1
99:59:59,999 --> 99:59:59,999
Solve for x and y and we have the system of equations right here we have 2x-5 is equal to 14

2
99:59:59,999 --> 99:59:59,999
and negative 6x + 3y is equal to negative 42.

3
99:59:59,999 --> 99:59:59,999
so we could try to solve this by elimination and maybe we want to

4
99:59:59,999 --> 99:59:59,999
let's see if we can eliminate our y variables first. We have a 3y here and a negative y up here

5
99:59:59,999 --> 99:59:59,999
and we could it won't eliminate if you just add 3y plus negative y

6
99:59:59,999 --> 99:59:59,999
that won't eliminate but if we could turn this negative y into a -3y then it would cancel out

7
99:59:59,999 --> 99:59:59,999
with the 3y and the best way to turn a negative y into a -3y is to multiply this entire top equation

8
99:59:59,999 --> 99:59:59,999
by 3

9
99:59:59,999 --> 99:59:59,999
so lets do that, let's multiply then we can get some space over on the left

10
99:59:59,999 --> 99:59:59,999
let's multiply this entire top equation

11
99:59:59,999 --> 99:59:59,999
by 3 so we're going to multiply by three. So when I multiply 2x by three I get 6x

12
99:59:59,999 --> 99:59:59,999
when I multiply -y by three I get -3y and then i multiply 14 by three

13
99:59:59,999 --> 99:59:59,999
3 times 14 is 42 right? 3 times 10 is thirty plus 12 is 42

14
99:59:59,999 --> 99:59:59,999
and then we can add both of these equations something interesting should already maybe be showing up on

15
99:59:59,999 --> 99:59:59,999
your radar but let's add both of these equations. I have the left hand side

16
99:59:59,999 --> 99:59:59,999
-6x+6x well those cancel out we get 0

17
99:59:59,999 --> 99:59:59,999
we have 3y minus 3y those cancel out so you get another 0

18
99:59:59,999 --> 99:59:59,999
and then finally we get -42 plus 42 well that's 0 so we end up with just 0=0

19
99:59:59,999 --> 99:59:59,999
which is clearly true which is clearly true, but it's not putting any constraints on the x or y

20
99:59:59,999 --> 99:59:59,999
and that's because whenever you have a situation like this where you just get something that's obviously

21
99:59:59,999 --> 99:59:59,999
true 0=0, 1=1 or 5=5 what we're dealing with in this situation is where both of our constraints,

22
99:59:59,999 --> 99:59:59,999
both of our equations are actually the same equation and this right here is a dependent system

23
99:59:59,999 --> 99:59:59,999
this is a dependent system

24
99:59:59,999 --> 99:59:59,999
when you see it right over here and you multiply it by 3 you got 6x-3y is equal to 42

25
99:59:59,999 --> 99:59:59,999
if we then multiply it by -1 you would get the exact same equation as the second equation

26
99:59:59,999 --> 99:59:59,999
you would get -6x plus 3y is equal to -42.

27
99:59:59,999 --> 99:59:59,999
Another way to think about it if you want to go from the first equation to the second equation

28
99:59:59,999 --> 99:59:59,999
you just multiply both sides of the equation by -3

29
99:59:59,999 --> 99:59:59,999
so both of these constraints are the same constraints. they're just kind of a scaled up multiple of eachother

30
99:59:59,999 --> 99:59:59,999
so if you were to graph them and I might as well just graph them for you

31
99:59:59,999 --> 99:59:59,999
right here, this first equation right here is 2x-y is equal to 14.

32
99:59:59,999 --> 99:59:59,999
so you subtract 2x from both sides, and you would get on the left hand side:

33
99:59:59,999 --> 99:59:59,999
you're left with just -y on the right hand side you have negative 2x + 14

34
99:59:59,999 --> 99:59:59,999
now multiply both sides by negative one and you get y=2x-14