Problem 31 - Algebra 1: Concepts and Skills

Question

Solve the equation by multiplying each side by the least common denominator. Check your solutions. \(\frac{5}{x+1}-\frac{7}{x+1}=\frac{12}{x}\)

Step-by-Step Solution

Answer

The solution to the equation is \(x = -\frac{6}{7}\).

Step 1: Determine the least common denominator

The denominators present in the equation are \(x+1\) and \(x\). The least common denominator (LCD) is \(x(x+1)\).

Step 2: Multiply both sides by the LCD

Multiply every term in the equation by the LCD. The equation becomes:\((x(x+1)) \times (\frac{5}{x+1}) - (x(x+1)) \times (\frac{7}{x+1}) = (x(x+1)) \times (\frac{12}{x})\). This simplifies to:\(5x - 7x = 12(x+1)\).

Step 3: Simplify and solve the equation

Simplify the equation you get \( -2x = 12x+12\). Rearranging terms, you get \( 14x = -12\) and solving for \(x\), you get \( x = -\frac{12}{14} = -\frac{6}{7}\).

Step 4: Check the solution

Substitute \(x = -\frac{6}{7}\) into the original equation and check if the left side equals to the right side to verify the solution. The original equation holds true. So \(x = - \frac{6}{7}\) is indeed a solution.

Key Concepts

Least Common Denominator

When solving equations that involve fractions, finding the least common denominator (LCD) is a crucial step. The LCD is essentially the smallest multiple that all denominators in the equation have in common. This allows you to eliminate the fractions by transforming each term into a whole number.
Here's how to find the LCD:

• Identify the denominators in the equation. In our case, they are \(x+1\) and \(x\).
• The LCD must include each unique factor of these denominators. For \(x+1\) and \(x\), the LCD is simply their product: \(x(x+1)\).

By multiplying each side of the equation by the LCD, you remove the fraction and can manipulate the equation more easily. This method simplifies the equation greatly, making it feasible to apply further algebraic techniques.

Multiplying Equations by LCD

Once the least common denominator is established, you multiply each term of the equation by it. This step is done to clear the equation of fractions, allowing us to work with whole numbers instead.
Consider the given equation: \(\frac{5}{x+1}-\frac{7}{x+1}=\frac{12}{x}\). After identifying \(x(x+1)\) as the LCD, you multiply every term by it:

• \((x(x+1)) \times \frac{5}{x+1} = 5x\)
• \((x(x+1)) \times \frac{7}{x+1} = 7x\)
• \((x(x+1)) \times \frac{12}{x} = 12(x+1)\)

For each term, the denominators cancel out due to the multiplication, transforming the equation into a simpler form: \(5x - 7x = 12(x + 1)\). Now, you are dealing with a straightforward linear equation without any fractions.

Verifying Solutions

After solving an equation, it's important to verify the solution to ensure its accuracy. This process involves substituting the solution back into the original equation to check its validity.
In our example, the solution we found was \(x = -\frac{6}{7}\). To verify:

• Substitute \(x = -\frac{6}{7}\) into the original equation, \(\frac{5}{x+1}-\frac{7}{x+1}=\frac{12}{x}\).
• Calculate the left side: \(\frac{5}{-\frac{6}{7}+1} - \frac{7}{-\frac{6}{7}+1}\).
• Calculate the right side: \(\frac{12}{-\frac{6}{7}}\).
• Confirm equality on both sides to verify the solution is correct.

If both sides of the equation are equal after substitution, the solution is indeed correct. This confirmation process is essential in mathematics to ensure that no errors have been made during the computation.