## Calcmathlete 3 years ago Note: This is not a question. Tutorial for Solving Systems of Linear Equations (Two Variables)

1. Calcmathlete

2. Calcmathlete

So Long! Any revisions?

3. Limitless

Gaussian-Jordan elimination.

4. Limitless

Cramer's rule

5. Calcmathlete

Oh ok. I didn't know that that was the official name ;)

6. Limitless

No--those are excluded.

7. Limitless

You should include those.

8. Calcmathlete

I knew about Cramer's Rule, but I wanted to keep this on an Algebra I level. If i added Cramer's Rule and stuff, it would be borderline Algebra II since most classes don't teach Cramer's Rule until Algebra II if at all.

9. Limitless

Who cares if it's Algebra I or Algebra II? Knowledge is knowledge. Cramer's rule involves determinants. Gaussian-Jordan elimination involves matrices and row operations. Neither is that hard.

10. Calcmathlete

I suppose. THe tutorial was just so long and it would've taken a while teaching determinants and all of that. Maybe next time. Thanks for the suggestion though :)

11. mathslover

it looked great .. keep it up calcmathlete. gr8

12. waterineyes

There is one more method... Can I suggest??

13. Calcmathlete

@waterineyes Sure @mathslover Thank you :)

14. waterineyes

@Calcmathlete when two lines are parallel then they have infinitely many solutions and not no solution..

15. Calcmathlete

Really? I thought that since parallel lines never intersect and a solution is an intersection, there would be no solution?

16. waterineyes

Yes.. That is why they are parallel lines and can have many infinitely solutions.. I will give the examples of both..

17. waterineyes

What can you say about these two lines having equations: 2x + 3y = 9 And, 4x + 6y = 18 Now, according to you they have no solution??

18. Calcmathlete

I thought so? Parallel lines never intersect no? And since they never intersect, there are no solutions? I thought it was infinitely many solutions when it was the same line.

19. waterineyes

I will show how how you can tell about the Nature Of Roots..

20. Calcmathlete

Wait, I just realized. The equations you gave are the same line right? That would be an infinitely many solutions. Oops

21. waterineyes

NATURE OF ROOTS: $ax_1 + by_1 + c_1 = 0$ $ax_2 + by_2 + c_2 = 0$ 1) UNIQUE SOLUTION: $\huge \color{blue}{\frac{a_1}{a_2} \ne \frac{b_1}{b_2}}$ 2) INFINITELY MANY SOLUTIONS: $\huge \color{violet}{\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}}$ 3) NO SOLUTION: $\huge \color{green}{\frac{a_1}{a_2} = \frac{b_1}{b_2} \ne \frac{c_1}{c_2}}$

22. waterineyes

Now I will tell about that other Method...

23. Calcmathlete

Alright. Thanks! I learn something new everyday! :) So to make sure I have everything completely understood. Would this system be infinitely many or no solution? x + y = 3 2x + 2y = 4

24. waterineyes

No solution..

25. waterineyes

Parallel Lines + Overlapping Lines = Infinitely Many Solutions.. Parallel Lines + Non Overlapping Lines = No Solution..

26. Calcmathlete

Ok. That's what I thought. But using that logic, my original example in my tutorial: 2x + y = 9 4x + 2y = 4 I said that it was no solution. It was in actuality infinitely many solutions?

27. waterineyes

It is having no solution..

28. Calcmathlete

Then what was the original problem? Oh wait. You're original comment says that you were just gonna show another method. My bad :)

29. waterineyes

Yes I will.. Do you know the name of this method??

30. waterineyes

The name of this method is : SOLVING SIMULTANEOUS EQUATIONS BY CROSS MULTIPLICATION... Any Idea if you have heard this before??

31. Calcmathlete

No? I've only learned Graphing, Substitution, Elimination, and Cramer's Rule. Also I'm starting to learn Gaussian Elimination. I'm interested now :)

32. waterineyes

Can you wait?? It will take time to explain to write and to give an example too.. May be I have to draw it too to explain more..

33. Calcmathlete

Alright. np :)

34. waterineyes

SOLVING SIMULTANEOUS EQUATIONS BY CROSS MULTIPLICATION METHOD: This method is simple one and not too hard but one must know this method.. Firstly we have to write the given equations in Standard Form that is: $\huge ax_1 + by_1 + c_1 = 0$ $\huge ax_2 + by_2 + c_2 = 0$ Then, make the following fractions: $\frac{x}{(b_1c_2 - b_2c_1)} = \frac{y}{(c_1a_2 - c_2a_1)} = \frac{1}{(a_1b_2 - a_2b_1)}$ Once you get this, firstly to find x, equate First Fraction with the Last Fraction.. And to get y, equate second and the Last Fraction..

35. waterineyes

I will give you an example that will make sense of this method..

