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erica123
one integer is 8 less than 4 times another integer. the product of the two integers is 60. What are the two integers? show work
Call the one integer x. call the other one y x = 4y - 8 xy = 60 Solve this.
can you help solve it?
Substitute the x=4y-8 into the second equation
you will get (4y -8)y = 60 solve for y. then solve for x
Here's the process, my little sugarplum: Pick your favorite equation and your favorite variable in that equation. Solve for that variable. Now, use that to substitute into the OTHER equation. This will give you an equation with just one variable in it =) Solve for that variable, and you should just get a number. Now, you know one variable, so you can look back at either of the first equations and use that variable to solve for the other one.
i got y=-15 and x=-68?
im not positive that thats correct
no =( I don't think so.
i dont know what i did wrong :/
Show me your work, okay?
ok 60=(4y-8)y 60=4y-8y 60/-4 = -4y/-4 y=-15
The problem lies in line 2
x= 4(-15)-8 x= -60-8 x=-68
distribute, sweet thang.
whoops! forgot the to square it. but then there are no like terms to combine how would we solve 60= 4y^2-8y
Well, that's a whole 'nother kind of problem, and I'm guessing you've actually had a lot of practice with it, but it can be tricky, I understand. The most common ways are factoring or quadratic formula. Sound familiar?
yes we have to move the 60 over so it becomes a zero so then we have the values a b c for quadratic equation
Good =) And actually, if you want to try factoring, it works nicely for this one. But I'm a fan of the quad formula, since it always works.
Yes =) Take out a common factor of 4 first though.
It makes it a lot easier.
Nooo show me your factoring, please.
4? isnt is 2 and 2 or 4 and 1?
noope. show me your factoring.
from 4y^2 -8y-60 = 0
... okay you aren't good at factoring. That's okay. A lot of people aren't. Use the quadratic formula.
I only say that because it's a whole different issue. We can work on it another time =)
ok after i do the quadratic equation i will get 2 different answers and those will be the answer to the problem right?
Re-read the problem and tell me if we have answered the question yet.
i did but i don't know won't that only find x and not y?
Right. Well actually, it finds y. Right?
We wrote our two equations. We substituted into one of them and got an equation with just one variable, y. We're now using the quadratic formula to solve for that variable. It's a little bit confusing because we're getting two possibilities for that variable, but we're still only solving for the one variable. We don't know the second variable yet.
yes it seems so confusing i got fractions for both answers -1/12 and 1/20 is that correct?
=/ I don't know what mistake you made.
\[\frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-(-8) \pm \sqrt{(-8)^2-4(4)(-60)}}{2(4)}\]
You probably just messed up on a sign somewhere. Did you plug in like that?
oh i see what i did im sorry i was doing 2ac instead of 2a
Good. Except we were solving for y. It's important because of the next step.
When I go back and plug into the original equation, I want to make sure I plug into the right variable.
which variable do i plug into and do i use the 5 or -3?
Okay, when you do this kind of problem, it's not important which variable you solve for first. You picked y. That's fine. What's not fine is that halfway through the problem, you started calling it x. Do you understand? You were solving for y, and then you randomly renamed it x. This will cause problems. You solved for y. Don't rename it and there won't be any confusion. So now you know y, so obviously that's what you can substitute in for.
sorry and yes we know 2 answer to y so then which one would you substitute for x?
Now, I understand that it's a little confusing that you get two answers for y. Try not to be confused by it. It just means that there are two possible answers to the question. One is when y=5 and the other is when y = -3. Whichever one you pick, you'll get an x value that goes along with that y value.
ok i chose 5 and plugged it into x=4y-8 and i got 12
Good =) So one possible solution is: x=12 y =5 Understand? Find the other possible solution.
the other possible solution is 55?
Woah there. Where'd 55 come from? You makin' guesses or somethin'?
wait ! lol sorry im using the wrong equation
I have no idea what you're doing. Lol.
That's not wrong... but it's not a complete answer.
but isnt it x= 12, -20
The question asks you what the two variables are. The two variables are x and y. It turns out this problem has two possible answers, but a good answer should give me an x value and a y value.
but theres 2 x and y values
Ooooh, goodness. How to explain this...
lol i dont know what should i do? should i just leave 2 values for both variables?
Alright, stop worrying so much about the correct answer and let's just consider the question, okay?
The question is talking about a couple of numbers. We called them x and y. And it told us something about those numbers.
It told us that that their product is 60. We wrote an equation about that. And it told us that one number was 8 less than 4 times the other. We wrote an equation about that.
Two solution sets. (5,12) and (-3, 20) That's all you need to know
And we're just trying to figure out two numbers that those things are true for.
So our answer should be two numbers. We happened to name them x and y, so our answer will look like x= y=
Well, it turns out, there's more than one possible pair of numbers.
yup so i just leave those 2 answers?
Right. And what are those two answers?
mmmm, I don't like how you wrote that.
Is x=5 y = -20 a solution?
Try plugging it in if you need to check.
Sorry, my mistake. Is y = 5 x=-20 a solution?
That's what I meant to ask.
wait no it isnt i just tried it out
Okay so... is y=5 still good?
It's not good when x=-20, we decided.
Alright, why is it still good?
because when we plug it in -20=4(5)-8 it gives us 12 which is not equal to -20
Okay, that tells me why y=5 x=-20 is not a solution. Is y =5 bad in general then? Is x=-20 bad?
if we plug in -3 it will work
Ahhhhhh my point isn't that they are bad COMPLETELY. My point is just that they don't work TOGETHER.
oh sorry lol so i can just leave both numbers?
Your answer really needs to be two numbers that work together. What are two numbers that work together?
Great. That's one answer.
As a SECOND. SEPARATE answer
you could do 5 and 12
So, to clarify, you wrote y = 5, -3 x = 12, -20 Which doesn't make it obvious which number goes with which. If you write it this way, it's much more clear y=5 x=12, and y=-3 x=-20
I know it probably seems unimportant, but writing it the first way really shows that you don't understand the question or your answer.
no it does seem important, thanks so much once again for your help it really means alot =D but i must go because its getting pretty late here
Well, if the problem was written as an actual real-life problem, it would be easier to explain why it was important. For example, maybe these numbers we're trying to solve for are x=pack of skittles purchased y = number of snickers bars purchased My answer would say something like "Oh, you EITHER bought 5 packs of skittles and 12 snickers bars OOOOR you bought -3 packs of skittles and -20 snickers bars" (shhh please ignore the fact that you can't buy a negative numer of snicker bars) Your answer would say something like, "You bought 5 packs of skittles and -3 packs of skittles. And the number of snickers bars you bought is 12. And also -20."
oh yeahhh! i like how you related it to real life lol and yeah so it really is possible to have negatives in our answer as long as it works together
Yeah, we can have negatives =) If we're solving real life problems, a lot of times we'll get negative answers. Those negative answers might make sense, or they might not and we might ignore them.
For example, if our problem was about money, then the positive answer might mean we made money and the negative answer might mean we lost money. However, if the problem we're solving is about how many children someone has and we got one negative answer and one positive answer, well we'd take the positive one and ignore the negative one because there is no way to have negative children.
yeah i see where you're going at and what you'tr trying to say. im sorry to end this convo but i have to leave to get some sleep because im up early in the morning tomorrow
goodnight and thanks so much once again! you have been an amazing help!