91Ó°ÊÓ

When solving equations involving fractions, it's important to simplify them as much as possible. Here, you might have noticed that the fractions on both sides of the equation share the same denominator, which simplifies things greatly. Always look for common denominators, as this allows you to combine the fractions into one.

In this exercise, we start with the equation: \( \frac{k}{k-4} - 5 = \frac{4}{k-4} \).

Since both fractions share the denominator \(k-4\), we can combine them by putting them under a single fraction, making the equation easier to manage:
\[ \frac{k - 5(k-4)}{k-4} \]\[ = \frac{-4k + 20}{k-4} = \frac{4}{k-4} \].

Look for opportunities like this in your problems to simplify and combine fractions whenever possible.

After combining and simplifying fractions so that they appear over a common denominator, observe that both sides of the equation have the same denominator. This step simplifies the problem further, as you can now ignore the common denominator and focus on the numerators.

In this problem, we took:
\[ \frac{-4k + 20}{k-4} = \frac{4}{k-4} \].

You can safely set the numerators equal to each other since the denominators are the same:
\[ -4k + 20 = 4 \].

This allows you to solve the equation just like any basic algebraic equation, which helps in focusing entirely on the numerical values without the fraction components.

After isolating the variable and solving for it, always substitute the value back into the original equation to check if it results in a valid solution. Sometimes, what appears to be a solution might actually create a situation that is undefined, such as a division by zero.

In our example, after solving for \( k = 4 \), substituting it back into the original equation results in:
\[ \frac{4}{4-4} - 5 = \frac{4}{4-4} \].

This creates a division by zero, which is undefined in mathematics. Therefore, \( k = 4 \) is an invalid solution, and this specific equation has no valid solutions.

Always make it a habit to verify your solutions to catch such invalid or extraneous solutions.