Problem 53 - Algebra for College Students

Question

Solve each equation. $$\frac{5}{x-3}=\frac{x}{x-3}$$

Step-by-Step Solution

Answer

x = 5

Step 1: Analyze the Equation

Given the equation \(\frac{5}{x-3}=\frac{x}{x-3}\), observe that both fractions have the same denominator \(x-3\).

Step 2: Remove the Denominators

Because the denominators are the same, we can equate the numerators directly: \(\frac{5}{x-3} = \frac{x}{x-3} \rightarrow 5 = x\).

Step 3: Solve for x

From the previous step, we see that \(x = 5\).

Step 4: Verify the Solution

Substitute \x = 5\ back into the original equation to check for correctness: \(\frac{5}{5-3} = \frac{5}{5-3} \rightarrow \frac{5}{2} = \frac{5}{2}\), which is true.

Key Concepts

Equating Numerators

When solving rational equations, one effective method is to equate the numerators when the denominators are the same. For instance, consider the equation \(\frac{5}{x-3}=\frac{x}{x-3}\). Both sides have the denominator \(x-3\). This allows us to remove the denominators temporarily and directly set the numerators equal to each other: \(5 = x\). This simplifies our equation and makes solving it easier.

Here are some steps to remember when equating numerators:

• Ensure that the denominators are exactly the same.
• Set the numerators equal to each other and solve the resulting equation.
• Always remember to verify your solution to make sure it doesn't make the original denominator zero.

Verifying Solutions

After finding a potential solution for the equation, it's important to verify that it truly satisfies the original equation. For our example, we found \(x = 5\). Substituting \(x = 5\) back into the original equation, \(\frac{5}{x-3} = \frac{x}{x-3}\), we get:

\(\frac{5}{5-3} = \frac{5}{5-3}\)

Since the left-hand side and the right-hand side of the equation are equal (\(\frac{5}{2} = \frac{5}{2}\)), our solution \(x = 5\) is verified.

Verification steps:

• Substitute the solution back into the original equation.
• Simplify both sides to ensure they are equal.
• If they match, your solution is correct. If not, reevaluate your steps.

Removing Denominators

In some rational equations, you can simplify the solving process by directly removing denominators. This method is particularly useful when the denominators on both sides of the equation are the same.

For the equation \(\frac{5}{x-3} = \frac{x}{x-3}\), notice that the denominators are \(x-3\). This similarity allows us to remove the denominators temporarily, leading to the simpler equation \(5 = x\).

Steps to remove denominators:

• Check if the denominators on both sides of the equation are identical.
• Once confirmed, set the numerators equal to each other.
• Solve the resulting simpler equation without the denominators.
  
• Remember to check if your solution introduces any restrictions related to the denominator.

By following these steps, you solve the equation more quickly and efficiently. Removing denominators significantly reduces the complexity of rational equations.