RD Sharma Solutions Class 10 Pair Of Linear Equations In Two Variables Exercise 3.4

Exercise 3.4

Q.1: x + 2y + 1 = 0 and 2x – 3y – 12 = 0

Soln:

x+2y+1 =0 …………………………………….(i)

2x-3y-12=0………………………………….. (ii)

Here a₁= 1 , b₁= 2 , c₁= 1

a₂= 2 , b₂= -3 , c₂= -12

By cross multiplication method,

\(\frac{x}{-24+3}=\frac{-y}{-12-2}=\frac{1}{-3-4}\)

\(\frac{x}{-21}=\frac{-y}{-14}=\frac{1}{-7}\)

Now,

\(\frac{x}{-21}=\frac{1}{-7}\)

=x= 3

And,

\(\frac{-y}{-14}=\frac{1}{-7}\)

=y=-2

The solution of the given system of equation is 3 and -2 respectively.

Q.2: 3x + 2y + 25 = 0, 2x + y + 10 = 0

Soln:

3x+2y+25 =0 …………………………………….(i)

2x+y+10=0………………………………….. (ii)

Here a₁= 3 , b₁= 2 , c₁= 25

a₂= 2 , b₂= 1 , c₂= 10

By cross multiplication method,

\(\frac{x}{20-25}=\frac{-y}{30-50}=\frac{1}{3-4}\)

\(\frac{x}{-5}=\frac{-y}{-20}=\frac{1}{-1}\)

Now,

\(\frac{x}{-5}=\frac{1}{-1}\)

=x= 5

And,

\(\frac{-y}{-20}=\frac{1}{-1}\)

=y=-20

The solution of the given system of equation is 5 and -20 respectively.

Q.3: 2x + y = 35, 3x + 4y = 65

Soln:

2x+y= 35 ……………………………….(i)

3x+4y=65…………………………….. (ii)

Here a₁= 2 , b₁= 1 , c₁= 35

a₂= 3 , b₂= 4 , c₂= 65

By cross multiplication method,

\(\frac{x}{-65+140}=\frac{-y}{-130+105}=\frac{1}{8-3}\)

\(\frac{x}{75}=\frac{-y}{-25}=\frac{1}{5}\)

Now,

\(\frac{x}{75}=\frac{1}{5}\)

=x= 15

And,

\(\frac{-y}{-25}=\frac{1}{5}\)

=y=5

The solution of the given system of equation is 15 and 5 respectively.

Q.4: 2x – y – 6 = 0, x – y – 2 = 0

Soln:

2x-y= 6 ……………………………….(i)

x-y=2…………………………….. (ii)

Here a₁= 2 , b₁= -1 , c₁= 6

a₂= 1 , b₂= -1 , c₂= 2

By cross multiplication method,

\(\frac{x}{2-6}=\frac{-y}{-4+6}=\frac{1}{-2+1}\)

\(\frac{x}{-4}=\frac{-y}{2}=\frac{1}{-1}\)

Now,

\(\frac{x}{-4}=\frac{1}{-1}\)

=x= 4

And,

\(\frac{-y}{2}=\frac{1}{-1}\)

=y=2

The solution of the given system of equation is 4 and 2 respectively.

Q5: \(\frac{x+y}{xy}=2\), \(\frac{x-y}{xy}=6\)

Soln:

\(\frac{x+y}{xy}=2\)

= \(\frac{1}{x}+\frac{1}{y}=2\) ……………………………….. (i)

\(\frac{x-y}{xy}=6\)

= \(\frac{1}{x}-\frac{1}{y}=6\) ………………………………… (ii)

Taking \(\frac{1}{x}\) =u

Taking \(\frac{1}{y}\) =v

= u+v=2 …………………………… (iii)

= u-v= 6 ……………………………. (iv)

By cross multiplication method,

\(\frac{u}{6-2}=\frac{-v}{6+2}=\frac{1}{-1-1}\)

\(\frac{u}{4}=\frac{-v}{8}=\frac{1}{-2}\)

Now,

\(\frac{u}{4}=\frac{1}{-2}\)

=u= -2

And,

\(\frac{-v}{-8}=\frac{1}{-2}\)

=v=4

\(\frac{1}{u}\) =x=\(\frac{-1}{2}\)

\(\frac{1}{v}\) =y=\(\frac{1}{4}\)

The solution of the given system of equation is \(\frac{-1}{2}\)  and \(\frac{1}{4}\) respectively.

