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Chapter 7: Problem 19

Solve. If no solution exists, state this. $$\frac{n+2}{n-6}=\frac{1}{2}$$

Short Answer

Expert verified
The solution is \( n = -10 \).

Step by step solution

01

Isolate the variable

Start by cross-multiplying to get rid of the fraction. Multiply both sides of the equation by \(2(n-6)\). This will eliminate the denominators.
02

Simplify the equation

After cross-multiplying, simplify the equation: \[ 2(n + 2) = 1(n - 6) \] which simplifies to \[ 2n + 4 = n - 6. \]
03

Solve for the variable

Move all terms involving \(n\) to one side and constant terms to the other side: \[ 2n - n = -6 - 4 \] simplifies to \[ n = -10. \]
04

Verify the solution

Plug the value of \(n\) back into the original equation to verify: \[ \frac{-10+2}{-10-6} = \frac{-8}{-16} = \frac{1}{2} \] which confirms the solution is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cross-Multiplication
When faced with a rational equation, cross-multiplication helps eliminate fractions.
For the given problem, \( \frac{n+2}{n-6} = \frac{1}{2} \), cross-multiplication involves multiplying both sides by the denominators of each fraction to make them disappear.
This creates a new equation without fractions, making it much easier to solve. For this exercise, we multiply both sides by \( 2(n - 6) \), leading us to: \[ 2(n + 2) = 1(n - 6). \] Cross-multiplication is your first step in solving rational equations.
Isolating Variables
Now that we've cross-multiplied, we can focus on isolating the variable, which in this case is \(n\).
To do this, we need to simplify the equation: \[ 2(n + 2) = n - 6. \] First, distribute the \(2\) across the \(n + 2\):
\[ 2n + 4 = n - 6. \] Next, we want all terms with \(n\) on one side and constant terms on the other side:
\[ 2n - n = -6 - 4. \] This equation simplifies further to: \[ n = -10. \] Achieving this balance is crucial 鈥� you need to move all variable terms to one side and constants to the other.
Simplification
Simplification helps break down complex equations into easier ones.
After cross-multiplication and isolating the variable, simplifying the equation is key to finding the value of \(n\).
Let's take our equation after distributing: \[ 2n + 4 = n - 6. \] We simplify it by moving all terms involving \(n\) to one side and constants to the other:
\[ 2n - n = -6 - 4. \] Simplifying both sides, you get \( n = -10 \).
Using clear, methodical steps in simplification avoids confusion and helps accurately solve for the variable.
Variable Verification
Once you've found a solution, it's vital to verify that it satisfies the original equation.
This step confirms your solution is correct, ensuring no calculation mistakes were made.
For our solution \( n = -10 \), substitute it back into the original equation: \[ \frac{-10 + 2}{-10 - 6} = \frac{-8}{-16} = \frac{1}{2}. \] Clearly, \[ \frac{1}{2} = \frac{1}{2} \] confirms our solution.
Always substitute your variable back into the initial equation to verify 鈥� this locks in your answer as accurate and true.

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