Problem 105 - Introductory Algebra for College Students

Question

Then rewrite the right side of the equation to correct the error that now exists. $$\frac{1}{x}+\frac{2}{5}=\frac{3}{x+5}$$

Step-by-Step Solution

Answer

The error correction leads to the equation \(\frac{1}{x} + \frac{2}{5} = \frac{3}{5x}\), which simplifies and yields the solution \(x = -1\).

Step 1: Correct The Error

The mistake lies in the right side of the equation i.e., \(\frac{3}{x+5}\). It should be the sum of the fractions on the left side - which means the denominator should be 'x' multiplied with 5, not 'x' plus 5. So, the correct equation then becomes: \(\frac{1}{x}+ \frac{2}{5} = \frac{3}{5x}\).

Step 2: Solve The Equation

First, let's find a common denominator, which is 5x in this case. Multiply each term with '5x' on both sides to clear the denominators. This will simplify the equation to: \(5 + 2x = 3\).

Step 3: Simplify Further And Solve For x

From \(5 + 2x = 3\), subtract '5' from both sides to get '2x = -2'. Then, divide each side by '2' to solve for 'x'. Thus, we get \(x = -1\).

Key Concepts

Understanding Fraction Addition

When it comes to introductory algebra, fraction addition is a fundamental concept. It involves adding the values of two or more fractions together. Fractions are parts of a whole, and they provide a way to represent numbers in a different form. The key point in adding fractions is that they must have a common denominator. This means that the number at the bottom of each fraction must be the same. For example, in order to add \(\frac{1}{x}\) and \(\frac{2}{5}\), they need to be rewritten with a common denominator. In solving algebra equations, this ensures that each fraction part is comparable and the operation can be carried out correctly.

The process involves:

• Identifying the denominators of each fraction.
• Finding a common number that both denominators divide evenly into, called the least common denominator (LCD).
• Adjusting each fraction so they all share this common denominator before performing the addition.

This step is essential for error-free fraction addition and accurately finding the result of two or more fractions put together.

Finding the Common Denominator

A crucial part of algebraic operations with fractions is finding a common denominator. This concept is critical in ensuring that fractions involved in addition, subtraction, or any equation work is consistent. The common denominator is essentially the least common multiple (LCM) of all the denominators in the problem.

In our exercise, the denominators \(x\) and \(5\) need to be combined to find a least common denominator, which would be \(5x\). This ensures both fractions are compatible for combination. The steps to find the common denominator are as follows:

• List the multiples of each denominator.
• Identify the smallest multiple that is common to both lists, which becomes the common denominator.
• Adjust each original fraction based on the common denominator by multiplying the numerator and the denominator of each fraction by the appropriate factor.

By achieving this, you harmonize the fractions, making it possible to accurately solve fractional equations by doing calculations on equal footing.

Steps to Solving Linear Equations

Solving linear equations involves finding the value of the variable that makes the equation true. In introductory algebra, this usually means working with a single variable, as in the equation \(5 + 2x = 3\). The following general steps outline how to solve such equations:

• Simplify Each Side: Combine any like terms on either side of the equation to make it easier to manipulate.


• Isolate the Variable: Use addition, subtraction, multiplication, or division to get the variable on one side of the equation and everything else on the other.


• Perform Operations: In our case, you subtract \(5\) from both sides to isolate terms involving \(x\). Then, divide both sides by the coefficient of \(x\) (here the coefficient is 2) to solve for \(x = -1\).

Each step requires careful operation to maintain the equation's balance, ensuring each side remains equal as transformations are made. Clearing fractions, combining like terms, and following these structured methods provide a clear path to finding the solution.