Q.6: ax+by=a-b, bx-ay=a+b

Soln:

ax+by=a-b……………………………….(i)

bx-ay=a+b…………………………….. (ii)

Here a₁= a , b₁= b , c₁= a-b

a₂= b , b₂= -a , c₂= a+b

By cross multiplication method,

\(\frac{x}{-ab-b^{2}+ab-a^{2}}=\frac{-y}{-a^{2}-ab-b^{2}+ab}=\frac{1}{-a^{2}-b^{2}}\)

\(\frac{x}{-b^{2}-a^{2}}=\frac{-y}{-a^{2}-b^{2}}=\frac{1}{-a^{2}-b^{2}}\)

Now,

\(\frac{x}{-ab-b^{2}+ab-a^{2}}=\frac{1}{-a^{2}-b^{2}}\)

=x= 1

And,

\(\frac{-y}{-a^{2}-ab-b^{2}+ab}=\frac{1}{-a^{2}-b^{2}}\)

=y=-1

The solution of the given system of equation is 1 and -1 respectively.

Q.7: x+ay-b=0, ax-by-c=0

Soln:

x+ay-b=0 ……………………………….. (i)

ax-by-c=0………………………………. (ii)

Here a₁= 1 , b₁= a , c₁= -b

a₂= a , b₂= -b , c₂= -c

By cross multiplication method,

\(\frac{x}{-ac-b^{2}}=\frac{-y}{-c+ab}=\frac{1}{-a^{2}-b}\)

Now,

\(\frac{x}{-ac-b^{2}}=\frac{1}{-a^{2}-b}\)

=x= \(\frac{b^{2}+ac}{a^{2}+b}\)

And,

\(\frac{-y}{-c+ab}=\frac{1}{-a^{2}-b}\)

=y=\(\frac{-c+ab}{a^{2}+b}\)

The solution of the given system of equation is  \(\frac{b^{2}+ac}{a^{2}+b}\) and \(\frac{-c+ab}{a^{2}+b}\)  respectively.

Q8

ax+by=a²

bx+ay=b²

Soln:

ax+by=a²……………………………….(i)

bx+ay=b²…………………………….. (ii)

Here a₁= a , b₁= b , c₁= a²

a₂= b , b₂= a , c₂= b²

By cross multiplication method,

\(\frac{x}{-b^{2}+a^{2}}=\frac{-y}{-ab^{2}-a^{2}b}=\frac{1}{a^{2}-b^{2}}\)

Now,

\(\frac{x}{-b^{2}+a^{2}}=\frac{1}{a^{2}-b^{2}}\)

=x= \(\frac{a^{2}+ab+b^{2}}{a+b}\)

And,

\(\frac{-y}{-ab^{2}-a^{2}b}=\frac{1}{a^{2}-b^{2}}\)

=y=-\(\frac{-ab(a-b)}{(a-b)(a+b)}\)

The solution of the given system of equation is  \(\frac{a^{2}+ab+b^{2}}{a+b}\) and \(\frac{-ab(a-b)}{(a-b)(a+b)}\) respectively.

