\frac{1}{0}\) which is undefined. Hence, \(x = 4\) is not a solution because it makes the equation undefined.

Key Concepts

Solution Verification

When we want to verify if a particular value is a solution to an equation, we substitute that value into the equation and simplify. If our substituted value makes both sides of the equation equal, then it is indeed a solution.To see how this works, let's take an example from our exercise: when we substitute \( x = 2 \) into the given equation \( \frac{1}{x} - \frac{1}{x-4} = 1 \), it simplifies to \( \frac{1}{2} - \frac{1}{2-4} = \frac{1}{2} + \frac{1}{2} = 1 \).

• Both sides of the equation equal 1, confirming \( x = 2 \) is a solution.
• This verification process is straightforward and crucial for checking potential solutions.

Remember that solution verification is always about substitution and simplification to see if the equation balances out.

Undefined Expression

In algebra, an undefined expression occurs when something in the equation cannot be calculated or doesn't exist within the realm of standard mathematics, like dividing by zero.Consider substituting \( x = 4 \) into our equation \( \frac{1}{x} - \frac{1}{x-4} = 1 \). Substitution leads to \( \frac{1}{4} - \frac{1}{0} \), and here the term \( \frac{1}{0} \) arises, which is undefined.

• Division by zero is undefined because there is no number that, when multiplied by 0, gives a result other than zero.
• Because of this, \( x = 4 \) cannot be a solution to the equation since it results in an undefined calculation.

Always be on the lookout for conditions that might lead to undefined expressions, especially in terms involving division.

Algebraic Substitution

Algebraic substitution is a fundamental technique where expressions or numbers are replaced within an equation to simplify or evaluate it.In our example, to check if certain values are solutions, we substitute these values into the equation. For instance:

• Substitute \( x = 2 \) into \( \frac{1}{x} - \frac{1}{x-4} = 1 \) to check if the equation holds true for this value.
• If when substituting a value it leads to a true statement, as it did here with \( 1 = 1 \), then the substitution confirms the value is a solution.

Understanding how to properly substitute and simplify is essential. It helps identify valid solutions and avoid misleading results, such as undefined expressions. Mastery in substitution techniques bolsters overall problem-solving abilities in algebra.