Methods 3/4 Help

[Cammmeron!]

Hey Guys, I just need some help with a few matrices questions that I'm having trouble solving. Any help would be appreciated and If you could show me the working out and explain how to go about solving the question that would help me out a lot. The questions are from Essential Mathematical Methods CAS, Chapter 2, Exercise 2F. Thanks in advance.

1. Find the value of m for which the simultaneous equations have no solutions.

(m+3)x+my=12
(m-1)x+(m-3)y=7

2.find the value of m for which the simultaneous equations have a. an infinite number of solutions and b.have no solution.

3x+my=5 and (m+2)x+5y=m

[Bad Student]

1. For the simultaneous equations to have no solutions, the determinant must equal zero.

This elements in this matrix come from the coefficients of x and y in the equations.





2. For the simulataneous equations to have infinite solutions or no solutions, we make the determine equal zero.





When m = -5, the equations become
and
If you rearrange for y, you will see that they are the same line, therefore they will have infinite solutions.

When m = 3, the equations become
and
If you rearrange for y, you will see that these lines have the same gradient, but differrent y-intercepts. Therefore, they will never touch so these equations have no solutions.

[Cammmeron!]

Thanks Heaps for the help Bad Student . If you don't mind, could you help me another two questions? if you could, could you just explain what the questions are asking of me and explain how to answer them as well as show working out, that way I'm not just copying your answers and not learning anything. That would be much appreciated.

3.
(a). Solve the simultaneous questions 2x-3y=4 and x+ky=2 where k is a constant.
(b). Find the Value of k for which there is not a unique solution.
 
4. Find the Values of b and c for which the equations x+5y=4 and 2x+by=c have:
(a). A unique solution
(b). An infinite set of solutions
(c). No solutions

Could you also explain to me what is meant by "a unique solution", "no solutions" and "infinite solutions"?

[Bad Student]

I'll try to answer question 4. I'm not sure how to explain question 3.

A unique solution means that the lines intersect once. Infinite solutions means that the lines are exactly the same. No solutions means that the lines never intersect.

(a) For a unique solution, the lines need to have different gradients. So the determinant must not equal 0.





(b) For infinite solutions or no solutions, the gradients need to be the same. We can use our answer from part (a) to find the value of b where the gradient is the same.



After that we need to find the value of c when the equations are the same.



If we multiply equation (1) by 2, the equations are the same.
Therefore,

(c) For no solutions, the lines need to have the same gradient but different vertical translations. So to make the vertical translation of the graphs different, we let  .

[Cammmeron!]

Thank you so much for your help Bad Student, I really appreciate it. If there are any other forum users able to help me with question 3 in my last post that would be really helpful, an explanation and working out would also be appreciated .

3.
(a). Solve the simultaneous questions 2x-3y=4 and x+ky=2 where k is a constant.
(b). Find the Value of k for which there is not a unique solution.

[b^3]

a)

b) Now if we rearrange into y=mx+c form we can compare the gradients and y intercepts.

For 'no unique solution' the two lines have to be parallel, this means that there is either no solutions or infinite solutions. So our y intercept won't matter in this case but we are looking for the value of k that makes the two lines have the same gradient. That is

[Cammmeron!]

Thank you b^3, but how would I solve that question using a matrix? because the chapter that question is from has to do with Matrices. Sorry for not mentioning that before hand.

[b^3]

If you write the two equations out in matrix form, then if we have a unique solution then the determinate of the 2x2 matrix will be non-zero, so if we want something other than aunique solution, we can make the determinate equal to zero and solve for k (note that when the determinate is zero we either have no solutions or infinite solutions).

[Cammmeron!]

Thanks b^3, managed to get the right answers . Moving on now, I've just got 2 more questions that I need to finish but they involve 3x3 matrices. Just so I know, Is it necessary in the Methods course to know how to deal with 3x3 matrices? or should I just use my calculator to get the answer?

[Aegis]

I just used my calculator to deal with 3x3 matrices so I would say no but anyone can correct me if I am wrong

[BubbleWrapMan]

Yeah, you just need to calc them, nothing by hand