An education fund allocates RM10,000 to help 100 students from form three, form five and form six in a school. Each form three student receives RM80, each form five student receives RM120 and each form six student receives RM160. The number of form three students that receive the aid is twice the number of form five students.
(a) Form a system of linear equations from the above information.
(b) Hence, determine the number of students for each form that receive the aid by using matrices.
Form three students --- x
form five students ---- y
form six students ----- 100-x-y
80x + 120y + 160(100-x-y) = 10000
80x + 120y + 16000-160x-160y = 10000
-80x - 40y = -6000
2x + y = 150
but it said:
x = 2y
2(2y) + y = 150
y = 30
x = 2y = 60
remaining students = 100-30-60 = 10
I did not use matrices on such a simple problem because solving it was so much easier using the above method,
but.... if you have to use matrices
our two equations are:
2x + y = 150
x - 2y = 0
2 1 150
1 -2 0 ---->
4 2 300
1 -2 0 --->
1 -2 0
4 2 300 ---> add
1 -2 0
5 0 300
1 -2 0
1 0 60
from the last x=60
in 1st: 60-2y=0
-2y=-60
y=30
etc
(a) To form a system of linear equations, we need to assign variables to the unknown quantities and express their relationships.
Let's assume the number of form five students receiving aid is x.
Given that the number of form three students receiving aid is twice the number of form five students, we can express the number of form three students as 2x.
The total number of students receiving aid is 100, so we have:
x + 2x + x = 100
Simplifying the equation, we get:
4x = 100
Now, we can determine the value of x:
x = 100 / 4
x = 25
So, the number of form five students receiving aid is 25, and the number of form three students is 2x, which is 2 * 25 = 50.
(b) Let's determine the number of students for each form using matrices.
We can set up a matrix equation to represent the given information:
[ 80 120 160 ] [ f3 ] [ 5000 ]
[ f5 ] = [ 3000 ]
[ f6 ] [ 2000 ]
The first matrix represents the amounts given to each student from form three, form five, and form six. The second matrix represents the unknowns, which are the number of students in each form. The third matrix represents the allocated amount of RM10,000.
To solve for the unknown matrix, we can use matrix multiplication and solve the resulting system of equations.
Let's call the first matrix A, the second matrix X, and the third matrix B.
AX = B
In order to solve for X, we need to find the inverse of A:
A^(-1) * AX = A^(-1) * B
X = A^(-1) * B
Let's calculate A^(-1):
[ 80 120 160 ]^(-1) = [ 1/80 -1/120 1/160 ]
[ 0 1/120 -1/160 ]
[ 0 0 1/160 ]
Now, we can calculate X:
X = [ 1/80 -1/120 1/160 ] * [ 5000 ]
[ 0 1/120 -1/160 ] [ 3000 ]
[ 0 0 1/160 ] [ 2000 ]
Simplifying the multiplication, we get:
X = [ 50 ]
[ 25 ]
[ 12.5 ]
Therefore, the number of students in form three, form five, and form six receiving aid are 50, 25, and 12.5, respectively. Since the number of students must be whole numbers, we can round 12.5 to the nearest whole number, which is 13.
So, the final result is that 50 students from form three, 25 students from form five, and 13 students from form six receive the aid.