Solve the following pairs of linear equations
(i) x - y = a + b
ax+ by = a^2 - 6^2
Answers
Step-by-step explanation:
px+qy=p−q ....(1)
qx−py=p+q ....(2)
Multiplying (1) by p and (2) by q we get (3) and (4) respectively as
p
2
x+pqy=p
2
−pq ....(3)
q
2
x−pqy=pq+q
2
...(4)
Adding (3) and (4) we get
(p
2
+q
2
)x=(p
2
+q
2
)
⇒x=1
Substituting value of x in (1), we get
p+qy=p−q
qy=−q
y=−1
Hence, x=1,y=−1
(ii) ax+by=c ...(1)
bx+ay=1+c ...(2)
Multiplying (1) by b and (2) by a we get (3) and (4) respectively as
abx+b
2
y=bc ...(3)
abx+a
2
y=a+ac ...(4)
Subtracting (4) from (3), we get
(b
2
−a
2
)y=(b−a)c−a
y=
(b
2
−a
2
)
(b−a)c−a
Substituting value of y in (i), we get
ax+b×
(b
2
−a
2
)
(b−a)c−a
=c
ax=c−
(b
2
−a
2
)
(b−a)bc−ab
ax=
(b
2
−a
2
)
c(b
2
−a
2
)−b
2
c+abc+ab
x=
a(b
2
−a
2
)
cb
2
−ca
2
−b
2
c+abc+ab
x=
a(b
2
−a
2
)
ab+abc−ca
2
x=
(b
2
−a
2
)
b−bc−ca
=
(b
2
−a
2
)
b+c(b−a)