Any other function of x.
I.e. Let
f(x)=x3 and
g(x)=2x+1 so intersections will be given by the condition where
f(x)=g(x)Because you
can't equate them in their given forms, how would you do it in this case?? If you wish to make them equal to each other, you would need to rearrange both equations for either x or y before you can make them equal to each other. But even then, this is
still called substitution as per my example with f(x) and g(x).
So following from the way you'd been taught, you would need to rearrange
x+2y−13=0⇒y=−21x+213 and then equate this to
y=2x which as I said is still called substitution.
For simplicity and neater working however, it is much better to just substitute for one variable into the equation. This is particularly useful when it comes to higher order polynomials, logarithms, trigonometry, etc...
Please show me a good example where your method works out and I will perfectly point out at which stage you make a substitution just to make it clear.
There are various methods for obtaining a point of intersections; ie simultaneous equations, or substitution. It just so happens that substitution is the most common form of finding these intersections and you're allowed to substitute because the variables (in both equations) are equivalent at the points of intersections; therefore you are allowed to replace them.
Well you better get used to this as you do it throughout the entire A-Level maths and you need to understand WHY you do this, as I have explained above.