Question

3x + 8y = 12
2x + 2y = 3

Part A: Create an equivalent system of equations by replacing one equation with the sum of that equation and a multiple of the other. Show the steps to do this.

Part B: Show that the equivalent system has the same solution as the original system of equations.

Answer 1

Ok first you want to use one equation and you need to revert it back to the y= form, do you understand?

Answer 2

You mean make it read like \[y = z - x\]?

Answer 3

Yes , so which equation are you going to use? You can choose one or the other it wouldnt matter.

Answer 4

I kinda want to start with the top one; \[3x +8y = 12\]

Answer 5

Ok so what steps would you need to take to isolate y?

Answer 6

Move 3x to the other side.

\[3x + 8y = 12 \]
\[-3x = 12 - 3x \]
\[8y = 12 - 3x\]?

Answer 7

Now you need to completely isolate y so what would you do? Divide 8.

Answer 8

\[\frac{ 8y = 12 - 3x }{ 8 } = y = 1.5 - .375x\]

Answer 9

Oops sorry this site is acting weird... Yea thats what you would get

Answer 10

Yeah, its being laggy today. Okay, so now what?

Answer 11

@StudiousA ?

Answer 12

Ok im back, soo sorry it was acting reallydumb, anyways I think that method would be too confusing so why not use the elimination method, have you learned that yet?

Answer 13

I don't think so.

Answer 14

now you replace the value of y in the other equation

Answer 15

2x + 2 (1.5-0.375x)=3

Answer 16

2x + 3- 0.75x=3
2x-0.75x=3-3
1.25x=0
x=0

Answer 17

So very sorry, i think my internet is acting dumb or something, anyways...
The elimination method simply uses the negated value of the second equation x coefficent over the x coefficient of the first equation and you multiply those to their respective equations....

-2 (3x +8y=12)
3 (2x+2y=3)

You distribute and get

-6x-16y=-24
6x+6y=9

You then add those equations add get

-10y=-15

Isolate y

y=1.5

Now you have to input the value of y into each equation

Answer 18

is this still part A? o.o

Answer 19

When you input y to both equations you get

3x +8(1.5)=12
3x+12=12
3x=0

2x+2y=3
2x+2(1.5)=3
2x=0

Answer 20

I'm sorry, I'm lost...

Answer 21

Yes that part was part a

Answer 22

Its okay so that whole thing i wrote was part a

Answer 23

Now part b says
Part B: Show that the equivalent system has the same solution as the original system of equations.


So usingthe information you found out about 3x and 2x you simply input, as of this moment do you understand how you got the values for. 2x and 3 x?

Answer 24

What part is confusing to you at this point? I can go back and explain.

Answer 25

Basically the elimination thing. I haven't learned that before.

Answer 26

Ok well let me see what other method i can use to explain it.

Answer 27

Okay.

Answer 28

Why dont i explain what i did

Answer 29

Its called the elimination method because you are trying in this case to ELIMINATE Aa value in this case x

Answer 30

Okay. That makes more sense, so what did we eliminate?

Answer 31

So the first equation of 3x+8y=12 has the value of 3x and the second one is 2x+2y=3 where the value of x is 2x so you want to find values that would cancel out 3x an 2x, and to do this you want one value to be negated so it can cancel out

Answer 32

So, basically finding multiples? like fractions? 6?

Answer 33

So lets do it part by part, if 3x were to cancel out from 2x you would need to find a common value which is 6 so 3 x -2 and 2 x 3

Answer 34

Exactly! Good job! Okay now that we have established that as mentioned it needs to cancel out so you would have to negate one value in this case instead of. 3x x 2 it would be 3x x -2

Answer 35

So far so good?

Answer 36

Yep!

Answer 37

Ok and when doing one thing to a part of the equation you need to do it to all parts soo....

Answer 38

I read slow, 1st percentile processing speed. So if I take a bit to reply, it's because I have to read it several times before it clicks. Just saying*

Answer 39

Yea its fine, no problem haha

Answer 40

Okay, so we do that to both sides of the = sign, right?

Answer 41

so,
\[4y = 4 - 6x \]??

Answer 42

Exactly

Answer 43

:D and thats the answers for just part A, right?

