Problem 87 - Elementary Algebra

Question

For exercises 87-90, the completed problem has one mistake. (a) Describe the mistake in words, or copy down the whole problem and highlight or circle the mistake. (b) Do the problem correctly. Problem: Solve: \(\frac{11}{x}+\frac{13}{12}=1\) Incorrect Answer: Least common denominator is \(12 x\). $$ \begin{aligned} 12 x\left(\frac{11}{x}+\frac{13}{12}\right) &=1 \\ 12 x\left(\frac{11}{x}\right)+12 x\left(\frac{13}{12}\right) &=1 \\ 132+13 x &=1 \\ \frac{-132}{0+13 x} &=-131 \\ \frac{13 x}{13} &=\frac{-131}{13} \\ x &=-\frac{131}{13} \end{aligned} $$

Step-by-Step Solution

Answer

The error is in the distribution. The correct answer is \(x = -132\).

Step 1: Identifying the Mistake

There's an error in handling the equation after multiplying both sides by the least common denominator. The incorrect approach incorrectly distributes the denominator.

Step 1: Find the Least Common Denominator

Find the least common denominator (LCD) of the fractions. Here, the LCD for \(\frac{11}{x} + \frac{13}{12} = 1\) is \(12 x \).

Step 2: Clear the Fractions

Multiply every term in the equation by the LCD \(12 x \). \[\begin{aligned} 12 x \times \frac{11}{x} + 12 x \times \frac{13}{12} &= 12 x \times 1 \ \]

Step 3: Simplify

Simplify each term: \[\begin{aligned} 12 \times 11 + x \times 13 &= 12 x \ 132 + 13x &= 12x \]

Step 4: Isolate the Variable

Rearrange the equation to isolate the variable \(x\): \[ 132 + 13x = 12x \ 132 = 12x - 13x \ 132 = -x \ x = -132 \]

Step 5: Verify Solution

Substitute \(x = -132\) back into the original equation to ensure it satisfies the equation. \[\frac{11}{-132} + \frac{13}{12} \] should equal \(1\). The calculated values will demonstrate that this solution is correct.

Key Concepts

Least Common Denominator

When solving rational equations, it's crucial to find the Least Common Denominator (LCD). This helps in combining fractions by giving them a common basis. In the equation \( \frac{11}{x} + \frac{13}{12} = 1 \), the LCD is \ 12x \ because it's the smallest expression that both denominators, \ x \ and \ 12 \, can divide into without leaving a remainder. Once you identify the LCD, you can rewrite each fraction using this common denominator to simplify the equation.

Clearing Fractions

Clearing fractions means getting rid of the denominators by multiplying every term by the LCD. This makes the equation easier to work with. In our problem, multiply every term by \( 12x \):

\[ 12x \times \frac{11}{x} + 12x \times \frac{13}{12} = 12x \times 1 \]

Simplifying, we get:

\[ 12 \times 11 + x \times 13 = 12x \]

This step ensures that each term is free of fractions and we can now handle a simpler linear equation.

Isolating Variables

Once the equation is free of fractions, isolate the variable to solve for it. In our simplified equation \[ 132 + 13x = 12x \], move all terms involving \( x \) to one side and constants to the other:

\[ 132 + 13x - 12x = 0 \]

This further simplifies to:

\[ 132 = -x \]

Finally, solving for \( x \) yields:

\[ x = -132 \]

Always remember to perform inverse operations to isolate the variable.

Verifying Solutions

It's always important to verify your solution by plugging it back into the original equation. Substitute \( x = -132 \) into the original equation:

\[ \frac{11}{-132} + \frac{13}{12} = 1 \]

Calculate each term separately:

\[ \frac{11}{-132} = -\frac{11}{132} \]

\[ \frac{13}{12} \] remains as it is.

Add these together:

\[ -\frac{11}{132} + \frac{13}{12} \]

Convert to a common denominator of \ 132 \:

\[ -\frac{11}{132} + \frac{143}{132} = \frac{132}{132} = 1 \]

This confirms that our solution is correct and the equation holds true.