Find b such that (4x - b)/(x - 5) = 3 has a solution set given by...

Question

Find b such that (4x - b)/(x - 5) = 3 has a solution set given by {Ø}.

Accepted Answer

Hello today we're going to be finding a value of A. That allows this equation to produce the solution set of null. Well what does it mean to have a solution set of null. Well what that means is that there's no solution that exists for the equation or better yet the equation is inconsistent. So what that means is that we need to find the value of A. That makes this equation inconsistent. So the equation given to us is the quantity seven X minus a over x minus three is equal to one. Now what we can go ahead and do with the numerator is divide out seven. So factoring out seven is going to give us seven times X minus a over seven over X -3 equal to one. Now we want to find a value of A. That allows this numerator quantity to match this denominator X minus three. The reason why is because if we can find a value of A. That makes the numerator quantity x minus three we can get seven times x minus three over x minus three equal to one. And that's going to reduce to give us seven equal to one which is the inconsistent value. So we're gonna go ahead and take the quantity X minus a over seven and set it equal to x minus three. And what we're going to do is just go ahead and sell for a now. So in order to do that I'm gonna go ahead and subtract X from both sides of this equation. Doing so is going to leave me with negative 8/ equal to negative three. And I want to go ahead and get a by itself so I need to get rid of this denominator negative seven. I can go ahead and multiply both sides of the equation by negative seven. Doing so is going to leave me with a equal to negative seven times negative three which is positive 21. And this is gonna be the value of A. That causes this equation to be inconsistent And let's just go ahead and double check that if we allow a to be 21 then we get the equation seven X minus 21 over X minus three is equal to one. Factoring out of seven from the numerator is going to give us seven times x minus three over x minus three is equal to one. We can go ahead and remove the x minus three from both the numerator and denominator And that leaves us with seven is equal to one which is our inconsistent system. So a equal to 21 is the solution to this problem. And that means that the answer to this problem is going to be D. So I hope this video helps you understanding how to find the value of A. And I'll go ahead and see you all in the next video

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