Solve the following pairs of linear equations.

Solve the following pairs of linear equations by cross-multiplication method:

(i) x – y = a + b
ax + by = a² – b²
(ii) 2bx + ay = 2ab
bx – ay = 4ab.

Solution:

(i) x – y = a + b
ax + by = a² – b²
We can write it as
x – y – (a + b) = 0
ax + by – (a² – b²) = 0
By cross multiplication method
x/ [a² – b² + b (a + b)] = y/ [- a (a + b) + a² – b²] = 1/ (b + a)
By further calculation
x/ (a² – b² + ab + b²) = y/ (-a² – ab + a² – b²) = 1/ (a + b)
So we get
x/ [a (a + b)] = y/ [-b (a + b)] = 1/ (a + b)
x = a (a + b)/ (a + b) = a
y = [-b (a + b)]/ (a + b) = – b
Therefore, x = a and y = – b.
(ii) 2bx + ay = 2ab
bx – ay = 4ab
We can write it as
2bx + ay – 2an = 0
bx – ay – 4ab = 0
By cross multiplication method
x/ (- 4a²b – 2a²b) = y/ (-2ab² + 8ab²) = 1/ (-2ab – ab)
By further calculation
x/ -6a²b = y/6ab²= 1/-3ab
So we get
x = -6a²b/ -3ab = 2a
y = 6ab²/-3ab = – 2b
Therefore, x = 2a and b = – 2b.

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