O-Level Additional Math

[iFruit]

SKT:

Hi,

Given that x and y satisfy the simultaneous equations
mx + (m-1)y = 10,
(m-2)x + 3my = 20.

(a) If the equations have no unique solution, find the values of m.
(b) If the equations have no solutions, find the value of m.

TIA.

Using Cramer’s rule

x = [30m - 20(m-1)]/ [3m² – (m² -3m +2)] = (10m+20)/ (2m² + 3m -2)

y = [20m -10(m-2)] / (2m² + 3m -2) = (10m+20)/ (2m² + 3m -2)


Recall that equation has
1) infinite solutions when (10m+20) = 0 and (2m²+ 3m -2)=0
2) No solutions when (10m+20) # 0 and (2m² + 3m -2)=0

(2m² + 3m -2)= (2m-1)(m+2) =0----> m=-2 or m=1/2


When m=-2, 10m+20 = 0,
when m=1/2, 10m+20 = 25


So infinite solutions when m=-2
No solutions when m=1/2

[atutor2001]

SKT:

Hi,

Given that x and y satisfy the simultaneous equations
mx + (m-1)y = 10,
(m-2)x + 3my = 20.

(a) If the equations have no unique solution, find the values of m.
(b) If the equations have no solutions, find the value of m.

TIA.

Below is an ELEMENTARY approach

For linear equation : y = mx +c

(a) 2 lines will have no unique solution if they are the same line i.e. both lines have the same gradient, m and the same y-intercept, c

mx + (m-1)y = 10 that is: y = -mx/(m-1) +10/(m-1)
(m-2)x + 3my = 20 that is: y = -(m-2)/(3m) + 20/(3m)

If the gradients are the same :

-mx/(m-1) = -(m-2)/(3m)
3m² = m²-2m-m+2
2m²+3m-2 = 0
(2m-1)(m+2) = 0
m = 1/2 or -2

If the y-intercepts are the same :

10/(m-1) = 20/(3m)
30m = 20m - 20
m = -2

Therefore, there is no unique solution if m = -2

(b) 2 lines will have no solution if they are parallel i.e. same gradient but different y-intercept.

Therefore, there is no solution if m = 1/2

[SKT]

Hi,


However, the answer provided for (a) is 1/2, -2, wonder what's the meaning of \"no unique solution\". Are the two lines parallel or collinear or both?

TIA

[iFruit]

SKT:

Hi,


However, the answer provided for (a) is 1/2, -2, wonder what's the meaning of \"no unique solution\". Are the two lines parallel or collinear or both?

TIA

That is right.

No Unique solution means, either no solution or infinite solutions.

when m=-2, there are infinite solutions (when it's the same line)

so both 1/2 and -2 are answers for (a).

[atutor2001]

SKT:

Hi,


However, the answer provided for (a) is 1/2, -2, wonder what's the meaning of \"no unique solution\". Are the two lines parallel or collinear or both?

TIA

NO UNIQUE solution means these 2 lines are collinear (2 lines overlapping each other, so every point on the lines is a solution or can say there are infinite number of solutions)
So answer for (a) is -2 ONLY

NO solution means these 2 lines are parallel but NOT collinear (these 2 lines will never meet which is why there is no solution)
So answer for (b) is 1/2

[iFruit]

atutor2001:

SKT:

Hi,


However, the answer provided for (a) is 1/2, -2, wonder what's the meaning of \"no unique solution\". Are the two lines parallel or collinear or both?

TIA

NO UNIQUE solution means these 2 lines are collinear (2 lines overlapping each other, so every point on the lines is a solution or can say there are infinite number of solutions)
So answer for (a) is -2 ONLY

NO solution means these 2 lines are parallel but NOT collinear (these 2 lines will never meet which is why there is no solution)
So answer for (b) is 1/2

A system of linear equations will have one of the three possible types of solutions

1) Unique solution
2) Infinite solutions
3) no solution

when you say \"no unique solution\" (the opposite of 1 in the list ) it should cover the cases of both 2 and 3 above.

so for (a) in the question, both 1/2 and -2 are the answers, IMHO.

