Question

The table below shows four systems of equations:

System 1:
4x - 5y =2
3x -y = 8

System 2:
4x - 5y = 2
3x - 2y = 1

System 3:
4x - 5y = 2
3x - 8y = 4

System 4:
4x - 5y = 2
10x - 9y = 4

Which pair of systems will have the same solution? (4 points)


System 1 and system 3, because the second equation in system 3 is obtained by adding the first equation in system 1 to two times the second equation in system 1

System 1 and system 3, because the second equation in system 3 is obtained by adding the first equation in system 1 to three times the second equation in system 1

System 2 and system 4, because the second equation in system 4 is obtained by adding the first equation in system 2 to two times the second equation in system 2

System 2 and system 4, because the second equation in system 4 is obtained by adding the first equation in system 2 to three times the second equation in system 2

Answer 1

@noseboy908

Answer 2

I'm here, I'm just working it out for myself first.

Answer 3

thats fine!

Answer 4

Let's get right in it, then. Recall yesterday how we were discussing every now and then how one moves parts of an equation around?

Answer 5

yes

Answer 6

Great! So for each set of equations we'll have to choose one of the equations to set it in the form of "x=..", or "y=...", so start by choosing an an equaion from the first set to change to one of those two forms. (Try to pick the one that will be the simplest to move around)

Answer 7

I'll choose the second equation

Answer 8

Great choice. Now, writing out your steps, tell me what form the second equation can be put in so that x or y is on its own side.

Answer 9

3x - y = 8
+y +y
3x = y +8

Answer 10

While you can use that method, I'd highly suggest setting y on its own side instead of x. This is because x has a 3 attached to it (via multiplication), and the y is on its own already in a way.

Answer 11

so then:
3x - y = 8
-3x -3x
-y = 8 - 3x

Answer 12

And now finish the last step in simplifying here. There is a '-1' that y is being multiplied by at present.

Answer 13

*by y at present

Answer 14

so the equation will be y = 8 - 3x?

Answer 15

Well, while you removed the -1 from the y, you have to divide the elements of the right side by it as well. (Just flip the signs from subtraction to addition or from addition to subtraction).

Answer 16

so then y = 8 - 3x?

Answer 17

That's the same equation you just gave me that I said you needed to change.

Answer 18

soon I meant y = 8 + 3x

Answer 19

*sorry

Answer 20

Just one more step! You have to change the 8 too!

Answer 21

y = -8 + 3x

Answer 22

There ya go. Now we have our revised formula to work with. Now let's take the first formula, which is 4x-5y=2, and plug in what you have equal to y in the above equation into the y variable of the first equation. Understand?

Answer 23

I'm not sure but would it be 4x - 5( -8 + 3x) = 2?

Answer 24

Exactly! It's great that you know that for this problem. Now, use your skills of equation simplification and simplify this equation as much as possible, showing your work please. :)

Answer 25

ok

Answer 26

4x - 5(-8 + 3x) = 2
4x - 40 - 15x = 2
+40 +40
4x - 15x = 42

Answer 27

I think I did something wrong because I don't know how to do the rest

Answer 28

wait I see what I did wrong

Answer 29

4x - 5(-8 + 3x) = 2
4x + 40 - 15x = 2
-40 -40
4x - 15x = -38

Answer 30

I still don't understand it

Answer 31

You are doing fine! Just keep adding and subtracting for now

Answer 32

4x - 15x = -38
-11x = -38?

Answer 33

As odd as it sounds, that's correct. Now, wait just a moment, as I wish to verify my math for the equations. You could get a drink of preference if you so desire.

Answer 34

ok

Answer 35

Thanks for the patience. We're at the point where we established that the the first set of equations can be simplified to -11x=38. Now, for the next step, there's one more level of simplification you can go to. See if you can simplify the one more time.

Answer 36

x = -3.45?

Answer 37

While your math is correct, you are expected to leave it as an improper fraction in simplest form. What is that?

Answer 38

38/11?

Answer 39

sorry -38/11

Answer 40

Your first answer there is correct. So the outcome of the first equation is "x = 38/11". Got it?

Answer 41

why wouldn't it be -?

Answer 42

You divided a negative number by a negative number. In every circumstance that comes out positive.

Answer 43

sorry I didn't realize 38 was a negative

Answer 44

but I got it now

Answer 45

Great, so let's move on to the second set of equations. What are they?

Answer 46

4x - 5y = 2
3x - 2y = 1

Answer 47

should I do all the work I did for the first set and show it to you?

Answer 48

For the first set?

Answer 49

yes

Answer 50

Since we've already worked through the first set I feel no need to see it again.

Answer 51

ok so let me figure the second set out and i'll give you the answers so you can see if their right.

Answer 52

Great! Though please post your work too. And also note that since we solved for x in the first set, you have to ultimately solve for x here as well. (That means solving for y in the first equation you choose)

Answer 53

3x -2y = 1
-3x -3x
-2y = 1 - 3x which would become 2y = -1 + 3x
Is this good so far?

Answer 54

Yep, keep going.

Answer 55

4x - 5y = 2
4x - 5( -1 + 3x) = 2
4x + 5 - 15x = 2
-5 -5
4x - 15x = -3
-11x = -3
11/3

Answer 56

You forgot to divide the elements of the right side by 2 before substituting it into the equation.

Answer 57

sorry but what do you mean?

Answer 58

At the point where you said
"3x -2y = 1
-3x -3x -2y = 1 - 3x
which would become 2y = -1 + 3x", you still have to divide EVERYTHING by 2 in order to get y on its own.

Answer 59

then it would be y = - 0.5 + 1.5

Answer 60

Remember what I said about using irregular fractions instead of decimals.

Answer 61

y = - 1/2 + 3/2?

Answer 62

There ya go. Now redo your math with that in mind.

Answer 63

4x - 5y = 2
4x - 5( -1/2 + 3/2x) = 2
4x + 5/2 - 15/2x = 2
-5/2 -5/2
4x - 15/2x = -1/2
-3/2 = -1/2
Would we simplify this more?

Answer 64

Just a minor error was made. go back to the step of "4x - 15/2x = -1/2", and recheck the math you did for the left side.

Answer 65

-7/2 = -1/2

Answer 66

Yes, just don't forget the fact that it's -7/2(x)! Now, final step here! Solve for x!

Answer 67

-7/2 = -1/2
x = 7

Answer 68

Try again. Remember to flip the fraction on the left when you multiply it by the fraction on the right.

Answer 69

I got the same answer

Answer 70

it's -7/2 * 2/1 which equals 7 right?

Answer 71

Your "-1/2" magically turned into "2/1"!

Answer 72

when you divide by fractions aren't you supposed to flip them?

Answer 73

the second one?

Answer 74

Only the fraction that is being moved from one side to another gets flipped. This is because you have to flip it to cancel it out on the side it's already on, and you have to do with the other side what you do to the other.

Answer 75

so 2/7?

Answer 76

ugh sorry 4/7