Solve simultaneous linear equations: (i) px + qy = p – q

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Solve simultaneous linear equations:

(i) px + qy = p – q
qx – py = p + q
(ii) x/a – y/b = 0
ax + by = a2 + b2.

Solution:

(i) px + qy = p – q …. (1)
qx – py = p + q ….. (2)
Now multiply equation (1) by p and (2) by q
p2x + pqy = p2 – pq
q2x – pqy = pq + q2
By adding both the equations
(p2 + q2) x = p2 + q2
By further calculation
x = (p2 + q2)/ (p2 + q2) = 1
From equation (1)
p × 1 + qy = p – q
By further calculation
p – qy = p – q
So we get
qy = p – q – p = – q
Here
y = -q/q = – 1
Therefore, x = 1 and y = – 1.
(ii) x/a – y/b = 0
ax + by = a2 + b2
We can write it as
x/a – y/b = 0
Taking LCM
(bx – ay)/ab = 0
By cross multiplication
bx – ay = 0 …… (1)
ax + by = a2 + b2 ….. (2)
Multiply equation (1) by b and equation (2) by a
b2x – aby = 0
a2x + aby = a2 + ab2
By adding both the equations
(a2 + b2)x = a2+ ab2 = a (a2 + b2)
So we get
x = a (a2 + b2)/ a2 + b2 = a
From equation (2)
b × a – ay = 0
By further calculation
ab – ay = 0
ay = ab
So we get
y = ab/a = b
Therefore, x = a and y = b.

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