Write an equation of a line passing through the point representing solution of the pair of linear equations x+y=2 and 2x-y=1, How many such lines can we find?

Explanation

Given pair of linear equations is `" " x+y-2=0 " " ...(i)`
and `" " 2x-y-1=0 " " ...(ii)`
On comparing with ax+by+c=0, we get
`a_(1)=1, b_(1)=1` and `c_(1)=-2 " " ` [from Eq. (i) ]
`a-(2)=2, b_(2)=-1` and `c_(2)=-1 " " `[from Eq. (ii)]
Here, `" "(a_(1))/(a_(2))=(1)/(2), (b_(1))/(b_(2))=(1)/(-1)`
and `" " (c_(1))/(c_(2))=(-2)/(-1)=(2)/(1) rArr (a_(1))/(a_(2))!=(b_(1))/(b_(2))`
So, both lines intersect at a point. Therefore, the pair of equations has a unique solution.
Hence, these equations are consistent.
Now, `" " x+y=2 rArr y=2-x`
If If x=0, then y=2 and if x=2, then y=0
`{:(x,0,2,),(y,2,0,),("Points",A,B,):}`
and `" " 2x-y-1=0 rArr y=2x-1`
If x=0, then y=-1, if `x=(1)/(2)`, then y=0 and if x=1, then y=1
`{:(x,0,1//2,1),(y,-1,0,1),("Points",C,D,E):}`

Plotting the points A (2,0) and B (0,2), we get the straight line AB. Plotting the points C (0, -1) and D(1/2, 0), we get the straight line CD. The lines AB and CD intersect at E (1,1).
Hence, infinite lines can pass through the intersection point of linear equations x+y=2 and 2x-y=1 i.e., E (1,1)like as y=x, 2x+y=3, x+2y=3. So on.