Question

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

Answer 1

Call the one integer x.
call the other one y


x = 4y - 8

xy = 60

Solve this.

Answer 2

can you help solve it?

Answer 3

Substitute the x=4y-8 into the second equation

Answer 4

you will get (4y -8)y = 60

solve for y.

then solve for x

Answer 5

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.

Answer 6

i got y=-15 and x=-68?

Answer 7

im not positive that thats correct

Answer 8

no =( I don't think so.

Answer 9

Try again.

Answer 10

i dont know what i did wrong :/

Answer 11

Show me your work, okay?

Answer 12

ok 60=(4y-8)y
60=4y-8y
60/-4 = -4y/-4
y=-15

Answer 13

The problem lies in line 2

Answer 14

x= 4(-15)-8
x= -60-8
x=-68

Answer 15

what the problem? :o

Answer 16

(4y-8)y = 4y^2 -8y

Answer 17

distribute, sweet thang.

Answer 18

whoops! forgot the to square it. but then there are no like terms to combine how would we solve
60= 4y^2-8y

Answer 19

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?

Answer 20

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

Answer 21

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.

Answer 22

Yes =) Take out a common factor of 4 first though.

Answer 23

2 and 2

Answer 24

It makes it a lot easier.

Answer 25

Nooo
show me your factoring, please.

Answer 26

4? isnt is 2 and 2 or 4 and 1?

Answer 27

noope. show me your factoring.

Answer 28

from 4y^2 -8y-60 = 0

Answer 29

4y*y ?

Answer 30

...
okay you aren't good at factoring. That's okay. A lot of people aren't. Use the quadratic formula.

Answer 31

ok

Answer 32

I only say that because it's a whole different issue. We can work on it another time =)

Answer 33

ok after i do the quadratic equation i will get 2 different answers and those will be the answer to the problem right?

Answer 34

Re-read the problem and tell me if we have answered the question yet.

Answer 35

i did but i don't know won't that only find x and not y?

Answer 36

Right. Well actually, it finds y. Right?

Answer 37

i think so?

Answer 38

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.

Answer 39

yes it seems so confusing
i got fractions for both answers
-1/12 and 1/20
is that correct?

Answer 40

=/ I don't know what mistake you made.

Answer 41

i dont know either :(

Answer 42

\[\frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-(-8) \pm \sqrt{(-8)^2-4(4)(-60)}}{2(4)}\]

Answer 43

You probably just messed up on a sign somewhere. Did you plug in like that?

Answer 44

oh i see what i did im sorry i was doing 2ac instead of 2a

Answer 45

Ah =)

Answer 46

x= 5 and x= -3?

Answer 47

Good. Except we were solving for y. It's important because of the next step.

Answer 48

When I go back and plug into the original equation, I want to make sure I plug into the right variable.

Answer 49

which variable do i plug into and do i use the 5 or -3?

Answer 50

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.

Answer 51

lol

Answer 52

sorry and yes we know 2 answer to y so then which one would you substitute for x?

Answer 53

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.

Answer 54

ok i chose 5 and plugged it into x=4y-8 and i got 12

Answer 55

Good =)
So one possible solution is:
x=12 y =5
Understand? Find the other possible solution.

Answer 56

the other possible solution is 55?

Answer 57

Woah there. Where'd 55 come from? You makin' guesses or somethin'?

Answer 58

whoops is it 63?

Answer 59

wait ! lol sorry im using the wrong equation

Answer 60

I have no idea what you're doing. Lol.

Answer 61

x= -20 ?

Answer 62

That's not wrong... but it's not a complete answer.

Answer 63

but isnt it x= 12, -20

Answer 64

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.

Answer 65

but theres 2 x and y values

Answer 66

Ooooh, goodness. How to explain this...

Answer 67

lol i dont know what should i do? should i just leave 2 values for both variables?

Answer 68

Kind of.

Answer 69

Alright, stop worrying so much about the correct answer and let's just consider the question, okay?

Answer 70

ok

Answer 71

The question is talking about a couple of numbers. We called them x and y. And it told us something about those numbers.

Answer 72

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.

Answer 73

Two solution sets. (5,12) and (-3, 20) That's all you need to know

Answer 74

And we're just trying to figure out two numbers that those things are true for.

Answer 75

Right?

Answer 76

right

Answer 77

So our answer should be two numbers. We happened to name them x and y, so our answer will look like
x=
y=

Answer 78

Well, it turns out, there's more than one possible pair of numbers.

Answer 79

yup so i just leave those 2 answers?

Answer 80

Right. And what are those two answers?

Answer 81

y= 5, -3
x= 12, -20

Answer 82

mmmm, I don't like how you wrote that.

Answer 83

why?

Answer 84

Is x=5 y = -20 a solution?

Answer 85

Try plugging it in if you need to check.

Answer 86

no i put y=5

Answer 87

and x= -20

Answer 88

Sorry, my mistake.
Is y = 5 x=-20 a solution?

Answer 89

That's what I meant to ask.

Answer 90

wait no it isnt i just tried it out

Answer 91

Oh my. =(

Answer 92

Okay so... is y=5 still good?

Answer 93

It's not good when x=-20, we decided.

Answer 94

yes

Answer 95

Alright, why is it still good?

Answer 96

because when we plug it in -20=4(5)-8
it gives us 12 which is not equal to -20

Answer 97

Okay, that tells me why
y=5 x=-20 is not a solution.
Is y =5 bad in general then?
Is x=-20 bad?

Answer 98

yes its bad the -20