Question

Solve by method of equating the co effiecients : x-y+2/2=x+y+7/3=2x-y-8/5

Answer 1

(x - y + 2) / 2  =  (x + y + 7) / 3  =  (2 x - y - 8) / 5  = k  (say)

x - y + 2 - 2 k = 0      -- (1)
x + y + 7 - 3 k  = 0    -- (2)
2x - y - 8 - 5 k = 0     -- (3)

Let  a and b be real values such that:
  a (x - y - 2 k +2) + b (x + y + 7 - 3k) = 2 x - y - 5 k - 8
  x (a + b) + y(-a +b) - k (2a + 3b) + 2a + 7b = 2 x - y - 5 k - 8    --- (4)

By method of equating coefficients, we equate:
    a + b = 2       -- (5)
    b - a = -1           --- (6)
    2 a + 7 b - 2 a k - 3 b k = - 5 k - 8
    k = (2 a + 7b + 8) / (2a + 3 b - 5)      -- (7)

Solving (5) and (6), we get:  b = 1/2    a = 3/2

Substituting these in (7):
         k = 29/(-1)   = - 29
so we have   x - y + 60 = 0     --- (8)
                    x + y + 94 = 0         --- (9)
                    2x - y + 137 = 0      --- (10)

m (x - y + 60) + n (x + y + 94) = 2 x - y + 137
x (m+n) + y (n - m) + 60 m + 94 n = 2 x - y + 137

Compare coefficients:    m + n = 2
                                      n - m = - 1
Solving them:       m = 3/2       n = 1/2
Substitute these in :       60 m + 94 n =  137,
 it is valid. so the solution is good.

Solving (8) and (9) and (10)  we get : x = - 77  and  y = -17