Download 8.Matrices + Simultaneous Equations

Matrices + Simultaneous
Equations
We can solve pairs of simultaneous
equations by using matrices.
4x-3y=11
 4 − 3  x  11

  =  
2x+y=3
 2 1  y   3 
Next multiply both sides of this matrix equation by it’s inverse.
1  1 3


10  − 2 4 
1  1 3  4 − 3  x  1  1 3 11


  = 
 
10  − 2 4  2 1  y  10  − 2 4  3 
 x  1  20   2 
  = 
 =  
 y  10  − 10   − 1
For the following questions say whether
they are consistent or not.
If consistent then solve them, using a
parameter if needed.
1. 2x+5y=12
 2 5  x  12 

  =  
4x-y=2
 4 − 1 y   2 
Multiply by the inverse
- 1  − 1 − 5  2 5  x  - 1  − 1 − 5 12 


  =

 
22  − 4 2  4 − 1 y  22  − 4 2  2 
 x  − 1  − 22   1 
  =

 =  
 y  22  − 44   2 
2. 3x – 5y = 23
9x – 15y = 10
The lines are parallel but do not coincide.
They are not the same line.
Inconsistent.
No solutions.
3. 12x – 6y = 24
4x – 2y = 8
The lines are parallel – in fact the same.
They coincide.
Infinite number of solutions.
2y = 4x-8
y = 2x-4
Using the parameter to gain solutions.
x=λ
y = 2λ-4
Work: page 39 Ex 1G
Exam type question