## erica123 Group Title 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 2 years ago 2 years ago

1. slaaibak Group Title

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

2. erica123 Group Title

can you help solve it?

3. slaaibak Group Title

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

4. slaaibak Group Title

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

5. SmoothMath Group Title

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 Group Title

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

7. erica123 Group Title

im not positive that thats correct

8. SmoothMath Group Title

no =( I don't think so.

9. slaaibak Group Title

Try again.

10. erica123 Group Title

i dont know what i did wrong :/

11. SmoothMath Group Title

12. erica123 Group Title

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

13. slaaibak Group Title

The problem lies in line 2

14. erica123 Group Title

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

15. erica123 Group Title

what the problem? :o

16. SmoothMath Group Title

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

17. SmoothMath Group Title

distribute, sweet thang.

18. erica123 Group Title

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 Group Title

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 Group Title

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 Group Title

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 Group Title

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

23. erica123 Group Title

2 and 2

24. SmoothMath Group Title

It makes it a lot easier.

25. SmoothMath Group Title

26. erica123 Group Title

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

27. SmoothMath Group Title

28. SmoothMath Group Title

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

29. erica123 Group Title

4y*y ?

30. SmoothMath Group Title

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

31. erica123 Group Title

ok

32. SmoothMath Group Title

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

33. erica123 Group Title

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 Group Title

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

35. erica123 Group Title

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

36. SmoothMath Group Title

Right. Well actually, it finds y. Right?

37. erica123 Group Title

i think so?

38. SmoothMath Group Title

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 Group Title

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

40. SmoothMath Group Title

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

41. erica123 Group Title

i dont know either :(

42. SmoothMath Group Title

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

43. SmoothMath Group Title

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

44. erica123 Group Title

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

45. SmoothMath Group Title

Ah =)

46. erica123 Group Title

x= 5 and x= -3?

47. SmoothMath Group Title

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

48. SmoothMath Group Title

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

49. erica123 Group Title

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

50. SmoothMath Group Title

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 Group Title

lol

52. erica123 Group Title

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

53. SmoothMath Group Title

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 Group Title

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

55. SmoothMath Group Title

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

56. erica123 Group Title

the other possible solution is 55?

57. SmoothMath Group Title

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

58. erica123 Group Title

whoops is it 63?

59. erica123 Group Title

wait ! lol sorry im using the wrong equation

60. SmoothMath Group Title

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

61. erica123 Group Title

x= -20 ?

62. SmoothMath Group Title

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

63. erica123 Group Title

but isnt it x= 12, -20

64. SmoothMath Group Title

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 Group Title

but theres 2 x and y values

66. SmoothMath Group Title

Ooooh, goodness. How to explain this...

67. erica123 Group Title

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

68. SmoothMath Group Title

Kind of.

69. SmoothMath Group Title

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

70. erica123 Group Title

ok

71. SmoothMath Group Title

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 Group Title

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 Group Title

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

74. SmoothMath Group Title

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

75. SmoothMath Group Title

Right?

76. erica123 Group Title

right

77. SmoothMath Group Title

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 Group Title

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

79. erica123 Group Title

yup so i just leave those 2 answers?

80. SmoothMath Group Title

Right. And what are those two answers?

81. erica123 Group Title

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

82. SmoothMath Group Title

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

83. erica123 Group Title

why?

84. SmoothMath Group Title

Is x=5 y = -20 a solution?

85. SmoothMath Group Title

Try plugging it in if you need to check.

86. erica123 Group Title

no i put y=5

87. erica123 Group Title

and x= -20

88. SmoothMath Group Title

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

89. SmoothMath Group Title

That's what I meant to ask.

90. erica123 Group Title

wait no it isnt i just tried it out

91. SmoothMath Group Title

Oh my. =(

92. SmoothMath Group Title

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

93. SmoothMath Group Title

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

94. erica123 Group Title

yes

95. SmoothMath Group Title

Alright, why is it still good?

96. erica123 Group Title

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

97. SmoothMath Group Title

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 Group Title

99. erica123 Group Title

wait no the 5 is bad

100. erica123 Group Title

if we plug in -3 it will work

101. SmoothMath Group Title

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

102. erica123 Group Title

oh sorry lol so i can just leave both numbers?

103. SmoothMath Group Title

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

104. erica123 Group Title

-3 and -20

105. SmoothMath Group Title

106. SmoothMath Group Title

107. SmoothMath Group Title

you could do 5 and 12

108. SmoothMath Group Title

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 Group Title

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 Group Title

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 Group Title

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 Group Title

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 Group Title

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 Group Title

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 Group Title

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 Group Title

Goodnight. =)

117. erica123 Group Title

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

118. SmoothMath Group Title

My pleasure =)