Using one variable vs two variables

Question

I'm in the process of of again learning algebra 1. But I ran across this problem the other day. The sum of two numbers is $10$ and the sum of their reciprocals is $5/12$. Find the numbers. Instinctively, I thought okay $x+x=10$ And when worked out $x=5$. Looked up the answer and quite different to mine. Did some research and found out the proper way to first solve for it should be $x+y=10$.

How is this so? I am confused. Beforehand thank you.

Answer 1

You (implicitly) assumed that the numbers were equal when when you wrote $x + x =10$ because $x = x$ for all numbers $x$. You should choose different letters for variables if you aren't sure they're the same.

Answer 2

$x$ is one variable. It can only hold one value at a time. So you can use it to represent only one unique number at a time.

The question talks about "two numbers". It never says the numbers are equal. They may or may not be equal. So we must use two different variables to represent them (say $x$ and $y$).

Just in case, the two numbers are equal (again, the question doesn't rule out that possibility), the solutions for $x$ and $y$ would be equal.

Here's an example of when we would use a single variable.

A number added to itself gives 10. Find the number.

Here, we are talking about a single number that is added to itself. So the equation would be $x + x = 10$.

Answer 3

You tried $5$, but the sum of the reciprocals $1/5+1/5 \ne 5/12$.

Hint: Did you try 3, 4? What are the factors of 12? These will help you figure out what $x$ and $y$ must be.

Answer:

If you choose $x=4$ and $y=6$ then $x+y = 4+6=10$ and $1/4+1/6=5/12.$