Pawan Prajapati
Last Activity: 3 Years ago
Solutions:
(i) px + qy = p – q……………(i)
qx – py = p + q……………….(ii)
Multiplying p to equation (1) and q to equation (2), we get
p2x + pqy = p2 − pq ………… (iii)
q2x − pqy = pq + q2 ………… (iv)
Adding equation (iii) and equation (iv),we get
p2x + q2 x = p2 + q2
(p2 + q2 ) x = p2 + q2
x = (p2 + q2)/ p2 + q2 = 1
From equation (i), we get
p(1) + qy = p – q
qy = p-q-p
qy = -q
y = -1
(ii) ax + by= c…………………(i)
bx + ay = 1+ c………… ..(ii)
Multiplying a to equation (i) and b to equation (ii), we obtain
a2x + aby = ac ………………… (iii)
b2x + aby = b + bc…………… (iv)
Subtracting equation (iv) from equation (iii),
(a2 – b2) x = ac − bc– b
x = (ac − bc– b)/ (a2 – b2)
x = c(a-b) –b / (a2+b2)
From equation (i), we obtain
ax +by = c
a{c(a−b)−b)/ (a2 – b2)} +by=c
ac(a−b)−ab/ (a2 – b2)+by=c
by=c–ac(a−b)−ab/(a2 – b2)
by=abc – b2 c+ab/a2-b2
y = c(a-b)+a/a2-b2
(iii) x/a – y/b = 0
ax + by = a2 + b2
x/a – y/b = 0
=> bx − ay = 0 ……. (i)
ax + by = a2 + b2 …….. (ii)
Multiplying a and b to equation (i) and (ii) respectively, we get
b2x − aby = 0 …………… (iii)
a2x + aby = a 3 + ab3 …… (iv)
Adding equations (iii) and (iv), we get
b2x + a2x = a 3 + ab2
x (b2 + a2) = a (a2 + b2) x = a
Using equation (i), we get
b(a) − ay = 0
ab − ay = 0
ay = ab,
y = b
(iv) (a – b)x + (a + b) y = a2 – 2ab – b2
(a + b)(x + y) = a2 + b2
(a + b) y + (a – b) x = a2− 2ab − b2 …………… (i)
(x + y)(a + b) = a 2 + b2
(a + b) y + (a + b) x = a2 + b2 ………………… (ii)
Subtracting equation (ii) from equation (i), we get
(a − b) x − (a + b) x = (a 2 − 2ab − b 2) − (a2 + b2)
x(a − b − a − b) = − 2ab − 2b2
− 2bx = − 2b (b + a)
x = b + a
Substituting this value in equation (i), we get
(a + b)(a − b) +y (a + b) = a2− 2ab – b2
a2 − b2 + y(a + b) = a2− 2ab – b2
(a + b) y = − 2ab
y = -2ab/(a+b)
(v) 152x − 378y = − 74
76x − 189y = − 37
x =(189y-137)/76……………..…(i)
− 378x + 152y = − 604
− 189x + 76y = − 302 ………….. (ii)
Using the value of x in equation (ii), we get
−189(189y−37/76)+76y=−302
− (189)2y + 189 × 37 + (76)2 y = − 302 × 76
189 × 37 + 302 × 76 = (189)2 y − (76)2y
6993 + 22952 = (189 − 76) (189 + 76) y
29945 = (113) (265) y
y = 1
Using equation (i), we get
x = (189-37)/76
x = 152/76 = 2
