Problem 39 - College Algebra and Calculus: An Applied Approach

Question

Solve the equation and check your solution. (Some equations have no solution.) $$ 10-\frac{13}{x}=4+\frac{5}{x} $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 3\)

Step 1: Get rid of fractions

Multiply each side by \(x\) to remove fractions: \[10x - 13 = 4x + 5\]

Step 2: Rearrange the equation

Rearrange the terms to gather all the terms on one side of the equation: \[10x - 4x = 13 + 5\]

Step 3: Simplify and solve

Simplify further to solve for \(x\): \[6x = 18\] Therefore, upon simplifying, we get: \(x = 3\)

Step 4: Check your solution

To confirm if \(x=3\) is the correct solution, substitute \(x=3\) into the original equation and see if both sides are equal: \[10 - \frac{13}{3} = 4 + \frac{5}{3}\] LHS = \(10 - \frac{13}{3} = \frac{30-13}{3} = \frac{17}{3}\); RHS = \(4 + \frac{5}{3} = \frac{12+5}{3} = \frac{17}{3}\) Since both sides are equal, the solution is correct.

Key Concepts

Checking Solutions

Once you've found a potential solution to an equation, it's important to verify if it satisfies the original equation. This step is called "checking the solution".
Checking ensures that the calculated value indeed works as a solution. Here's how to do it effectively:

• Substitute the potential solution back into the original equation.
• Calculate both sides of the equation separately.
• Confirm if both sides are equal. If they are equal, your solution is correct.
• If not equal, recheck your work for errors.

In our exercise, we calculated that \(x = 3\). By substituting \(x = 3\) back into the equation \(10-\frac{13}{x} = 4+\frac{5}{x}\), and confirming both sides as \(\frac{17}{3}\), we assured that \(x=3\) is indeed a valid solution.

Fractions in Equations

Equations often include fractions, which can make them look complicated at first. But don't worry, there's a straightforward method to handle them!
The first step is to eliminate fractions to simplify the equation, making it easier to solve. We can do this by multiplying every term by the least common denominator (LCD).

• In the exercise, the equation \(10-\frac{13}{x} = 4+\frac{5}{x}\) involves fractions \(\frac{13}{x}\) and \(\frac{5}{x}\).
• To eliminate these fractions, we multiply the entire equation by \(x\), because it's the common denominator.
• This results in a simpler equation: \(10x - 13 = 4x + 5\).

Creating an equation without fractions allows you to proceed more easily to algebraic manipulation and solving.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying equations to isolate the variable you want to solve for.
This manipulation involves moving terms from one side of the equation to the other. Here's how we approached it in our solution:

• We started with the equation: \(10x - 13 = 4x + 5\).
• The goal is to have all terms containing \(x\) on one side and constants on the other.
• Subtract \(4x\) from both sides, yielding \(6x = 18\).
• Finally, divide both sides by 6 to solve for \(x\): \(x = 3\).

By following these steps, algebraic manipulation helps in systematically arriving at the solution, making sure that the variable is isolated and solved accurately.