Solve the pair of equations px + qy = 2pq and qx + py = p2 + q2 by the cross multiplication method.
Answers
(i)px+qy=p−q;qx−py=p+q
(ii)ax+by=c;bx+ay=1+c
Medium
Solution
verified
Verified by Toppr
(i) px+qy=p−q…(i)
qx−py=p+q…(ii)
Multiplying (i) by p and (ii) by q we get (iii) and (iv) respectively as
p
2
x+pqy=p
2
−pq…(iii)
q
2
x−pqy=pq+q
2
…(iv)
Adding (iii) and (iv) we get,
(p
2
+q
2
)x=(p
2
+q
2
)
⇒x=1
Substituting value of x in (i), we get
p+qy=p−q
qy=−q
y=−1
Hence, x=1,y=−1
(ii) ax+by=c…(i)
bx+ay=1+c…(ii)
Multiplying (i) by b and (ii) by a we get (iii) and (iv) respectively as
abx+b
2
y=bc…(iii)
abx+a
2
y=a+ac…(iv)
Subtracting (iv) from (iii), 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
x=
(b
2
−a
2
)
b+c(b−a
Class 10
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>>Pair of Linear Equations in Two Variables
>>Reducing a Pair of Equations to Linear Form
>>Solve the following pair of...
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Solve the following pair of linear equations:
(i)px+qy=p−q;qx−py=p+q
(ii)ax+by=c;bx+ay=1+c
Medium
Solution
verified
Verified by Toppr
(i) px+qy=p−q…(i)
qx−py=p+q…(ii)
Multiplying (i) by p and (ii) by q we get (iii) and (iv) respectively as
p
2
x+pqy=p
2
−pq…(iii)
q
2
x−pqy=pq+q
2
…(iv)
Adding (iii) and (iv) we get,
(p
2
+q
2
)x=(p
2
+q
2
)
⇒x=1
Substituting value of x in (i), we get
p+qy=p−q
qy=−q
y=−1
Hence, x=1,y=−1
(ii) ax+by=c…(i)
bx+ay=1+c…(ii)
Multiplying (i) by b and (ii) by a we get (iii) and (iv) respectively as
abx+b
2
y=bc…(iii)
abx+a
2
y=a+ac…(iv)
Subtracting (iv) from (iii), 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
x=
(b
2
−a
2
)
b+c(b−a