## erica123 3 years ago 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

1. slaaibak

Call the one integer x. call the other one y x = 4y - 8 xy = 60 Solve this.

2. erica123

can you help solve it?

3. slaaibak

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

4. slaaibak

you will get (4y -8)y = 60 solve for y. then solve for x

5. SmoothMath

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.

6. erica123

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

7. erica123

im not positive that thats correct

8. SmoothMath

no =( I don't think so.

9. slaaibak

Try again.

10. erica123

i dont know what i did wrong :/

11. SmoothMath

12. erica123

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

13. slaaibak

The problem lies in line 2

14. erica123

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

15. erica123

what the problem? :o

16. SmoothMath

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

17. SmoothMath

distribute, sweet thang.

18. erica123

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

19. SmoothMath

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?

20. erica123

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

21. SmoothMath

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.

22. SmoothMath

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

23. erica123

2 and 2

24. SmoothMath

It makes it a lot easier.

25. SmoothMath

26. erica123

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

27. SmoothMath

28. SmoothMath

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

29. erica123

4y*y ?

30. SmoothMath

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

31. erica123

ok

32. SmoothMath

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

33. erica123

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

34. SmoothMath

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

35. erica123

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

36. SmoothMath

Right. Well actually, it finds y. Right?

37. erica123

i think so?

38. SmoothMath

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.

39. erica123

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

40. SmoothMath

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

41. erica123

i dont know either :(

42. SmoothMath

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

43. SmoothMath

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

44. erica123

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

45. SmoothMath

Ah =)

46. erica123

x= 5 and x= -3?

47. SmoothMath

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

48. SmoothMath

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

49. erica123

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

50. SmoothMath

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.

51. slaaibak

lol

52. erica123

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

53. SmoothMath

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.

54. erica123

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

55. SmoothMath

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

56. erica123

the other possible solution is 55?

57. SmoothMath

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

58. erica123

whoops is it 63?

59. erica123

wait ! lol sorry im using the wrong equation

60. SmoothMath

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

61. erica123

x= -20 ?

62. SmoothMath

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

63. erica123

but isnt it x= 12, -20

64. SmoothMath

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.

65. erica123

but theres 2 x and y values

66. SmoothMath

Ooooh, goodness. How to explain this...

67. erica123

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

68. SmoothMath

Kind of.

69. SmoothMath

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

70. erica123

ok

71. SmoothMath

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

72. SmoothMath

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.

73. slaaibak

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

74. SmoothMath

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

75. SmoothMath

Right?

76. erica123

right

77. SmoothMath

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

78. SmoothMath

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

79. erica123

yup so i just leave those 2 answers?

80. SmoothMath

Right. And what are those two answers?

81. erica123

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

82. SmoothMath

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

83. erica123

why?

84. SmoothMath

Is x=5 y = -20 a solution?

85. SmoothMath

Try plugging it in if you need to check.

86. erica123

no i put y=5

87. erica123

and x= -20

88. SmoothMath

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

89. SmoothMath

That's what I meant to ask.

90. erica123

wait no it isnt i just tried it out

91. SmoothMath

Oh my. =(

92. SmoothMath

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

93. SmoothMath

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

94. erica123

yes

95. SmoothMath

Alright, why is it still good?

96. erica123

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

97. SmoothMath

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

98. erica123

99. erica123

wait no the 5 is bad

100. erica123

if we plug in -3 it will work

101. SmoothMath

Ahhhhhh my point isn't that they are bad COMPLETELY. My point is just that they don't work TOGETHER.

102. erica123

oh sorry lol so i can just leave both numbers?

103. SmoothMath

Your answer really needs to be two numbers that work together. What are two numbers that work together?

104. erica123

-3 and -20

105. SmoothMath

106. SmoothMath

107. SmoothMath

you could do 5 and 12

108. SmoothMath

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

109. SmoothMath

I know it probably seems unimportant, but writing it the first way really shows that you don't understand the question or your answer.

110. erica123

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

111. SmoothMath

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."

112. erica123

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

113. SmoothMath

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.

114. SmoothMath

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.

115. erica123

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

116. SmoothMath

Goodnight. =)

117. erica123

goodnight and thanks so much once again! you have been an amazing help!

118. SmoothMath

My pleasure =)