Solve each system. (Hint: In Exercises 69–72, let

Question

1 / x = t

and

1 / y = u

.)

$\frac{2}{x} + \frac{3}{y} = 18$

$\frac{4}{x} - \frac{5}{y} = - 8$

Accepted Answer

Hey, everyone here we are asked to solve the following system of equations where X and Y are not equal to zero. Here we are given two equations. The first one is negative two divided by X plus three, divided by Y is equal to 12. And the second equation is eight divided by X plus nine, divided by Y is equal to 120. Here we have four answer choice options. Each of them being a solution set of some ordered pair answer A is the ordered pair, six and eight. Answer B is the ordered pair A and six answer C is 1/6 and 1/8 and answer D is 1/8 and 1/6. So here taking a look at our given system of equations, we see that it would be rather difficult to start solving these equations outright since we have our variables in the denominators. So we want to begin by simplifying our equations and we can do this by letting the variable U be equal to 21 over Y, one divided by Y. And then we will let t be equal to one divided by X. And so now with these two expressions for U and T, we can simplify our two equations. So our first equation becomes negative two times T plus three times U. And this is still equal to 12. And now for our second equation, we have eight times T plus nine times U is equal to 120. So now with our two simplified equations, we can go ahead and begin solving by first taking our first equation, which if we recall, we have negative two T plus three U is equal to 12. And we want to solve this equation for T. So solving for T, we first need to subtract three U from both sides. So we have negative two T is equal to minus three U. And now getting T alone, we divide both sides by the coefficient which is negative two. So now simplifying, we see that T is equal to the quantity of minus three U all divided by negative two. So now with this new expression that de describes our variable T, we can use the substitution method by substituting in this expression into our second equation and then solving for you. So recalling our second equation, we had eight T plus nine U is equal to 120. And now substituting in our new equation, we have eight times the quantity of 12 minus three U all divided by negative two. We then have plus nine U is equal to 120. Now just simplifying, we see that the negative two cancels with V eight. So we are left with negative four times the quantity of 12 minus three U plus nine U is equal to 120. Now just distributing in this negative four, we have negative 48 plus 12 U plus nine U is equal to 120. Now just adding 48 to both sides and combining like terms, we have 21 U is equal to 168. Now finishing up solving for you, we just divide both sides by the coefficient which is 21. So we find that U is equal to eight. So now that we have found our value for you, we can take this value and substitute it in the expression we found for T to directly solve for the value of T. So utilizing the equation where T is equal to 12 minus three U all divided by negative two and substituting in our value, we have T is equal to 12 minus three times our value of U which is eight all divided by negative two. Now simplifying we find that T is equal to six. So now with our value of U and our value of T, we can now subs our expressions that we originally had for U and T such that we can solve for the values of X and Y. So starting with T again, recalling that we had T is equal to one divided by X. So now we know that one divided by X is equal to six. So now just solving for X, we first multiply both sides by X. So we have that one is equal to 26 X. Now dividing both sides by six, we find that our value for X is 1/6. Now we can repeat this process with you, but here we need to recall that U is equal to one divided by Y. So here we know that one divided by Y is equal to eight. Now, solving for Y as we did for X, we find that Y what and our values for X and Y, so we can summarize our solution by writing it as a solution set of an ordered pair where the ordered pair is 1/6 and 1/8. So with our ordered pair that we have found, we see that we are left with answer choice C where again, we have the solution set of the ordered pair, 1/6 and 1/8. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

Solve each system. (Hint: In Exercises 69–72, let $1 / x = t$ and $1 / y = u$ .)
$\frac{2}{x} + \frac{3}{y} = 18$
$\frac{4}{x} - \frac{5}{y} = - 8$

Key Concepts

Substitution Method

Solving Systems of Linear Equations

Back-Substitution

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