Question

87x + 93y = 1092 and 87x + 93y = 12

Solve for x. Then solve for y.

Answer 1

So, what do you think your first step should be?

Answer 2

ok for the problem on the left, you have to cancel out to each side. 87x cancels 87x then 1092 divided by 87 = 12.5

Answer 3

12.5 would be x.. no?

Answer 4

sorry, hold on one second

Answer 5

If you are trying to solve for both x and y given these two equations, then there is no answer. 87x + 93y cant equal both 1092 and 12

Answer 6

yeah, i was gonna suggest a typo

Answer 7

so r u trying to solve for x in the equation the left

Answer 8

or for both x and y

Answer 9

ok let me give you the word problem. 1 sec

Answer 10

The octane rating of a gasoline is a measure of the amount of isooctane in the gas. Subaru recommends 91-octane gasoline for the 2008 Legacy 3.0 R. How much 87-octane gas and 93-octane gas should Kelsey mix in order to make 12 gal of 91-octane gas for her Legacy?

Answer 11

the teacher said this afterwords....... "On your last problem, your equation of 87x + 93y = 1092 is correct, but if you have 2 variables, you must have 2 equations. I only see one equation. The other equation should be the sum = 12."

Answer 12

Right

Answer 13

So, the X and Y in this equation represent the number of gallons of each 87-octane and 93-octane fuel.

Answer 14

So, we only want 12 gallons overall, no matter how much we mix with what; in that case, X + Y = 12.

Answer 15

But we also want the average of our new fuels to be 91 octane: 87 * X + 93 * Y = 12 * 91.

Answer 16

So, we've got the two equations to solve:

X + Y = 12
87 * X + 93 * Y = 1092

Answer 17

Now, where would we go from here?

Answer 18

right the two equations is what she was saying.

Answer 19

you have to get x by itself correct?

Answer 20

Right. We also have a choice of which equation to start out from by making X by itself; which one would you choose?

Answer 21

x first then y

Answer 22

OK, good. So how would you get X by itself on the first equation?

Answer 23

take 87x over 87 and x is by itself

Answer 24

then you have 1092 divided by 87

Answer 25

Right - in that case, you'd need to remember to divide Y by the same amount and move it over to the other side, too.

Answer 26

but when you do that you are left with 93y = 12.5x

Answer 27

divide 93 by 87 also?

Answer 28

But there's actually a much easier way to do it; what happens if you solve for X in the other equation, X + Y = 12?

Answer 29

easier way is good becuase i have tried this problem about 5 different times and i come up with a different answer every time

Answer 30

*because

Answer 31

lol

Answer 32

Half of math is finding the hidden shortcuts in problems.

Answer 33

So, how would you solve for X in X + Y = 12?

Answer 34

good question

Answer 35

Well, how would you get X by itself on one side?

Answer 36

take out y,

Answer 37

y divided by y cancels out then 12 divided by y

Answer 38

?

Answer 39

Well, are we trying to divide Y or subtract Y?

Answer 40

i thought you would divide. to equal it out. I c im wrong

Answer 41

If we were multiplying X and Y together to get 12, we'd definitely want to divide it out.

Answer 42

oooo ooooo oooo

Answer 43

so we are going to subtract now? first?

Answer 44

But since we're ADDING Y to X, we want to get rid of Y by doing the opposite of adding - subtracting.

Answer 45

Yep! You got it.

Answer 46

So what does the equation look like now after you did that?

Answer 47

ok.....
87x + 93y = 1092
-87x -87x
----------------------
93y = 1005x

Answer 48

close or did I really mess myself up?

Answer 49

Close! One big thing to remember in algebra is that different letters don't like each other.

Answer 50

So, we'd have to leave 1092 by itself - in this case, we'd leave it as 1092 - 87x.

Answer 51

and you get 1005

Answer 52

Well, the thing is that 87x means 87 multiplied by some number we don't know.

Answer 53

correct

Answer 54

X could be zero, X could be a billion - so unless we know what X is, we don't know what any number subtracted by it is, either

Answer 55

right

Answer 56

But we know that 1092 - 87 * X is exactly that - we want to leave it that way for now.

Answer 57

ok

Answer 58

So, if X was 1, then 1092 - 87 X would equal 1005

Answer 59

But if X was 0, then 1092 - 87X would equal 1092

Answer 60

ok

Answer 61

so x is 1

Answer 62

Del, I was reading this, and you're really good! You're like a legit teacher, the good kind who are patient and really care if you 'get it' or not. I applaud you and bedeck you with medals. :)

Answer 63

Aww, thanks :p.

Answer 64

yahoo go del

Answer 65

ok now the problem at hand

Answer 66

Right.

Answer 67

I think it might be a good idea to step back a bit. Remember when I said there's sometimes an easy way to figure things out in math?

Answer 68

yes

Answer 69

Let's go back to the first equation, X + Y = 12.

Answer 70

How would we get X by itself, just in this single and only equation?

Answer 71

subtract

Answer 72

Sorry, I just got lagged out.

Answer 73

ok i am lost

Answer 74

Right, so what would the new equation be?

Answer 75

So, we need to subtract Y from both sides of the equation X + Y = 12.

Answer 76

x + y = 12
-y -y
-------------
x = 11

Answer 77

x= 12

Answer 78

Close! Remember, letters and numbers don't like getting together willy-nilly.

Answer 79

Remember when I told you that you couldn't take 87X from 1092?

Answer 80

It's like that. Let's see what happens when we keep X, Y, and the numbers apart:

Answer 81

X + Y = 12
- Y -Y
--------------
X = 12 - Y

Answer 82

Now, pick any two numbers that add up to twelve.

Answer 83

6,6

Answer 84

OK, good. So, just for the heck of it, let X = 6 and Y = 6.

Answer 85

Then X + Y = 12, right?

Answer 86

yes

Answer 87

Right! So let's look at the new equation we just found out, X = 12 - Y. Then 6 = 12 - 6. It makes sense!

Answer 88

yes

Answer 89

Take another pair of numbers, 4 and 8. Since 4 + 8 = 12, then 4 = 12 - 8.

Answer 90

In a way, then, X + Y = 12 and X = 12 - Y are saying the same exact thing, just with the numbers shifted around.

Answer 91

ok

Answer 92

That's why they're equivalent.

Answer 93

Do you think you get why X = 12 - Y in the problem?

Answer 94

yes because y has to be away from x

Answer 95

Right - and we take Y away from X by subtracting it, because Y was added to X.

Answer 96

Congrats! We're done Part 1 of the problem! (2 more parts to go).

Answer 97

I'm sorry if I'm going a bit slow and not throwing the answer at you, I just want to make sure you get why you're doing stuff and not just doing it.

Answer 98

well whats confusing me is im used to see it layed out as i was typing it.. not in word form