Question

multiple choice
question 3
lin solved the equation $5(5 + n) = 4n - 5 - 9n$.
lin got $25 + 5n = -5n - 5$ and said there was no solution.
do you agree with lin?
a yes, the variables are the same.
b yes, the variables are not the same.
c no, the variables are not the same.

Explanation:

Step1: Simplify the equation

Start with \(25 + 5n=-5n - 5\). Add \(5n\) to both sides: \(25+5n + 5n=-5n-5 + 5n\), which simplifies to \(25 + 10n=-5\).

Step2: Solve for n

Subtract 25 from both sides: \(10n=-5 - 25\), so \(10n=-30\). Divide both sides by 10: \(n=\frac{-30}{10}=-3\). Since we can find a solution (\(n = - 3\)), Lin is wrong. The variables are the same (both have \(n\)), but we can still solve for \(n\). So we don't agree with Lin, and option A and B are wrong. Option C's reasoning is incorrect about variables not being the same (they are), but the conclusion "No" is correct? Wait, no, let's re - evaluate. Wait, the equation after Lin's step is \(25 + 5n=-5n - 5\). Let's solve it properly.
Add \(5n\) to both sides: \(25+10n=-5\). Subtract 25: \(10n=-30\), \(n = - 3\). So there is a solution. Lin said no solution, so we don't agree with Lin. Now look at the options:

• Option A: Says yes (agree with Lin) and reason variables are same. But we can solve it, so A is wrong.
• Option B: Says yes (agree with Lin) and reason variables not same. Variables are same, so B is wrong.
• Option C: Says no (disagree with Lin) but reason variables not same (which is wrong, variables are same). Wait, maybe the question's options have a mistake, but among the options, the correct conclusion is "No" (disagree with Lin). The reasoning in C is wrong, but the other options are more wrong. Wait, no, let's check again. Wait, the original equation: \(5(5 + n)=4n-5 - 9n\). Simplify right side: \(4n-9n=-5n\), so right side is \(-5n - 5\). Left side: \(25 + 5n\). So equation is \(25 + 5n=-5n - 5\). We can solve for \(n\) as shown. So Lin is wrong (there is a solution). So we don't agree with Lin. Now, the options:

Option A: Yes (agree) - wrong.
Option B: Yes (agree) - wrong.
Option C: No (disagree) - even though the reason is wrong (variables are same), it's the only option with "No". So we choose C.

Answer:

C. No, the variables are not the same.