Question

In each of the following pairs of linear equations determine whether they have a unique

solution, no solution or infinite number of solutions. In case there is a unique solution, find it:

a) 6x + 5y = 4 , 9x +7.5y = 6
b) x + 3y = 10
, -2x + 5y = 13

c) x - 2 y = 4
, 3x – 6y= 19​

Answer 1

Step-by-step explanation:

The given system of equations may be written as

2x+3y−7=0

6x+5y−11=0

The given system of equations is of the form

a

1

x+b

1

y+c

1

=0

a

2

x+b

2

y+c

2

=0

where, a

1

=2,b

1

=3,c

1

=−7 and a

2

=6,b

2

=5,c

2

=−11

We have,

a

2

a

1

=

6

2

=

3

1

and

b

2

b

1

=

5

3

Clearly,

a

2

a

1



=

b

2

b

1

So, the given system of equations has a unique solution.

To find the solution, we use the cross-multiplication method.

By cross-multiplication, we have

3×−11−5×−7

x

=

2×−11−6×−7

−y

=

2×5−6×3

1

⇒

−33+35

x

=

−22+42

−y

=

10−18

1

⇒

2

x

=

−20

y

=

−8

1

⇒x=

8

−2

=−

4

1

and y=

−8

−20

=

2

5

Hence, the given system of equations has a unique solution given by x=−

4

1

,y=

2

5

.