Show that the straight lines given by `x(a+2b)+y(a+3b)=a` for different values of `aa n db` pass through a fixed point.

Given equation of variable line is
x(a+2b)+y(a+3b) =a+b
`"or " a(x+y-1)+b(2x+3y-1) =0 " "(1)`
At least one of a and b will be non zero.
`"Let " a ne 0.`
So, equation (1) reduces to
`(x+y-1)+(b)/(a)(2x+3y-1)=0`
`"or " x+y-1+lambda(2x+3y-1)=0, "where" lambda=(b)/(a) " " (2)`
This is the equation of family of straight lines concurrent at point of intersection of lines
`x+y-1 = 0 " " (3)`
`"and " 2x+3y-1 = 0 " " (4)`
Solving (3) and (4), we get x=2, y=-1.
Hence, given variable lines pass through the fixed point (2,-1) for all values of a and b.