LM curve from money demand

The calculation is correct but the explanation is not simply 'equal the equations and solve for Y'.

Following Blanchard et al Macroeconomics: A European Perspective, 2nd ed. ch 5. by definition

LM curve represents all the combinations of output and interest rate for which the money market is in equilibrium, i.e. the demand for money equals the supply of money.

As given by your question the demand for money is given by $M/P$. Now in your question you do not explicitly label $a+bY−cr$ as a supply for money - but it looks like a supply for money. The reason why you equate the equations is because you want to equate supply and demand for money. If this would be given by some more complex relationship, for example you are given some extra equation for how $M$ or $P$ behaves, you would have to use that too and equating the two equations above would not be enough.

In addition, it is not necessarily to solve it for $Y$, you could as well solve it for $r$. In fact by convention in economics prices go on $y$-axis, so $r$ goes on $y$-axis and output $Y$ would go on $x$-axis, so if you would want to plot LM curve you would actually solve for $r$ not $Y$. This being said if you want to know what is effect of parameters and variables on output (which often is the case) it is not a mistake to solve it for $Y$ instead of $r$.