In Exercises 11–20, write an equation that expresses each relatio...

Question

In Exercises 11–20, write an equation that expresses each relationship. Then solve the equation for y.

x varies directly as z and inversely as the difference between y and w.

Accepted Answer

Welcome back. I am so glad you're here, we are given this relationship, we're told that X varies directly as the difference between Y and Z. Where y is greater than Z and X varies inversely as the sum of T and W. So let's write this expression out. Whenever we have variation, we start writing the variable that varies. So in this case X we say X equals the constant term K. And now when there's direct variation, the variables that vary directly. So this case the difference between Y and Z get multiplied by the constant term K. So it'll be K times the difference between Y and Z will write that as parentheses y minus Z. To show that we do that subtraction first and then that difference gets multiplied by K. For inverse variation, it will be K divided by the variables that vary inversely. So in this case the sum of T and W. Again we'll put parentheses there and we'll do that some first T plus W. And that some will be what the K is divided by. Alright, we've written that expression and now we are asked to solve the equation for Z. So let's start off by getting rid of the denominator on the right hand side. We will multiply both sides by T plus W. So the left hand side will read X times T plus W equals the right hand side will now just be K times Y minus Z. And we still want to get Z by itself. Let's divide both sides by the constant term K. So the left will now read X times T plus W over K equals now. Just y minus Z. Let's bring that Y over. We can subtract Y from both sides and that will leave us with X times T plus W over k minus Y equals a negative Z. And to get that Z completely by itself, we want to divide both sides by -1. And when we divide that first part, that X times T plus W divided by K by negative one, we'll just stick a negative in front of that. X. So negative X times T plus W over K. And negative Y divided by negative one is plus Y equals Z. We've got Z by itself, which is great. None of these look exactly like our answer choices. So we'll switch this y out, bring that in front. So we'll have Z equals y minus the numerator, X times T plus W. And the denominator K. We look at our answer choices and this matches with answer choice C. Well done, we'll catch you on the next one.

In Exercises 11–20, write an equation that expresses each relationship. Then solve the equation for y. x varies directly as z and inversely as the difference between y and w.

Key Concepts

Direct Variation

Inverse Variation

Solving for a Variable

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