Q9

\(\frac{5}{x+y}-\frac{2}{x-y}=-1\)

\(\frac{15}{x+y}+\frac{7}{x-y}=-10\)

Soln:

Let \(\frac{1}{x+y}=\) =u

Let \(\frac{1}{x-y}=\) =v

The given system of equations are :

5u-2v=-1

15u+7v =10

Here a₁= 5, b₁= -2 , c₁= 1

a₂= 15 , b₂= 7 , c₂= -10

By cross multiplication method,

\(\frac{u}{20-7}=\frac{-v}{-50-15}=\frac{1}{35+30}\)

\(\frac{u}{13}=\frac{-v}{-65}=\frac{1}{65}\)

Now,

\(\frac{u}{13}=\frac{1}{-65}\)

=u= \(\frac{1}{5}\)

\(\frac{1}{u}=\) =x+y

=x+y=5 ………………………..(i)

And,

\(\frac{-v}{-65}=\frac{1}{-65}\)

=v=1

\(\frac{1}{v}=\) =x-y

=x-y =1 ………………………… (ii)

Adding equation (i) and (ii)

2x= 6

=x=3

Putting the value of x in equation (i)

3+y=5

=y=2

The solution of the given system of equation is 3 and 2 respectively.

Q10

\(\frac{2}{x}+\frac{3}{y}=13\)

\(\frac{5}{x}-\frac{4}{y}=-2\)

Soln:

Let \(\frac{1}{x}\) =u

Let \(\frac{1}{y}\) =v

The given system of equations becomes:

2u+3v=13 ……………………………… (i)

5u-4v=-2…………………………………. (ii)

By cross multiplication method,

\(\frac{u}{6-52}=\frac{-v}{4+65}=\frac{1}{-8-15}\)

\(\frac{u}{-46}=\frac{-v}{69}=\frac{1}{-23}\)

Now,

\(\frac{u}{-46}=\frac{1}{-23}\)

=u= 2

\(\frac{1}{u}=\) =\(\frac{1}{x}\)

=x=\(\frac{1}{2}\)

And,

\(\frac{-v}{69}=\frac{1}{-23}\)

=v=3

\(\frac{1}{v}\)= \(\frac{1}{y}\)

=y =\(\frac{1}{3}\)

The solutions of the given system of equations are \(\frac{1}{2}\) and \(\frac{1}{3}\) respectively.

Q11

\(\frac{57}{x+y}+\frac{6}{x-y}=5\)

\(\frac{38}{x+y}+\frac{21}{x-y}=9\)

Soln:

Let \(\frac{1}{x+y}=\) =u

Let \(\frac{1}{x-y}=\) =v

The given system of equations are :

57u+6v=5

38u+21v =9

Here a₁= 57, b₁= 6 , c₁= -5

a₂= 38 , b₂=21 , c₂= -9

By cross multiplication method,

\(\frac{u}{-54+105}=\frac{-v}{-513+190}=\frac{1}{1193-228}\)

\(\frac{u}{51}=\frac{-v}{-323}=\frac{1}{969}\)

Now,

\(\frac{u}{51}=\frac{1}{969}\)

=u= \(\frac{1}{19}\)

\(\frac{1}{u}\) =x+y

=x+y=19 ………………………..(i)

And,

\(\frac{-v}{-323}=\frac{1}{969}\)

=v=\(\frac{1}{3}\)

\(\frac{1}{v}=\) =x-y

=x-y =3 ………………………… (ii)

Adding equation (i) and (ii)

2x= 22

=x=11

Putting the value of x in equation (i)

11+y=19

=y=8

The solution of the given system of equation is 11 and 8 respectively.

Q12

\(\frac{x}{a}-\frac{y}{b}=2\)

ax-by=a²-b²

Soln:

a₁=\(\frac{1}{a}\) , Let b₁=\(\frac{1}{b}\), Let c₁=-2

a₂=a ,  b₂=-b,  c₂=b²-a²

By cross multiplication method

= \(\frac{x}{\frac{b^{2}-a^{2}}{b}-2b}=\frac{-y}{\frac{b^{2}-a^{2}}{b}+2b}=\frac{1}{\frac{-b}{a}-\frac{a}{b}}\)

= \(\frac{x}{\frac{b^{2}-a^{2}-2b^{2}}{b}}=\frac{-y}{\frac{b^{2}-a^{2}+2b^{2}}{b}}=\frac{1}{\frac{-b^{2}-a^{2}}{ab}}\)

Now, \(\frac{x}{\frac{b^{2}-a^{2}-2b^{2}}{b}}=\frac{1}{\frac{-b^{2}-a^{2}}{ab}}\)

x= a

and, \(\frac{-y}{\frac{b^{2}-a^{2}+2b^{2}}{b}}=\frac{1}{\frac{-b^{2}-a^{2}}{ab}}\)

=y=b

Hence the solution of the given system of equation are a and b respectively.

