Question

Check whether the given pair of equations represent intersecting, parallel or coincident lines. Find the solution if the equations are consistent.
2x + y - 5 = 0
3x - 2y - 4 = 0

Answer 1

Given pair of equations are :
2x + y - 5 = 0...........(1)
3x - 2y - 4 = 0.........(2)

from equation (1),
$a_1=2,b_1=1,c_1=-5$
and from equation (2),
$a_2=3,b_2=-2,c_2=-4$

now, $\frac{a_1}{a_2}=\frac{2}{3}$
$\frac{b_1}{b_2}=\frac{1}{-2}$
$\frac{c_1}{c_2}=\frac{-5}{-4}=\frac{5}{4}$

we see, $\frac{b_1}{b_2}\neq\frac{c_1}{c_2}$
hence, pair of equations represent intersecting.


now, multiplying 2 with equation (1) and adding with equation (2)
2(2x + y - 5) + (3x - 2y - 4) = 0
4x + 2y - 10 + 3x - 2y - 4 = 0
7x - 14 = 0 => x = 2
put x = 2 in equation (1),
2 × 2 + y - 5 = 0
y = 1

hence, x = 2 and y = 1

Answer 2

Hi ,

Given : 2x + y - 5 = 0 ,

3x - 2y - 4 = 0 Comparing the given

equations with a1x + b1y + c1 = 0 and

a2x + b2y + c2 = 0

We have a1 = 2 , b1 = 1 , c1 = -5 ,

a2 = 3 , b2 = -2 , c2 = -4 ;

Now ,

a1/a2 = 2/3 ,

b1/b2 = 1/(-2 ) ,

Therefore ,

a1/a2 ≠ b1/b2 ,

So , the system of equations are

consistent and have unique solution .
_____________________________

2x + y - 5 = 0 => y = -2x + 5 ---( 1 )

3x - 2y - 4 = 0---( 2 )

Substitute ( 1 ) in equation ( 2 ) , we get

3x - 2( -2x + 5 ) - 4 = 0

=> 3x + 4x - 10 - 4 = 0

=> 7x - 14 = 0

=> 7x = 14

x = 14/7

x = 2

substitute x = 2 in equation ( 1 ) , we get

y = -2( 2 ) + 5

y = -4 + 5

y = 1

Therefore ,

x = 2 , y = 1

I hope this helps you.

: )