Answer 44

Wait a second...

Answer 45

Uh oh..

Answer 46

Oh i see what you did, sorry was answering to the previous thing. You need to use the original equations and multiply

Answer 47

So as we said before we are trying to cancel out the x values soo
-2 (3x+8y12)

Answer 48

And distribute the -2

Answer 49

Oh! I tried to combine them.

Answer 50

Oh

Answer 51

Sorry this internet again.... and I meant

-2(3x+8y=12)

Answer 52

So,

-2 (3x+8y12) =

-6x + -16y = -24?

Answer 53

Yea goo job! now do the same thing to the second equation but by using the value of 3 because you are trying to cancel the x value

Answer 54

oops haha i meant good job, not goo... human imperfection haha, we all have it

Answer 55

Ok so what would you get?

Answer 56

would I use 3 or -3?

Answer 57

-3, right?

Answer 58

-3 (2x + 2y = 3) =

-6x - 6y = -9?

Answer 59

Sorry this stupd wifi ugh anyways it would be 3, do you understand why?

Answer 60

Not really..

Answer 61

Well you are trying to eliminate the x value correct? so you need to find what will completely take it away when you add both equations together in this case its 3 because..

Answer 62

When -2 is multiplied or distributed to each value of the equation 3x +8y= 12 you get -6

Answer 63

So that I don't have -6 twice? Because I'm trying to eliminate it, so x = 0 when I add -6x and 6x together!

Answer 64

for the x value

and if you multiply the second equation of 2x +2y=3 by 3

Answer 65

by 3

Answer 66

Exactly because if it were -3 then you can"t cancel out the -6 it would add and become -12

Answer 67

as of this point are you understanding what to do for the first step?

Answer 68

so,
3(2x + 2y = 3) =

6x + 6y = 9

Answer 69

yes!!

Answer 70

so now add the equations

Answer 71

what are they?

Answer 72

I believe so.

I'm trying to eliminate x from both equations. correct?

also, after adding them , its -10y = 15

Answer 73

Omg thats correct, good job!
Now what you want to do is isolate y, so what would you do to get the value of y?

Answer 74

Divide, right?

Answer 75

exactly, so what would get? 15 divided by -10

Answer 76

I dropped my pretzel from excitement haha

Answer 77

\[\frac{-10y = 15}{-10}\]
\[y = 1.5\]

Answer 78

yes good, now what you are going to have to do is input that value in both original equations.

Answer 79

It must feel good being able to teach someone something they wouldn't otherwise understand. (:

Answer 80

so input the value of y which is 1.5 into both 3x+8y=12 and into 2x=2y=3

Answer 81

Okay, um. \[3x + 8(1.5) = 12\]? I don't think we ever got an x-value?

Answer 82

oops i meant 2x+2y=3

yea, i love helping others and when i see i actually had an impact on them, well the feeling is great

Answer 83

You will get an x value but not one in which the x is isolated, 3x is treated as a whole value and so is 2x

Answer 84

so first complete 3x+8(1.5)=12

Answer 85

what did you get for it?

Answer 86

Okay, let me see.

\[3x + 8(1.5) = 12\]

\[3x + -12 = 12\]

and..
\[2x + 2(1.5) = 3\]
\[2x + 3 = 12\]

Answer 87

not exactly, do you want me to walk you through it?

Answer 88

Oops! I messed up at the end, it should be
\[2x + 3 = 3\]

Answer 89

yea the value for the second is correct 2x+3=3

Answer 90

but 8 x 1.5 isnt -12, it is... be careful while putting it in the calculator or doing mentally, whichever you are doing

Answer 91

And that shouldnt be -12..

Answer 92

yea, so what is it

Answer 93

I typed it wrong, but calculated correctly. Lol

The equations are:

\[3x + 12 = 12\]
\[2x + 3 = 3\]

Answer 94

Correct! Good job! ok now as I mentioned before both 3x and 2x are treated as a single value so..

Now isolate 3x from the first equation

Answer 95

What do you do?

Answer 96

I subtract 12 from both sides, to get 3x on its own, but then its 3x = 0

Answer 97

Correct!!! Now using the same concept do that to 2x in the second equation, what do you get?

Answer 98

2x = 0