Q13

\(\frac{x}{a}+\frac{y}{b}=a+b\)

\(\frac{x}{a^{2}}+\frac{y}{b^{2}}=2\)

Soln:

Here, a₁=\(\frac{1}{a}\) , Let b₁=\(\frac{1}{b}\), Let c₁=-(a+b)

a₂=\(\frac{1}{a^{2}}\),  b₂=\(\frac{1}{b^{2}}\),  c₂=-2

By cross multiplication method

=\(\frac{x}{\frac{-2}{b}+\frac{a}{b^{2}}+\frac{1}{b}}=\frac{-y}{\frac{-2}{a}+\frac{1}{a}+\frac{b}{a^{2}}}=\frac{1}{\frac{-1}{ab^{2}}-\frac{-1}{a^{2}b}}\)

= \(\frac{x}{\frac{a-b}{b^{2}}}=\frac{-y}{\frac{-a-b}{a^{2}}+\frac{1}{a}+\frac{b}{a^{2}}}=\frac{1}{\frac{-1}{ab^{2}}-\frac{-1}{a^{2}b}}\)

Now, \(\frac{x}{\frac{a-b}{b^{2}}}=\frac{1}{\frac{-1}{ab^{2}}-\frac{-1}{a^{2}b}}\)

=x=a²

\(\frac{-y}{\frac{-a-b}{a^{2}}+\frac{1}{a}+\frac{b}{a^{2}}}=\frac{1}{\frac{-1}{ab^{2}}-\frac{-1}{a^{2}b}}\)

=y=b²

The solution of the given system of equation are  a²  and b² respectively.

Q14

\(\frac{x}{a}=\frac{y}{b}\)

ax+by=a²+b²

Soln:

Here, a₁=\(\frac{1}{a}\) , Let b₁=\(\frac{1}{b}\),  c₁=0

Here, a₁=a ,  b₂=b, Let c₁=-(a²+b²)

By cross multiplication method

\(\frac{x}{\frac{a^{2}+b^{2}}{b}}= \frac{y}{\frac{a^{2}+b^{2}}{a}}=\frac{1}{\frac{a}{b}+\frac{b}{a}} \)

Now, \(\frac{x}{\frac{a^{2}+b^{2}}{b}}=\frac{1}{\frac{a}{b}+\frac{b}{a}} \)

=x=a

And \( \frac{y}{\frac{a^{2}+b^{2}}{a}}=\frac{1}{\frac{a}{b}+\frac{b}{a}} \)

=y=b

The solution of the given system of equations are  a and b respectively.

Q15

2ax+3by=a+2b

3ax+2by=2a+b

Soln:

The given system of equation is

2ax+3by=a+2b …………………………… (i)

3ax+2by=2a+b…………………………….. (ii)

Here a₁= 2a, b₁= 3b , c₁= -(a+2b)

a₂= 3a , b₂=2b , c₂= -(2a+b)

By cross multiplication method

\(\frac{x}{-4ab+b^{2}}=\frac{-y}{-a^{2}+4ab}=\frac{1}{-5ab}\)

Now,

\(\frac{x}{-4ab+b^{2}}=\frac{1}{-5ab}\)

=x= \(\frac{4a-b}{5a}\)

And, \(\frac{-y}{-a^{2}+4ab}=\frac{1}{-5ab}\)

=y=\(\frac{4b-a}{5b}\)

The solutions of the system of equations are \(\frac{4a-b}{5a}\) and \(\frac{4b-a}{5b}\).

