# Eliminating the variable when solving equations leads to $\emptyset$

I came across this question while answering an intermediate algebra textbook:

A student tried to solve the equation $8x = 7x$ by dividing both sides by x, obtaining $8 = 7$. He gave the solution set to $\emptyset$. Why is this incorrect?

As we know it the correct answer is x = $0$. Can you give any possible explanations to the student why is this so?

• Anything can happen when you divide by zero. So don't divide by zero. – Jacky Chong Oct 1 '16 at 4:49
• Dividing by "$x$" is only valid if $x$ is not zero. – Math1000 Oct 1 '16 at 4:50
• As others have pointed out, the division by $x$ is unjustified. We can still subtract $7x$ from both sides of the equation though. This directly shows $x=0$. – Glare Oct 1 '16 at 4:52
• You can post you answers as answers instead of comments. – Ralf Rafael Frix Oct 1 '16 at 5:28
• $Ax=Bx\iff [(x\ne 0 \land A=B)\lor (x=0)].$ If $A\ne B$ then $[(x\ne 0\land A=B)\lor (x=0)] \iff x=0.$ – DanielWainfleet Oct 1 '16 at 20:34

We can solve the equation $$8x = 7x$$ by subtracting $7x$ from each side of the equation to obtain $$x = 0$$ Substituting $x = 0$ into the original equation reveals that it is a valid solution, so the solution set is $S = \{0\}$, which is not empty.
Division by zero is undefined. When you divide by a variable, you are implicitly assuming that the variable does not equal zero. That is not a valid assumption in this problem since zero is the only real number that satisfies the equation $8x = 7x$.
In my recent reply it is shown $x=0,$ and division by $0$ is not allowed, and if done we can see the differences between big quantities compared to $\infty$ are not sensitive and would be wrong in comparision..