Interpreting expressions with multiple variables: Resistors | Modeling | Algebra II | Khan Academy

We're told an electronic circuit has two resistors with resistances r1 and r2 connected in parallel. The circuit's total resistance r sub t, or rt, is given by this formula:

Suppose we increase the value of r1 while keeping r2 constant. What does the value of r sub t do? Does it increase, decrease, or stay the same? So pause the video and see if you can answer this question.

All right, now let's work through this together. Some of you might be familiar with the idea of an electronic circuit and resistors and what they represent, but you really don't need to understand that in order to understand what's going on in this expression.

There's some quantity r sub t that's equal to one over, and then in the denominator we have one over r1 plus one over r2. So if we increase the value of r1 while keeping r2 constant, what happens? This is going to increase, and r2 is going to be constant.

So one way to think about it is we have two variables here, especially in this denominator, but really in this entire expression. If r2 is going to be constant, we really just have to focus our analysis on r1. If r2 is constant, that means it's just a number. It could be 2, it could be 5, it could be pi, whatever, but that is not going to change as we increase the value of r1.

So let's think about what's happening here. If r sub 1 increases, then what does that do to 1 over r1? Well, if you increase the denominator, then you are going to decrease the reciprocal of that. So that means that this whole thing right over here is going to decrease.

Now, if 1 over r1 is decreasing, what’s going to happen to 1 over r1 plus 1 over r2? Will this entire expression increase or decrease? Well, this part is staying constant. r2 is constant. So 1 over r2 is constant. Just imagine r2 could be 2 or 3, so this should just be one half or one third or whatever it is.

However, this part of the expression is going down. So if you're taking the sum of two things, one part's going down and the other part's constant, then that means this whole thing is going to be going down. So the entire denominator of this entire thing is going down.

Now, the entire denominator is going down. If 1 over r1 plus 1 over r2—if this whole thing is going down, what's going to happen to the reciprocal of that, 1 over (1 over r1 plus 1 over r2)? Well, if something is going down, the reciprocal of that is going to go up.

If you get smaller and smaller denominators, one over that is going to be a larger and larger value. So, the value of rt increases if r1 increases and r2 is constant. For those of you who know about resistance, which is really how well a current can flow through a circuit, that will also make intuitive sense. But you don't need to understand resistance to analyze this mathematically.