Q16

5ax+6by=28

3ax+4by=18

Soln:

The systems of equations are:

5ax+6by=28 …………………………. (i)

3ax+4by=18……………………………. (ii)

Here a₁= 5a, b₁= 6b , c₁= -(28)

a₂= 3a , b₂=4b , c₂= -(18)

By cross multiplication method

\(\frac{x}{4b}=\frac{-y}{-6a}=\frac{1}{2ab}\)

Now,

\(\frac{x}{4b}=\frac{1}{2ab}\)

=x= \(\frac{2}{a}\)

And, \(\frac{-y}{-6a}=\frac{1}{2ab}\)

=y=\(\frac{3}{b}\)

The solution of the given system of equation is \(\frac{2}{a}\) and \(\frac{3}{b}\).

Q17

(a+2b)x+(2a-b)y=2

(a-2b)x+(2a+b)y=3

Soln.

The given system of equations are :

(a+2b)x+(2a-b)y=2 ………………………. (i)

(a-2b)x+(2a+b)y=3………………………….. (ii)

Here a₁= a+2b, b₁= 2a-b , c₁= -(2)

a₂= a-2b , b₂=2a+b , c₂= -(3)

By cross multiplication method:

\(\frac{x}{-2a+5b}=\frac{y}{a+10b}=\frac{1}{10ab}\)

Now, \(\frac{x}{-2a+5b}=\frac{1}{10ab}\)

=x=  \(\frac{5b-2a}{10ab}\)

And \(\frac{y}{a+10b}=\frac{1}{10ab}\)

=y=\(\frac{a+10b}{10ab}\)

The solution of the system of equations are =x=  \(\frac{5b-2a}{10ab}\)

And =y=\(\frac{a+10b}{10ab}\) respectively.

Q18

\(x(a-b+\frac{ab}{a-b})=y(a+b-\frac{ab}{a+b})\)

x+y=2a²

Soln:

The given systems of equations are:

\(x(a-b+\frac{ab}{a-b})=y(a+b-\frac{ab}{a+b})\)

x+y=2a²

From equation (i)

\(x(\frac{a^{2}+b^{2}-2ab+ab}{a-b})-y(\frac{a^{2}+b^{2}+2ab-ab}{a+b})\)

= \(x(\frac{a^{2}+b^{2}-ab}{a-b})-y(\frac{a^{2}+b^{2}+ab}{a+b})\) ……………….. (iii)

From equation (ii)

X+y-2a²=0

Here a₁= \((\frac{a^{2}+b^{2}-ab}{a-b})\) , b₁= –\((\frac{a^{2}+b^{2}+ab}{a-b})\) , c₁=0

a₂=1, b₂=1 , c₂=-2a²

By cross multiplication method:

\(\frac{x}{2a^{2}(\frac{a^{2}+b^{2}+ab}{a+b})}=\frac{-y}{(-2a^{2})(\frac{a^{2}+b^{2}-ab}{a-b})}=\frac{1}{\frac{2a^{3}}{(a-b)(a+b)}}\)

Now, \(\frac{x}{2a^{2}(\frac{a^{2}+b^{2}+ab}{a+b})}=\frac{1}{\frac{2a^{3}}{(a-b)(a+b)}}\)

=x=\(\frac{a^{3}-b^{3}}{a}\)

And  \(\frac{-y}{(-2a^{2})(\frac{a^{2}+b^{2}-ab}{a-b})}=\frac{1}{\frac{2a^{3}}{(a-b)(a+b)}}\)

=y=\(\frac{a^{3}+b^{3}}{a}\)

The solutions of the given system of equations are \(\frac{a^{3}-b^{3}}{a}\) and \(\frac{a^{3}+b^{3}}{a}\) respectively.

