Fsc Part 1 Mathematics (Complete Solution)

# Q3 Solve the following linear equations.

 (i)   2x1-3x2=5 5x1+x2=4

SOLUTION:

 2 -3 5 1

 x1 x2

=

 5 4

Let   A=

 2 -3 5 1

; X=

 x1 x2

;    B=

 5 4

AX=B

=> X=A-1B

Then A-1=
lAl=

 2 -3 5 1

=2+15 =17≠0

 1 3 -5 2

SO,  A-1=
=
 1 17

 1 3 -5 2

=

 1/17 3/17 -5/17 2/17

A-1B=

 1/17 3/17 -5/17 2/17

 5 4

=

 17/17 -17/17

=

 1 -1

Therefore X=A-1B

 x1 x2

=

 1 -1

Thus x1=1 and x2=-1

 (ii)   4x1+3x2=5 3x1-x2=7

SOLUTION:

 4 3 3 -1

 x1 x2

=
<
 5 7

Let   A=

 4 3 3 -1

; X=

 x1 x2

;    B=

 5 7

AX=B

=> X=A-1B

Then A-1=
lAl=

 4 3 3 -1

=-4-9 =-13≠0

 -1 -3 -3 4

SO,  A-1=
= -
 1 13

 -1 -3 -3 4

=

 1/13 3/13 3/13 -4/13

A-1B=

 1/13 3/13 3/13 -4/13

 5 7

=

 26/13 -13/13

=

 2 -1

Therefore X=A-1B

 x1 x2

=

 2 -1

Thus x1=2 and x2=-1

 (iii)   3x-5y=1 -2x+y=-3

SOLUTION:

 3 -5 -2 1

 x y

=
<
 1 -3

Let   A=

 3 -5 -2 1

; X=

 x y

;    B=

 1 -3

AX=B

=> X=A-1B

Then A-1=
lAl=

 3 -5 -2 1

=3-10 =-7≠0

 1 5 2 3

SO,  A-1=
= -
 1 7

 1 5 2 3

=

 -1/7 -5/7 -2/7 -3/7

A-1B=

 -1/7 -5/7 -2/7 -3/7

 1 -3

=

 -14/7 -7/7

=

 2 1

Therefore X=A-1B

 x y

=

 2 1

Thus x1=2 and x2=1