Question

Solve the system of equations by substitution. What is the solution for x?
2x + y = 1
4x + 2y = −2

x = 0

x = 2

There is no x value as there is no solution.

x can be any value as there is an infinite number of solutions.

Answer 1

Hello, welcome to open study!

I can help you through the question. This is asking if there is a point in the (x,y) plane where these 2 lines cross. If there is, that point (x,y) is a solution to the system.
If there is not a point where they cross, they are parallel lines, and there are no solutions (x,y). If they are both the same line, overlap each other, then there will be an infnite number of solutions where the lines intersect, being that they are the same line.

Answer 2

inconsistent, dependent, independent

Those words sound familiar from class?

Answer 3

Yes, thanks so much for your help! @DanJS

Answer 4

@TheatreKid

Answer 5

if you want to go through the rest of it , let me know

Answer 6

there is a quick way to tell what the answer is, i can see they are parallel

Answer 7

Can you please help with one more if you don't mind?

Answer 8

Oh yes sure!

Answer 9

You want to know an easy way to tell the answer to these probs?

Answer 10

sure!

Answer 11

that would be awesome

Answer 12

Well it will tell you if they are parallel (no solutions), or the same line (infinite solutions) real fast...here are the equations again

2x + y = 1
4x + 2y = −2

Answer 13

reload page for thoseweird things to go away

Answer 14

Try this, multiply the first equation , everything, by 2. As long as you do that to both sides, the equation doesnt change.

Answer 15

2(2x+y) = 2*1 ---------> 4x + 2y = 2
right?

Answer 16

Yes thats right

Answer 17

k, now we have this for the system,

4x + 2y = 2
4x + 2y = -2

notice the left hand side...what happened?

Answer 18

everything doubled...

Answer 19

right, everything in the first equation we doubled, now look at both equations together

4x + 2y = 2
4x + 2y = -2

The left sides are exactly the same!

Answer 20

If you have this situation, the lines are parallel, and there are zero solutions

Answer 21

Ahhhh... that makes sense.

Answer 22

If you solved for y = mx+b form, they both will have the same slope, but different y-intercepts....parallel lines, zero solutions

Answer 23

so looking at the original problem

2x + y = 1
4x + 2y = -2

i think to myself, if i doubled 2x, i get the 4x in the second equation. Now if i double the y term also, does that equal the y term in the second equation. In this case it does, so the lines are parallel

Answer 24

In general,
If you can multiply an equation by a constant number, to get the left hand side of the second equation, the lines are parallel, and have no solutions.

Answer 25

Make sense?

Answer 26

Wow, thank you! That helps.

Answer 27

One small step further

Answer 28

Yup!

Answer 29

If you do the exact same thing, and BOTH sides of the equations become the same, then they are the same line and have infinitely many solutions

Answer 30

I knew that one, thanks for the reminder.

Answer 31

In this case the right sides werent the same, BUT Lets pretend the first equation had a -1 instead of the +1 on the right

2x + y = 1
4x + 2y = -2

Answer 32

2x + y = -1
4x + 2y = -2

multiply the first equation by 2, and you get the second equation, they are the same line, infinitely many solutions

Answer 33

So that is an easy, off the cuff, sort of way to tell real fast if you have parallel lines, or the same line.

Answer 34

There is a way to tell if they are perpendicular too that is pretty quick too , if you want to know that

Answer 35

Thank you for your help with that! Would you mind helping me with one more question instead of telling me about the way to find if they are perpendicular or not, because I don't need to know that right now, plus I think it's in my lessons, but thank you :)

Answer 36

sure

Answer 37

Can you help me to solve this system of equations. x+3y = -1 2x + 2y = 6

Answer 38

For future reference when you come to it,

a*x + b*y = c
d*x + e*y = f

a,b,c,d,e,f are all numbers in the 2 equations for a line.

IF (a*d) + (b*e) = 0 - the lines are perpendicular

Answer 39

x+3y = -1
2x + 2y = 6

There are a few ways to solve this ,, i would multiply the first equation by -2 first, then add the two equations together

Answer 40

-2x - 6y = +2 Multiplied first equation by (-2)
2x + 2y = 6
-----------------
0x - 4y = 8 Add them together, Notice -2x+2x = 0x

so -4y = 8
y = -2

Answer 41

Now just use that y = -2 in one of the equations to get the x value

Answer 42

Thank you so so so much!!

Answer 43

This will help me not only with these questions but with others as well; so thanks.

Answer 44

awesome, you can fan me so you can tell when i am online if you want :)

Answer 45

feel free to ask questions whenever i am on here, you can type in @DanJS into the question and it will get my attention