How to solve algebraic equation

Question

How can I solve an algebraic equation with only two variables. I have tried through graphical method but can't get it correct.Here it is: $$3x-y=2x+3y$$

Answer 1

You can simplify the expression: $$3x-y=2x+3y\implies 3y+y=3x-2x\implies 4y=x$$ Hence the solutoin of your equation are all the point of the graphic given by the $\color{red}{y=\frac x4}$

That is this: $\hspace{6cm}$

Answer 2

An equation of this form denotes the intersection between two planes in $\mathbb{R}^3$, namely the set

$$S = \{ (x,y) \mid 3x - y = 2x + 3y \}$$

In general, the intersection between two planes can be the empty set (if the planes are parallel), a line or a plan (if the two planes coincide).

By looking at their normal vectors, we immediately see that the planes $z = 3x - y$ and $z = 2x +3$ do not coincide and are not parallel.

It is somewhat difficult to solve it graphically, as finding $S$ involves drawing the planes $z = 3x - y$ and $z = 2x + 3y$.

A better approach is to note that if $3x - y = 2x + 3y$, then $x = 4y$ and therefore $y = \frac{1}{4}x$. This is the equation defining $S$.

Answer 3

There are two unknowns but a single equation, so that the problem is (once) undeterminate.

You can set one of the variables to an arbitrary value, and solve for the other.

For instance, choose $y$ as the independent variable and draw

$$x=4y.$$