85295james 3 years ago Solve the system of equations by graphing

1. 85295james

-1/3x + y = –1 y = 4+1/3 x

2. theEric

Do you know what it means to solve them by graphing? I'm unsure! I'm into calculus 3 and I can't think of what it means. Does this sound familiar? You're looking for a point (x,y) so that $\frac{1}{3}x+y=-1y=4+\frac{1}{3}x$

3. theEric

Because I would immediately think to use algebra. I'm looking online for a refresher. If you multiplied by (-1), though... $(-1)\frac{1}{3}x+(-1)y=y=(-1)4+(-1)\frac{1}{3}x$

4. theEric

But that would be changing values. I'm not sure right now, sorry, but I will research how to solve a syste of equations by graphing now.

5. theEric

Wait.... I should've asked - are those two separate equations? I thought they were, and then I decided maybe they weren't..

6. theEric

If you have two line equations and they share just one (x,y) point, then they intersect on a graph. You just have to be able to plot the lines on grid paper and look for the intersection. For really random lines you will need to calculate, but for math problems designed for learning, you should be fine with just graphing.

7. theEric

So $-\frac{1}{3}x + y = -1$and$y=4+\frac{1}{3}x$ ?

8. theEric

THAT I can do for sure! :)

9. theEric

That would be a system of equations...

10. theEric

Sorry I delayed due to uncertainty. I'll guide you through it now. First, you should know how to plot a line, given it's equation. Second, you should rearange each equation so that you can easily plot it (I recommend making your equations look like y=mx+b) Third, you need to plot both lines. Fourth, you can the see where they intersect. So look at what the point is.

11. theEric

Of the four steps, where would you like to start? You need to do all four, but maybe you already know how to plot a line with it's equation. If you don't, let me know, please!

12. theEric

Actually there is a 5th step. You have to check to make sure that the (x,y) point you found to be the intersection actually IS on both lines buy using eliassaab's method for substituting.

13. 85295james

i got three pics to choos from and i dont know how to load them

14. 85295james

B. no solution C. no solution D. infinitely many solutions

15. theEric

Once I put the equations into y=mx+b form, which I will do in a moment, you will see that it must be "no solution". Once you can plot the two equations, you will see why!

16. theEric

For now, think about intersections. As simple as they are. They are the point or points where two things meet! If they never meet, then there are no intersections. If they are together at every point, there are infinately many points of intersection.

17. 85295james

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18. 85295james

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19. theEric

I'll trust that you can algebraically manipulate your equations to look like y=mx+b. $-\frac{1}{3}x+y=-1\rightarrow y=\frac{1}{3}-1$

20. 85295james

no

21. theEric

and $y=\frac{1}{3}x + 4$

22. theEric

No you can't do the algebra? Well lets look at the one line's equation: $-\frac{1}{3}x+y=-1$

23. theEric

Now, you have a goal. You want that "y" to be on its own side, all alone.

24. theEric

There's a rule of thumb, "if you do something to one side, do it to the other". This is so you do the same thing to all sides. By changing each side in the same way, each side will be different from what it was before. BUT each changed side will be equal to each other, so you know it's a legitamate equation! Here's an example, to help you understand what I mean. $5=2+3$ add 10 to both sides $5+(10) = 2+3+(10)$ and solve $15=2+3+10$ $15=15$

25. theEric

Since both sides are equal, each side is okay to work with. And the variables are still the same too. $x=1$add 10 to both sides$x+(10) = 1+(10)$$x+10=11$x still equals 1!

26. theEric

So changing both sides is a great thing to do! I was adding 10 in those examples. To your equation, try adding $\frac{1}{3}x$ to both sides. Here is the equation again: $-\frac{1}{3}x+y=1$

27. theEric

Tell me when your done, so I know you've seen the result, and we can talk about why I told you to add (1/3)x.

28. theEric

Sorry I've taken so long!

29. 85295james

it is cool

30. theEric

Thanks! So, have you done the addition? What did you get?

31. theEric

$-\frac{1}{3}x+\frac{1}{3}x+y=-1+\frac{1}{3}x$is what you want to simplify, here. I took the liberty of adding to both sides :P Just to be clear! :)

32. theEric

Okay, I'll go on without you... But I hope that you'll ask me about anything you don't understand or that you see might be wrong! I got this much, much more understandable equation,$y=-1+\frac{1}{3}x=\frac{1}{3}x-1$ which looks like$y=mx+b$ which is easier to plot by hand.

33. 85295james

so is it C

34. theEric

Nope! Just look at either equation, and the number multiplying x, known as the slope, is $\frac{1}{3}$ The slope is, you go up one unit (along y) and you go over 3 units (along x). The slope goes up and to the right. So look for that.

35. theEric

Since they habe the same slope, they are parallel, and thus cannot intersect.

36. theEric

Thank you!

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