[atutor2001]

iFruit:

atutor2001:

[quote=\"SKT\"]Hi,


However, the answer provided for (a) is 1/2, -2, wonder what's the meaning of \"no unique solution\". Are the two lines parallel or collinear or both?

TIA

NO UNIQUE solution means these 2 lines are collinear (2 lines overlapping each other, so every point on the lines is a solution or can say there are infinite number of solutions)
So answer for (a) is -2 ONLY

NO solution means these 2 lines are parallel but NOT collinear (these 2 lines will never meet which is why there is no solution)
So answer for (b) is 1/2

A system of linear equations will have one of the three possible types of solutions

1) Unique solution
2) Infinite solutions
3) no solution

when you say \"no unique solution\" (the opposite of 1 in the list ) it should cover the cases of both 2 and 3 above.

so for (a) in the question, both 1/2 and -2 are the answers, IMHO.[/quote]I agree with you if we are using mathematical definition whereby if the coefficient determinant is zero, then there is no unique solution. But this is beyond O level elementary math. O level elementary math only cover the concept of parallel and collinear. I remembered by \"no unique solutions\", the question would mean that \"there are solutions but the solutions are not unique\". Anyway, I have check with my youngest kid on his understanding and he concurred. Hope we are right.

Hope this link can help with regard to the coverage of math at elementary level.

http://www.freeganita.com/en/cogeo/7_4.htm

[iFruit]

atutor2001:

iFruit:

[quote=\"atutor2001\"]
NO UNIQUE solution means these 2 lines are collinear (2 lines overlapping each other, so every point on the lines is a solution or can say there are infinite number of solutions)
So answer for (a) is -2 ONLY

NO solution means these 2 lines are parallel but NOT collinear (these 2 lines will never meet which is why there is no solution)
So answer for (b) is 1/2

A system of linear equations will have one of the three possible types of solutions

1) Unique solution
2) Infinite solutions
3) no solution

when you say \"no unique solution\" (the opposite of 1 in the list ) it should cover the cases of both 2 and 3 above.

so for (a) in the question, both 1/2 and -2 are the answers, IMHO.

I agree with you if we are using mathematical definition whereby if the coefficient determinant is zero, then there is no unique solution. But this is beyond O level elementary math. O level elementary math only cover the concept of parallel and collinear. I remembered by \"no unique solutions\", the question would mean that \"there are solutions but the solutions are not unique\". Anyway, I have check with my youngest kid on his understanding and he concurred. Hope we are right.

Hope this link can help with regard to the coverage of math at elementary level.

http://www.freeganita.com/en/cogeo/7_4.htm[/quote]Hi atutor2001,

I hate to argue over such a trivial matter ( and you may be right about o'level coverage), but if you look at the questions closely,

(a) If the equations have no unique solution, find the values of m.
(b) If the equations have no solutions, find the value of m.


The question clearly expects multiple answers for (a) and a single answer for (b).

In addition, the question (a) is asking for the values of m for which there is no unique solution. Then you have to list all the values of m for which there is no unique(single) solution.

Just my 2 cents.

Regards.

[atutor2001]

iFruit:

Hi atutor2001,

I hate to argue over such a trivial matter ( and you may be right about o'level coverage), but if you look at the questions closely,

(a) If the equations have no unique solution, find the values of m.
(b) If the equations have no solutions, find the value of m.


The question clearly expects multiple answers for (a) and a single answer for (b).

In addition, the question (a) is asking for the values of m for which there is no unique solution. Then you have to list all the values of m for which there is no unique(single) solution.

Just my 2 cents.

Regards.

Hi iFruit

My apology, I didn't notice the \"values\" in the question. That being the case, my interpretation of the question is wrong. Then both 1/2 and -2 should be the answer for (a).

Regards.

[SKT]

Hi iFruit and atutor2001,


Thank you very much for the clear explanation.

Regards.