Q19

bx+cy=a+b

\(-ax(\frac{1}{a-b}-\frac{1}{a+b})+cy(\frac{1}{b-a}+\frac{1}{b+a})=\frac{2a}{a+b}\)

Soln:

The system of equation is given by :

bx+cy=a+b ……………………………………. (i)

\(-ax(\frac{1}{a-b}-\frac{1}{a+b})+cy(\frac{1}{b-a}+\frac{1}{b+a})=\frac{2a}{a+b}\)…..(ii)

From equation (i)

bx+cy-(a+b) =0

From equation (ii)

\(-ax(\frac{1}{a-b}-\frac{1}{a+b})+cy(\frac{1}{b-a}+\frac{1}{b+a})-\frac{2a}{a+b}=0\)

= \(x(\frac{2ab}{(a-b)(a+b)})+y(\frac{2ac}{(b-a)(b+a)})-\frac{2a}{a+b}=0\)

= \(\frac{1}{a+b}(\frac{2abx}{a-b}-\frac{2acy}{a-b}-2a)=0\)

= \(\frac{2abx}{a-b}-\frac{2acy}{a-b}-2a=0\)

= 2abx-2acy-2a(a-b) =0 …………………………. (iv)

By cross multiplication

= \(\frac{x}{-4a^{2}c}=\frac{-y}{4ab^{2}}=\frac{-1}{4abc}\)

Now, \(\frac{x}{-4a^{2}c}=\frac{-1}{4abc}\)

=x=\(\frac{a}{b}\)

And,

= \(\frac{-y}{4ab^{2}}=\frac{-1}{4abc}\)

=y=\(\frac{b}{c}\)

The solution of the system of equations are \(\frac{a}{b}\) and \(\frac{b}{c}\)

Q20

(a-b)x+(a+b)y=2a²-2b²

(a+b)(x+y) =4ab

Soln.

The given system of equations are :

(a-b)x+(a+b)y=2a²-2b² ………………………….. (i)

(a+b)(x+y) =4ab …………………………. (ii)

From equation (i)

(a-b)x+(a+b)y-2a²-2b² =0

= (a-b)x+(a+b)y-2(a²-b²) =0

From equation (ii)

(a-b)x+(a-b)y-4ab=0

Here, a₁= a-b , b₁ = a+b , c₁=-2(a²+b²)

Here, a₂= a+b , b₂ = a+b , c₂=-4ab

By cross multiplication method

\(\frac{x}{2(a+b)(a^{2}-b^{2}+2ab)}=\frac{-y}{2(a-b)(a^{2}+b^{2})}=\frac{1}{-2b(a+b)}\)

Now,

\(\frac{x}{2(a+b)(a^{2}-b^{2}+2ab)}=\frac{1}{-2b(a+b)}\)

=x= \(\frac{2ab-a^{2}+b^{2}}{b}\)

And, \(\frac{-y}{2(a-b)(a^{2}+b^{2})}=\frac{1}{-2b(a+b)}\)

=y=\(\frac{(a-b)(a^{2}+b^{2})}{b(a+b)}\)

The solution of the system of equations are \(\frac{2ab-a^{2}+b^{2}}{b}\) and \(\frac{(a-b)(a^{2}+b^{2})}{b(a+b)}\) respectively.

Q21

a²x+b²y=c²

b²x+a²y=d²

Soln:

The given system of equations are :

a²x+b²y=c² ………………………………….. (i)

b²x+a²y=d²……………………………………… (ii)

Here, a₁= a² , b₁ = b² , c₁=-c²

Here, a₂= b² , b₂ = a² , c₂=-d²

By cross multiplication method

= \(\frac{x}{-b^{2}d^{2}+a^{2}c^{2}}=\frac{-y}{-a^{2}d^{2}+b^{2}c^{2}}=\frac{1}{a^{4}-b^{4}}\)

Now,

\(\frac{x}{-b^{2}d^{2}+a^{2}c^{2}}=\frac{1}{a^{4}-b^{4}}\)

=x= \(\frac{a^{2}c^{2}-b^{2}d^{2}}{a^{4}-b^{4}}\)

And, = \(\frac{x}{-b^{2}d^{2}+a^{2}c^{2}}=\frac{-y}{-a^{2}d^{2}+b^{2}c^{2}}=\frac{1}{a^{4}-b^{4}}\)

=y= \(\frac{a^{2}d^{2}-b^{2}c^{2}}{a^{4}-b^{4}}\)

The solution of the given system of equations are \(\frac{a^{2}c^{2}-b^{2}d^{2}}{a^{4}-b^{4}}\) and \(\frac{a^{2}d^{2}-b^{2}c^{2}}{a^{4}-b^{4}}\) respectively.

