Question

For what value of p and q will the following pair of linear equations have infinitely many solutions?
4x + 5y = 2;
(2p + 7q)x + (p+8q)y = 2p- q + 1

Answer 1

@zepdrix

Answer 2

They'll have infinitely many solutions `when they are both the same line`.
That means in the second equation,
we need `the stuff on x` to be 4, ya?

Answer 3

\[\large\rm 2p+7q=4\]

Answer 4

So then `the stuff on y`, needs to be how much?

Answer 5

@zepdrix Sorry, I lost connection ;-;

Answer 6

Oh I forgot I had OpenStudy open XD haha
glad it gave me the orange notice

Answer 7

5, the stuff on y should be 5

Answer 8

\[\large\rm p+8q=5\]k good.

Answer 9

xD

Answer 10

Ty ty

Answer 11

And the stuff on the right should be 2,\[\large\rm 2p- q + 1=2\]But we really only need two equations to solve this,
so use whichever two are the easiest.

Answer 12

\[\large\rm 2p+7q=4\]\[\large\rm p+8q=5\]Remember how to solve a system like this? :o

Answer 13

Yeah, there are many methods ;-;

Answer 14

Gimme a min, I'll solve it

Answer 15

I got q = 6/9

Answer 16

Same... hmm these are weird numbers :p

Answer 17

Yup, quite weird ;-;

Answer 18

P = 24/63

Answer 19

Im not so sure about p

Answer 20

Hmm thinking

Answer 21

For infinite solutions,
we need the same y-intercept and slope.
Everything needs to be the same.

So all three of these equations must hold true,\[\large\rm 2p+7q\qquad=4\]\[\large\rm ~~p+8q\qquad=5\]\[\large\rm 2p-~~q + 1=2\]
Using only the first two equations,
you should end up with \(\large\rm q=\frac23\) and \(\large\rm p=-\frac13\).
But this doesn't hold true for the other equation :((
Hmmm.

Is `no solutions` an option for your question?

Answer 22

If you replace the p's with x's,
and the q's with y's,
then the system of coefficients looks like this,\[\large\rm 2x+7y\qquad=4\]\[\large\rm ~~x+8y\qquad=5\]\[\large\rm 2x-~~y + 1=2\]If you graph this system it looks like this, https://www.desmos.com/calculator/paysje4onr
Notice that there is no `point` where all three lines intersect?
Ya no solution :(

Answer 23

Nope, they are asking for the values which will make it have infinite solutions

Answer 24

There are no values.
I don't think I made a mistake... but it's possible.
Lemme see if someone smart is around >.< hmm

Answer 25

Grr noone is online right now D:

Answer 26

;-;

Answer 27

It's cool, mate. I'll wait for somebody. Ty for your efforts! I appreciate it :) @zepdrix

Answer 28

@Divyankasc A lil help, sis? ;-;

Answer 29

I'm assuming we gotta do a1/a2 = b1/b2 and then cross multiply?

Answer 30

Yep

Answer 31

So I did that and I got 6p= -3q

Answer 32

so p = -3q/6

Answer 33

Then substitute in the b1/b2 = c1/c2 part, I guess @Divyankasc

Answer 34

4x + 5y = 2;
(2p + 7q)x + (p+8q)y = 2p- q + 1

For infinitely many solutions,
a1/a2 = b1/b2 = c1/c2

a1/a2 = b1/b2
→ 4/2p + 7q = 5/p + 8q
4(p + 8q) = 5(2p + 7q)
4p + 32q = 10p + 35q
6p = -3q
q = -2p ....(1)


a1/a2 = c1/c2
→ 4/2p + 7q = 2/2p - q +1
4(2p - q + 1) = 2(2p + 7q)
4p - 4q + 1 = 4p + 14q
18q = 1
q = 1/18

From (1),
1/18 = -2p
p =-1/36




Please check it again ↑↑↑
Because I do many mistakes while solving

Answer 35

Mistake! Wait

Answer 36

Okiee

Answer 37

4x + 5y = 2;
(2p + 7q)x + (p+8q)y = 2p- q + 1

For infinitely many solutions,
a1/a2 = b1/b2 = c1/c2

a1/a2 = b1/b2
→ 4/2p + 7q = 5/p + 8q
4(p + 8q) = 5(2p + 7q)
4p + 32q = 10p + 35q
6p = -3q
q = -2p ....(1)


a1/a2 = c1/c2
→ 4/2p + 7q = 2/2p - q +1
4(2p - q + 1) = 2(2p + 7q)
4p - 4q + 4 = 4p + 14q
18q = 4
q = 4/18 = 2/9

From (1),
2/9= -2p
p = -1/9

Answer 38

Does the answer match ?

Answer 39

I don't have the answer key :/

Answer 40

Ohk, I think I amcorrect now

Answer 41

Then too, wait :P

Answer 42

Gimme a min ;-;

Answer 43

I'll check

Answer 44

Ahhh! I am really really sorry -_-

Answer 45

4x + 5y = 2;
(2p + 7q)x + (p+8q)y = 2p- q + 1

For infinitely many solutions,
a1/a2 = b1/b2 = c1/c2

a1/a2 = b1/b2
→ 4/2p + 7q = 5/p + 8q
4(p + 8q) = 5(2p + 7q)
4p + 32q = 10p + 35q
6p = -3q
q = -2p ....(1)


a1/a2 = c1/c2
→ 4/2p + 7q = 2/2p - q +1
4(2p - q + 1) = 2(2p + 7q)
8p - 4q + 4 = 4p + 14q
4p - 18q = -4
2p - 9q = -4

From (1),
2p + 18p = -4
20p = -4
p = -1/5

From (1),
q = 2/5



This is the answer for sure!

Answer 46

I mean I am in so hurry that I did many mistakes

Answer 47

Happens to me all the time! It's cool, Divya! Thank you for your help :)

Answer 48

And please make sure that you haven't written the question wrong. Recheck it.

Answer 49

NP :)

Answer 50

The question's right, I checked it again. ;-;

Answer 51

So is the answer, I believe. It looks good to me. c;

Answer 52

Cool, then my answer is also right.. Meow :3

Answer 53

Yup.. :3

Answer 54

You made a mistake.
You wrote 4p - 18p = -4
then in the next line, u divided every term by 2 except -4.. @Divyankasc

Answer 55

It's fine, Divya! :3

Answer 56

Ah! Okay ^_^" Good that you find out my mistake