Evaluate x^2 - (xy - y) for x satisfying 3(x + 3)/5 = 2x + 6 and ...

Question

Evaluate x^2 - (xy - y) for x satisfying 3(x + 3)/5 = 2x + 6 and y satisfying - 2y - 10 = 5y + 18.

Accepted Answer

Hello. Today we're going to solve both equations for X and Y. And then using those values of X and Y to evaluate 36 X squared minus two X. Y plus Y squared. So let's go ahead and start by evaluating the equation for X. The equation for X is four times the quantity X plus one divided by seven equal to six X minus 1/3. Now, since we're solving for X we want to go ahead and set this equation equal to zero. In order to do that, I'm gonna go ahead and subtract negative six X. And add one third to both sides of this equation. Doing so is going to give me four times X plus 1/7 minus six X plus 1/3 equal to zero. Now we want to go ahead and combine this all into a single quantity. And in order to do that we want to go ahead and find the lowest common denominator. Well keep in mind that six X. Is technically over one. So the lowest common denominator between 71 and three Is going to be 21. So we want to go ahead and multiply each fraction by a quantity that allows the denominator to become 21. Well for the first quantity, that number is going to be three. Because if we multiply the denominator by three we get 21 but we must also multiply the numerator by three for negative six X over one. We want to multiply this by 21/ Because multiplying one by 21 is going to give us 21 and we have to do the same to the numerator and one third can be multiplied by 7/7 to give us 21 in the denominator. So multiplying these values together is going to give us three times four times the quantity X plus 1/21 minus six x times 21 which is going to give us 100 and 26 X over 21 and that's gonna be plus one times seven which is 7/3 times seven which is 21 equal to zero. Now that we have all the denominators over 21 we can go ahead and combine each of these fractions into a single quantity. Doing so is going to give us three times four times the quantity X plus one minus X plus seven. All of that over 21 equal to zero. And since this entire quantity is over 21 we can eliminate the denominator by multiplying everything by 21. Doing so is going to give us three times four times X plus one minus 126 X plus seven equal to zero. Now we need to go ahead and simplify this statement. Well three times four is going to give us 12 and we're going to need to distribute that quantity of 12 2 X plus one. Doing so is going to give us 12 x plus 12 minus 126 X plus seven equal to zero. And now we need to go ahead and combine like terms well negative X and 12 X can be combined and positive 12 and positive seven can be combined as well, combining the like terms together gives us negative X plus 19 equal to zero. And now we need to go ahead and sell for X. So we need to get X by itself in order to do that. We can subtract 19 from both sides of this equation. That's going to leave us with negative 114 X equal to negative 19. And in order to sell for X we're just going to divide both sides by negative 114. Doing so is going to give us X is equal to 19 over 114 and that can reduce to X is equal to 1/6. So we found our value for X. But now we need to go ahead and find our value for Y while the equation for y is Y plus three equal to native three minus Y. Now in order to solve this I can go ahead and subtract three from both sides of the equation. Doing so is going to give me why is equal to negative 6 -1. And to get why to one side of the equation, I can go ahead and add Y to both sides of the equation. Doing so is going to give me two y equal to negative six. And to solve for y, I just need to divide both sides by two and why is equal to negative three. Okay, so now we found our value for X and we found our value for Y. Just to go ahead and recap X is equal to 1/6 and Y is equal to -3. Now we need to go ahead and evaluate 36 X squared minus two X, Y plus y squared. And we're gonna evaluate this by plugging in our values for X and Y. Doing so is going to give us 36 times X which is 1/6 squared minus two times X which is 1/6 times y which is negative three plus y squared which is negative three squared. Now evaluating this 1/6 squared is the same thing as 1/6 times 1/6. So that's going to give us 36 times the quantity, 1/36 negative two times 1/6 is going to give us negative 2/6 which reduces to negative 1/3 and that's gonna be multiplied by negative three and negative three squared is the same thing. I say negative three times negative three which is going to give us positive nine Now, times 1/36 is going to give us 36/ and that can just reduce to one negative 1/ times negative three is going to give us positive 3/3, which reduces to one. So this entire equation reduces to one plus one plus nine And that is equal to 11 which is our final solution to this entire equation. And that means that the solution to this problem is going to be C. So I hope this video helps you understanding how to evaluate this type of problem, and I'll go ahead and see you all in the next video.

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