Question

Help with an algebra project?

Answer 1

Here's the background information...
Samantha and Jake are training for a marathon. To prepare for this marathon they have been training and tracking their progress periodically.

In the first week of training Jake ran an average of 9.50 minutes per mile. Later, in week four of training he ran an average of 6.75 minutes per mile.
In the first week of training Samantha ran an average of 8.75 minutes per mile. Later, in week five of training she ran an average of 5.75 minutes per mile.

Assuming that Jake and Samantha continue to train and improve their times at the same rate your task is to determine which week they will have the same average minutes per mile. We will assume that the relationship is linear as they will be training for a maximum of 10 weeks. To complete this task follow the steps below.
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This is the first question.

Determine the equation of a line in standard form that represents Jake’s training progress. His progress corresponds to the points (1, 9.50) and (4, 6.75).

And I put 11x + 12y = 125 down, but I'm not sure if it's right

Answer 2

(1, 9.50) and (4, 6.75)
with those two points, what did you get for the slope ?

Answer 3

I got -0.91667 or -11/12

Answer 4

ok, now we can do
y - 9.5 = -11/12(x - 1)
multiply both sides by 12 to get
12y - 12*9.5 = -11(x-1)
or
12y - 114= -11x + 11
add 11x to both sides
11x +12y -114= 11

add 114 to both sides
11x +12y = 125

yes you have the correct answer

Answer 5

Thank you!
The next part to the question is:
Determine the equation of a line in standard form that represents Samantha’s training progress. Her progress corresponds to the points (1, 8.75) and (5, 5.75).

And I got...
3x + 4y = 38
Is that right too?

Answer 6

I got -3/4 as my slope

Answer 7

yes, that works. as a check you can try either point in the equation and see if it "works"
for example, with x=1 and y= 8.75 you get
3*1+4*8.75 = 3 + 35 = 38
which shows it works.

Answer 8

Okay, perfect.

The next question is:
At what moment will Jake and Samantha have the same average minutes per mile?

And this is where I'm confused because I'm not sure about how to find the average.

Answer 9

the "y" value is the minutes per mile.
this question is asking at what week they both have the same speed (y value)

If we plot the two lines, they are asking where the lines meet.

Answer 10

You have two equations and two unknowns. You can solve using substitution
(which means solve for y and put that expression into the other equation. Or you canuse "elimination"

For example, multiply the 2nd equation by -3 (that means both sides, all terms)
then add the two equations
11x +12y = 125
3x + 4y = 38

Answer 11

Okay, when I did that I get x = 5.5 as my end result, right?

Answer 12

yes, at week 5.5 (which means in the middle of the 5th week), they both have the same speed

Answer 13

So I don't have to find the value of y at all for that question?

Answer 14

They did not ask for the y value, just when they y's are the same value.

Answer 15

Ohh, okay, that makes sense.

So the next question is:
If Jake and Samantha continue to train until week 13, what will their times be?

And I'm also stumped on this one. At the beginning, it gave the information of how the times changed over the span of their training, but I don't know how to find out what their new times would be on the 13th week of training.

Answer 16

week 13 means x=13
they want you to find y when x=13

Answer 17

How would I start to do that? Is this something like an algebraic sequence?

Answer 18

you can replace x with 13 in your equations.
then solve for y

Answer 19

For example
11x + 12y = 125 with x=13
11*13 + 12y = 125
143+ 12y = 125
add -143 to both sides.
can you finish ?

Answer 20

For the first equation I got this...
11(13) + 12y = 125
143 + 12y = 125
12y = -18
y = -1.5

And for the second I got this...

3(13) + 4y = 38
39 + 4y = 38
4y = -1
y = -0.25

Are those right?

Answer 21

yes, but of course those answers are not realistic. people don't take negative time to run a race

Answer 22

Right, that's what was confusing me. Thank you!

And the next part is this:

Do you believe a linear model best represents the relationship of the time of the runners and the weeks that passed? (Hint question 5). What do you think this says about problems in the real world? Justify your thoughts in 3-4 sentences.

So I'll of course figure out the sentences... but am I right in thinking that a linear model does represent it best? Since there are many factors that can be plugged into the equations?

Answer 23

Do you think people improve at a steady rate until they can run so fast it takes no time to finish the race ?

Answer 24

Oh... no... because like you said, it would be unrealistic. So if that were compared to "problems in the real works" like it says, then it a linear model wouldn't best represent the relationship. Right?

Answer 25

Though it is possible to use a linear model to predict how things will go, generally they work only for a short period. In this problem, the answers you got for x=13 are not possible, and that shows the model is not realistic (but it might be for a few weeks)

Answer 26

The real world is generally more complicated than a linear model, but we still use them a lot for "short periods".

Answer 27

Okay, that makes total sense.
And that's all the questions I have on the project. Thank you soo very much for you're help. It made a huge difference!

Answer 28

yw