Question

will give medal to best response

Answer 1

let m be the number of months for which they both add bumper stickers to their collection
so bradley will add 4 x m
and ryan will add 6 x m
now their total number of stickers are equal after m months
i.e. 23+4m = 9+6m
do you get this ?

Answer 2

im not sure

Answer 3

so you need to equate 23 + 4m = 9+ 6m
or 23- 9 = 6m - 4m
or 14 = 2m
or m = 14/2 = 7
so the answer would be 7 months

Answer 4

Bradley starts with 23 stickers. Ryan starts with 9. After 1 month, Bradley has 23 + 4, and Ryan has 9+6. After 2 months, Bradley has 23 + 4 + 4, and Ryan has 9 + 6 + 6. After 3 months, 23+4+4+4 and 9+6+6+6, and after \(m\) months, Bradley has \(23 + 4m\) and Ryan has \(9 + 6m\). The answer you seek is the value of \(m\) where \[23+4m = 9+6m\]

Answer 5

i got 1.4 i think im confused

Answer 6

what is the slope of the line that passes through the pair of points (-6, 8), and (2, 3)

Answer 7

You could also solve this by graphing lines and seeing where they intersect. In this graph, the x axis is the number of months, and the y axis is the number of stickers. The month where the two lines intersect is the month where they are equal

Answer 8

How did you come up with 1.4? Let's fix that before we go on to something else

Answer 9

i don't know i was combining like terms

Answer 10

are you talking about a different problem now, with the "what is the slope of the line..."?

Answer 11

no

Answer 12

okay. where did those points come from?

Answer 13

i divided 14 by 10

Answer 14

I agree that 14/10 = 1.4, but where did you come up with that?

Answer 15

do you see how to make a formula for the number of stickers Bradley has?

Answer 16

after \(m\) months?

Answer 17

yeah 23+4m=9+6m

Answer 18

well, that's both formulas, Bradley and Ryan. Bradley is the 23 + 4m part, Ryan is the 9 + 6m. So we need to solve that equation for \(m\). We solve it by getting \(m\) alone on one side of the equals sign.

Answer 19

combined like terms

Answer 20

\[23+4m = 9+6m\]

We can do just about anything so long as we do it to both sides of the equation. Why don't we start by collecting all of the \(m\) terms on one side. If we subtract \(4m\) from each side, we get:
\[23 + 4m - 4m = 9 + 6m - 4m\]which simplifies to \[23 = 9 + 2m\]right?

Answer 21

oh

Answer 22

this is really important to understand, so ask questions if it isn't crystal clear!

Answer 23

How could we go from \[23 = 9 + 2m\] to something where the right hand side only has terms with \(m\) and the left hand side only has numbers?

Answer 24

i did it backwards earlier

Answer 25

-9

Answer 26

subtract 9 from both sides? okay, what does that give you?

Answer 27

14 = 2m

Answer 28

14/2 = 7

Answer 29

good. and now to get the value of just 1 m instead of 2? yep, you got it!

Answer 30

m = 7

Answer 31

i guess you added 6m + 4m instead of subtracting, right?

Answer 32

yeah

Answer 33

and 23-9 = 14, then 14/10 = 1.4

this illustrates the importance of checking the answer!

first, does 1.4 months make sense for this problem? no, not really, because it is talking about whole months and whole stickers.

second, let's try it in the formula:

\[23+4m=9+6m\]\[23 + 4(1.4) = 9 + 6(1.4)\]\[23+5.6 = 9 + 8.4\]\[28.6 \ne 17.4\]

Answer 34

always, always check your work :-) I like to reread the problem before doing so, because that will often catch errors where I didn't read the problem correctly the first time.

Answer 35

okay may we do the other problem

Answer 36

and if it turns out you made a mistake, and you have time, it's valuable to go back and figure out why you made the mistake. look for patterns in your mistakes so you know what to watch out for in the future.

okay, what's the other problem?

Answer 37

what is the slope of the line that passes through the pair of points (-6, 8), and (2, 3)

Answer 38

gotcha. do you know the formula for the slope of a line going through two points?

Answer 39

maybe

Answer 40

if one point is \((x_1, y_1)\) and the other point is \((x_2,y_2)\), then the formula for the slope of a line through those two points is
\[m = \frac{y_2-y_1}{x_2-x_1}\]

it doesn't matter which point is which, so long as you are consistent — you have to use the same order for both x and y

Answer 41

slope of a line through (2,2) and (1,1) is the same as the slope of a line through (1,1) and (2,2), right?

Answer 42

yes

Answer 43

okay, pick one of those points and call it \((x_1, y_1)\), then pick the other and call it \((x_2, y_2)\), then plug them into the formula for \(m\) (slope)

Answer 44

?

Answer 45

do the math, what do you get?

Answer 46

remember, points are given as (x,y), not (y,x)

Answer 47

do i subtract multiply or divide

Answer 48

okay. pick a point to be \((x_1,y_1)\). What is it?

Answer 49

-6, 3

Answer 50

but that isn't one of our points...

Answer 51

-6, 8

Answer 52

"what is the slope of the line that passes through the pair of points (-6, 8), and (2, 3)"
So our points are

(-6,8)
and
(2,3)

-6, 8 isn't a point, you need the parentheses to indicate that it is a point

(-6, 8) is \((x_1,y_1)\) so \(x_1 = -6\) and \(y_1 = 8\)
with me so far?

Answer 53

yep

Answer 54

our other point is (2,3) so what are the values of \(x_2\) and \(y_2\)?

Answer 55

x2 = 2 and y2 = 3

Answer 56

good. now, the formula is
\[m = \frac{y_2-y_1}{x_2-x_1}\]and we have\[x_1=-6\]\[y_1=8\]\[x_2=2\]\[y_2=3\]Can you find the value of \(m\)?

Answer 57

-5/8

Answer 58

Yes, that's correct. as a visual check

the line slopes down and to the right, so the slope is negative

Answer 59

so im right

Answer 60

I said that, didn't I? :-)

Answer 61

again, important to check that you've done it correctly. notice how easy it would be to make a mistake with one of the signs and perhaps get the right number but the wrong sign

Answer 62

okay can you help with one more

Answer 63

probably, there's only one way to find out :-)

Answer 64

okay, do you know what point-slope form for a line equation is?

Answer 65

kinda

Answer 66

i just figured this out sorry for bothering you

Answer 67

oh, no bother...what did you get for the answer?