Identify the coordinates of Q and R using the midpoint formula and creating two equations for the following case:There are two equations: x - 2y -...

A point P is given with coordinates
P(1,1).

P(1,1) is the mid point of Q(x1, y1) and R(x2,y2),
where Q is on x-2y-4 and R is on x+y = 5.

Therefore by mid
point formula (x1+x2)/2 = 1 and (y1+y2)/2 = 1

x1+x2 = 2
..(1)and y1+y2 = 2 ..(2).

Since (x1, y1) lies on x-2y-4 =
0, or x-2y = 4 and (x2,y2) lies on x+y = 5, we have:

x1-2y1
= 4 ...(3) and  x2+y2 = 5....(4).

We solve the system of
equations (1), (2) (3) and (4):

(1)-(3): x2+2y1 = -2...
(5).

(4)-(2): x2-y1 =
3.........(6)

(5)-(6): 3y1 = -5. So y1 = -5/3. Put y1 =
-5/3 in (6), and we get x2 = 3+y1 = 3-5/3 = 4/3.

Put x2 =
4/3 in (1), x1+x2 = 2. So x1 =2-x2 =  2-4/3 = 2/3.

Put y1 =
-5/3 in (2): y1+y2 = 2. So y1 = 2-y2 = 2-(-5/3) =
11/3.

Therefore Q(x1,y1) = Q(2/3, -5/3) which lies on
x-2y-4 = 0 and R(x2,y2) = (4/3, 11/3) which lies on x+y = 5.