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Obtaining a function from an equation

Video transcript
For a given input value b, the function f outputs a value a to satisfy the following equation 4a plus 7b is equal to negative 52. So for a given input b, the function f, the function f will output an a that satisfies this relationship right over here for the a and the b. Write a formula for f of b in terms of b. So we want to do, we just want to solve for -- if we're given a "b", what "a" does that imply that we have to output? Or another way to think about it is -- let's just solve for a, or we could think about a as being a function of b. So let's write this. So we have 4a plus 7b -- is equal to negative 52. So I can solve for a in terms of b, that any b that I have -- Let's say these b's are on the right hand side I can put it in. I can substitute that value for b and I can just solve for a. I can solve for a that needs to be outputted. So let's do that. Let's solve for a. So I want to get all the a on I wanna just have an a leftover on the left hand side, and have everything else on the right hand side including the b's. So let's get rid of this b on the left hand side. And I can do that by subtracting 7b. Of course I wanna do that to both sides. I can't just do an equation and do an operation only on one side like that. So let's subtract, and we are left with we are left with -- the 7b's add up to zero. 7b minus 7b. We're left with 4a is equal to negative 52 minus 7b, minus 7b. Now, to isolate the a here, just to have an a here instead of 4a, we can divide both sides by 4. We can divide both sides by 4. So I'm gonna divide everything by 4. And on the left hand side, we got our goal. We are left with an a is equal to -- Now what's negative 54 divided by -- What is the negative 52 divided by 4? So let's think about it. 52 is 40 plus 12. 40 divided by 4 is 10. 12 divided by 4 is 3. So it's gonna be 13. Negative 13. So it's negative 13 minus 7/4 b minus 7/4 b. So given a "b", if you give me a "b", I can put that value right over here, and I can calculate what the corresponding a needs to be in order to satisfy this relationship. So if I want a formula for f of b in terms of b, I can say, look, you give me a "b", the output of our function, which is f or a. The output of our function is going to be -- is going to be negative 13 minus 7/4 b, because the output of our function needs to be an "a" that will satisfy -- that will satisfy this equation up here. So hopefully that helped.