How did he simplify this problem like this?

Question

$$\left(\frac{x}{12}+\frac{x}{18}\right)t=x$$

$$\left(\frac{1}{12}+\frac{1}{18}\right)t=1$$

I want a solid rule, how did he simplify the problem in the image like this!

Answer 1

If $$\left(\frac{x}{12}+\frac{x}{18}\right)t=x\tag{1}$$ then in general it is not true that $$\left(\frac{1}{12}+\frac{1}{18}\right)t=1\tag{2}$$ Counterexample: if $t=36$ and $x=0$, then you see that $(1)$ is equivalent to $0=0$ which is true, but $(2)$ is equivalent to $5=1$ which is not.

However, if you already know that $x\neq0$, then the implication is true since in that case $$ \left(\frac{x}{12}+\frac{x}{18}\right)t=x $$ is equivalent to (this is just a factorization) $$ \left(\left(\frac{1}{12}+\frac{1}{18}\right)t\right)x=x $$ and you can divide both sides (which are equal, after all!) by $x$ (since $x\neq0$... otherwise division by $0$ doesn't make sense!) to maintain equality and infer $$ \frac{\left(\left(\frac{1}{12}+\frac{1}{18}\right)t\right)x}{x}=\frac{x}{x} $$ which is the same as $$ \left(\left(\frac{1}{12}+\frac{1}{18}\right)t\right)\frac{x}{x}=\frac{x}{x} $$ or, since $\frac{x}{x}=1$ always, $$ \left(\left(\frac{1}{12}+\frac{1}{18}\right)t\right)\cdot1=1 $$ that is $$ \left(\frac{1}{12}+\frac{1}{18}\right)t=1 $$

Of course, at a certain level of mathematical maturity, all of these intermediate steps are very rarely all written out since they should be obvious.

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Note that if you have $$ \left(\frac{1}{12}+\frac{1}{18}\right)t=1\tag{3} $$ then it must always be true that $$ \left(\frac{x}{12}+\frac{x}{18}\right)t=x $$ (just multiply both sides of $(3)$ by $x$... this will always be a valid operation, unlike division by an a priori arbitrary real number $x$ which is not valid if $x=0$).

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In fact, a well-known false proof that $2=1$ makes use of an invalid division by $0$.

Suppose that $x=y\neq0$. Then \begin{align} x^2&=xy\\ x^2-y^2&=xy-y^2\\ (x+y)(x-y)&=y(x-y)\\ x+y&=y\\ 2y&=y\\ 2&=1 \end{align} right? No! From $$ (x+y)(x-y)=y(x-y)\tag{4} $$ you can't infer that $$ x+y=y $$ because dividing both sides of $(4)$ by $(x-y)$ and simplifying doesn't make sense since $x-y=0$.