Why do we need two equations to find two unknown quantities?

Any equation involving two variables gives one variable in terms of the other. For example, consider the equation

y+2x=1.

In this case we can rearrange to find y in terms of x, i.e.

y=1-2x.

So that if we know the value of x, we can calculate the value of y.

Alternatively, we could rearrange our original equation to give x in terms of y, instead:

x=(1-y)/2.

Now if we know the value of y we can calculate the value of x.

In this way we can find the value of one variable if we know the value of the other, and we can form lots of pairs of values which all satisfy the equation. So there isn't one particular answer, i.e. one particular pair of x,y values which satisfies the equation, since lots of different pairs will satisfy it.

To reduce the number of possible pairs to one, so to get a single unique pair of values for x and y, we need a further constraint on them, in other words another equation that they must also satisfy.

Then, provided this second equation doesn't contradict the first one, or just say the same thing in a different way, we will be able to find a unique pair of values for x and y.

Here's an example of the second equation contradicting the first:

First equation: x+y=2

Second equation: x+y=3.

Clearly there is no way that both these equations can be true!

Here's an example where the second equation says the same thing as the first in a different way:

First equation: x+y=2

Second equation : 2x + 2y =4.

If we can get the first equation by doing something to the second (here it's dividing it by 2) then we are getting no new information about x and y from the second equation, so we still can't find a unique pair of values for x and y.