36. waterineyes

I am taking your Elimination Method Example @Calcmathlete .. 2x - y = 5 3x + 2y = 5 First Step : Write them in Standard Form: 2x - y - 5 = 0 3x + y -5 = 0 Second Step: Identify the values of $$a_1, b_1, c_1, a_2, b_2, c_2$$.. Third Step: Plug in the values in the formula: $\frac{x}{(-1)(-5) - (1)(-5)} = \frac{y}{(-5)(3) - (-5)(2)} = \frac{1}{(2)(1) - (3)(-1)}$ $\frac{x}{10} = \frac{y}{-5} = \frac{1}{5}$ Third Step: To find x, equate: $\frac{x}{10} = \frac{1}{5}$ So, $$x = 2$$ To get y, equate: $\frac{y}{-5} = \frac{1}{5}$ $y = -1$ So, by Solving with Cross Multiplication Method, we get: $$(x, y) = (2, -1)$$...

37. waterineyes

Sorry there will not come 2y it is simply y in the second equation of first step..

38. waterineyes

Second equation that I have taken is: 3x + y = 5 and not: 3x + 2y = 5 Please note this point.. Solution is correct and is according to 3x + y = 5

39. Calcmathlete

Oh. Ok. I see where the name of cross multiplication comes from. I guess it's just a matter of remembering the order/pairings. Is there a specific time where this would be quicker than the other methods like how you know when to use substitution and elimination?

40. waterineyes

We always use Substitution Method when in any of the equations given, The coefficient of either y or x is 1 it becomes easy to solve.. Like: 3x + 2y = 13 x + y = 5 Now just find x or y from second equation and put it in first equation you have y = 2 and x = 3.. For Elimination: If in the given equations: If you can get the same coefficient of x or y by multiplying something than you will use Elimination: Like: 5x + 4y = 18 10x + 7y = 34 See here when you multiply first equation by 2 then coefficient of x are becoming equal.. 10x + 8y + 36 10x + 7y = 34 Subtract them: y = 2 and so, x = 2.. You can use cross multiplication method also if you are good in multiplication and signs.. Because we always do mistakes with the signs...

41. waterineyes

I have one pictorial representation of cross multiplication method: |dw:1341466088360:dw|

42. waterineyes

|dw:1341466207150:dw|

43. Calcmathlete

I get it! :D Parts of it remind me a bit of Cramer's Rule and parts of it remind me of the Law of Sines.

44. Calcmathlete

Once again, thank you!

45. dpaInc

the author of kramer's rule...

46. agentx5

@dpaInc lol This is an interesting read btw, lots of good information. Want to put it all together into one collaborative document in a new topic/thread @Calcmathlete ? I want to give you a medal to afterall :-D (gave the previous one to @waterineyes for showing me something I didn't know)

47. zepp

Ishaan gave me an advice when I made my first tutorial, he told me to keep it short, a post that's too long would discourage some people to read through it, even if it's useful;, but good job! :D

48. waterineyes

@agentx5 But you know now that's the best thing.. If anyone has knowledge lets other light their candle on it..

49. Calcmathlete

I'll probably do that later when I get all of the other methods that I missed. In any case, it's long...lol @zepp Yeah. I noticed. Maybe I should pick a topic not as long like split this into just elimination method since that in itself is pretty long.

50. waterineyes

There is nothing to do with Long and short... If someone want to increase his or her knowledge and has interest then he or she will read the whole.. But the important thing is you should write the words or give the tutorial in a way that can be easily understandable by the reader... Reader gets bored only when he or she is not able to understand what you have written..

51. zepp

True that :)

Good Job,@Calcmathlete .

53. Trexy

Thanks @Calcmathlete

54. AnnaHood

so how would you do Solve, using the substitution method y = 2x + 6 3x – y = 6 A. There are an infinite number of solutions. B. (14, 12) C. (12, 30) D. There is no solution. I think it might be D but I'm not sure

55. iLorita

Can you help me with this one? Which is a solution of the equation? 2x – y = 5 @Calcmathlete

56. harleyk_97

@iLorita have you gotten the answer on this? I am on a problem and I don't know how to do it. its x-2y=2 and y=-2x+5 and I need to know how many solutions this equation has. I don't know though.

57. Marrivonne

What is the solution set of y = x2 + 2x + 7 and y = x + 7?

58. Marrivonne

{(0, 7), (-1, 6)} {(0, 7), (-7, 0)} {(0, 7), (1, 8)} {(-2, 0), (4, 0)}

59. waterineyes

Wow..!! I just want to confirm that above "waterineyes" is same as the one who is commenting now!! :P I am trying to remember when I had posted these things, silly me.!! :)

60. waterineyes

You have started asking questions on a Tutorial? :P Buddy, give some respect to this Tutorial... @Marrivonne ..