Question

practice for my test can someone please check if i got the answers right please

Answer 1

I'm checking them one by one and so far #1 and #2 are correct. Nice work

Answer 2

thanks i think the last one wrong

Answer 3

#3 is correct

Answer 4

and #4 is correct

Answer 5

#5 is incorrect

Answer 6

#6 is correct

Answer 7

#7 is correct

Answer 8

and #8 is incorrect

Answer 9

can you help me with 5 and 8 explain them to me please?

Answer 10

Ok we'll start with #5

Let's say we had one equation ax+by = c

now let's say that we multiply both sides of this equation by some number k to get


ax+by = c

k(ax+by) = k*c

kax + kby = kc


Would you agree that ax+by = c and kax + kby = kc have the same solutions (ie they intersect each other at an infinite number of points)?

Answer 11

yeah

Answer 12

So if we can find the determinants D, Dx and Dy, we can determine what they equal when we get an infinite number of solutions (which is what #5 is asking for)

Answer 13

So we have the system

ax+by = c
kax + kby = kc


which means that to set up the coefficient determinant, we write the coefficients in the matrix like so


| a b |
| ka kb |


What is that determinant?

Answer 14

Do you know how to find the determinant above?

Answer 15

no

Answer 16

Simply multiply the stuff in the main diagonal (a and kb) and subtract off the terms in the off diagonal (ka and b)


So a*kb - ka*b = akb - akb = 0


This means D = 0

Answer 17

You found this, but this isn't the whole story.

Answer 18

soo....D, Dx and Dy will equal zero.

Answer 19

Yes, here's why

to find Dx, replace the first column with the column c and kc to get


| c b |
| kc kb |


then compute the determinant

Dx = c*kb - kc*b = ckb - ckb = 0

So Dx = 0

Answer 20

ohh i get it :) thankss

Answer 21

same thing for Dy, but now the second column

| a c |
| ka kc |

Dy = a*kc - ka*c = akc - akc = 0

So Dy = 0 as well

Answer 22

that's great

Answer 23

now onto #8

Answer 24

write the coefficients in a matrix


| 4 -3 |
| 2 1 |


Now replace the second column with the right hand side to get


| 4 -14 |
| 2 -2 |


and then compute the determinant


Dy = 4*(-2) - 2*(-14)

Dy = -8 + 28

Dy = 20

Answer 25

So somewhere along the way, you cut something in half...that's my guess anyway.

Answer 26

ohh i understand that I did wrong

Answer 27

that's great

Answer 28

thats for your help @jim_thompson5910

Answer 29

you're very welcome