(i) Given pair of linear equations is
`" " 3x-y-5=0 " " ...(i)`
and `" " 6x-2y-p=0 " " ...(ii) `
On comparing with ax+by+c=0, we get
`" " a_(1)=3, b_(1)=-1`
and `" " c_(1)=-5 " " `[from Eq. (i)]
`" " a_(2)=6, b_(2)=-2`
and `" " c_(2)=-p" " `[from Eq. (ii)]
Since, the lines represented by these equations are parallel, then
`(a_(1))/(a_(2))=(b_(1))/(b_(2)) !=(c_(1))/(c_(2))`
`(3)/(6)=(-1)/(-2) !=(-5)/(-p)`
Taking last two parts, we get `(-1)/(-2)!=(-5)/(-p)`
`rArr " " (1)/(2)!=(5)/(p)`
`rArr " " P!=10`
Hence, the given pair of linear equations are parallel for all real values of p except 10 i.e., both lines are parallel to each other.
`:. " " (a_(1))/(a_(2))=(b_(1))/(b_(2))!=(c_(1))/(c_(2)) rArr (-1)/(p)=(p)/(-1)!=(-1)/(-1)`
Taking last two parts, we get
`(p)/(-1)!=(-1)/(-1)`
`rArr " " p!=-1`
Taking first two parts, we get
`(-1)/(p) = (p)/(-1)`
`rArr " " p^(2)=1`
`rArr " " p=+- 1`
but `p!=-1`
`:. " " p=1`
Hence, the given pair of linear equations has no solution for p=1.
(iii) Given, pair of linear equations is
-3x+5y-7=0
and `" " 2px-3y-1=0 " " ...(ii)`
On comparing with ax+by+c=0, we get
`a_(1)=-3, b_(1)=5`
and `" " c_(1)=-7 " " `[from Eq. (i)]
`a_(2)=2p, b_(2)=-3 " "` [from Eq. (ii)]
and `" " c_(2)=-1" " `[from Eq. (ii)]
Since , the lines are intersecting at a unique point i.e., it has a unique solution.
`:. " " (a_(1))/(a_(2))!=(b_(1))/(b_(2))`
`rArr " " (-3)/(2p)!=(5)/(-3)`
`rArr " " 9 !=10p`
`rArr " " p!=(9)/(10)`.
Hence, the lines represented by these equations are intersecting at a unique point for all real values of p except `(9)/(10)`
(iv) Given pair or linear equations is
`2x+3y-5=0 " " ...(i)`
and `" " px-6y-8=0 " " ...(ii)`
On comparing with ax+by+c=0, we get
`a_(1)=2, b_(1)=3`
and `" " c_(1)=-5 " " `[from Eq. (i)]
`a_(2)=p, b_(2)=-6`
and `" " c_(2)=-8 " " `[from Eq. (ii)]
Since, the pair of linear equations has a unique solution.
`:. " " (a_(1))/(a_(2))!=(b_(1))/(b_(2))`
`rArr " " (2)/(p)!=(3)/(-6)`
`rArr " " p!=-4`
Hence, the pair of linear equations has a unique solution for all values of p except - 4 i.e., `" " p in R - {-4,}`.
(v) Given pair of linear equations is
`2x+3y=7 " " ...(i)`
and `" " 2px+py=28-qy`
`rArr " " 2px+(p+q)y=28 " " ...(ii)`
On comparing with ax+by+c=0, we get
`a_(1)=2, b_(1)=3`
and `" " c_(1)=-7 " " `[from Eq. (i)]
`a_(2)=2p, b_(2)=(p+q)`
and `" " c_(2)=-28 " " `[from Eq. (ii)]
Since, the pair of equations has infinitely many solutions i.e., both lines are coincident.
`:. " " (a_(1))/(a_(2))=(b_(1))/(b_(2))=(c_(1))/(c_(2))`
`rArr " " (2)/(2p)=(3)/(p+q)=(-7)/(-28)`
Taking first and third parts, we get
`(2)/(2p)=(-7)/(-28)`
`rArr " " (1)/(p)=(1)/(4)`
`rArr " " p=4`
Again, taking last two parts, we get
`(3)/(p+q)=(-7)/(-28) rArr (3)/(p+q)=(1)/(4)`
`rArr " " p+q=12`
`rArr " " 4+q=12 " " `[`:'`p=4]
`:. " " q=8`
Here, we see that the values of p = 4 and q=8 satisfies all three parts.
Hence, the pair of equations has infinitely many solutions for the values of p = 4 and q= 8