Q23

2(ax-by+a+4b=0

2(bx+ay)+b-4a=0

Soln:

The given system of equation may be written as :

2(ax-by+a+4b=0 …………………….. (i)

2(bx+ay)+b-4a=0……………………. (ii)

Here, a₁= 2a , b₁ = -2b , c₁=a+4b

Here, a₂= 2b , b₂ = 2a , c₂=b-4a

By cross multiplication method

=\(\frac{x}{-2b^{2}+8ab-2a^{2}-8ab}=\frac{-y}{2ab-8a^{2}-2ab-8b^{2}}=\frac{1}{4a^{2}+4b^{2}}\)

= \(\frac{x}{-2b^{2}-2a^{2}}=\frac{-y}{-8a^{2}-8b^{2}}=\frac{1}{4a^{2}+4b^{2}}\)

Now, \(\frac{x}{-2b^{2}-2a^{2}}=\frac{1}{4a^{2}+4b^{2}}\)

=x= \(\frac{-1}{2}\)

And, \(\frac{-y}{-8a^{2}-8b^{2}}=\frac{1}{4a^{2}+4b^{2}}\)

=y=2

The solution of the given pair of equations are \(\frac{-1}{2}\) and 2 respectively.

Q24

6(ax+by)=3a+2b

6(bx-ay) = 3b-2a

Soln:

The systems of equations are

6(ax+by)=3a+2b ………………………… (i)

6(bx-ay) = 3b-2a …………………………. (ii)

From equation (i)

6ax+6by-(3a+2b)=0  ……………………… (iii)

From equation (ii)

6bx-6ay-(3b-2a) =0 …………………………… (iv)

Here, a₁= 6a , b₁ = 6b , c₁=-(3a+2b)

Here, a₂= 6b , b₂ = -6a , c₂=-(3b-2a)

By cross multiplication method

\(\frac{x}{-18(a^{2}+b^{2})}=\frac{-y}{12(a^{2}+b^{2})}=\frac{-1}{36(a^{2}+b^{2})}\)

Now, \(\frac{x}{-18(a^{2}+b^{2})}=\frac{-1}{36(a^{2}+b^{2})}\)

=x= \(\frac{1}{2}\)

And , \(\frac{-y}{12(a^{2}+b^{2})}=\frac{-1}{36(a^{2}+b^{2})}\)

=y=\(\frac{1}{3}\)

The solution of the given pair of equations are \(\frac{1}{2}\) and \(\frac{1}{3}\) respectively.

Q25

\(\frac{a^{2}}{x}-\frac{b^{2}}{y}=0\)

\(\frac{a^{2}b}{x}+\frac{b^{2}a}{y}=a+b\)

Soln:

The given systems of equations are

\(\frac{a^{2}}{x}-\frac{b^{2}}{y}=0\)

\(\frac{a^{2}b}{x}+\frac{b^{2}a}{y}=a+b\)

Taking \(\frac{1}{x}\) =u

Taking \(\frac{1}{y}\)=v

The pair of equations becomes:

a²u-b²v=0

a²bu+b²av-(a+b) =0

Here, a₁= a² , b₁ = -b² , c₁=0

Here, a₂= a²b , b₂ = b²a , c₂=-(a+b)

By cross multiplication method

= \(\frac{u}{b^{2}(a+b)}=\frac{v}{a^{2}(a+b)}=\frac{1}{a^{2}b^{2}(a+b)}\)

Now, \(\frac{u}{b^{2}(a+b)}=\frac{1}{a^{2}b^{2}(a+b)}\)

=x= \(\frac{1}{a^{2}}\)

And, \(\frac{v}{a^{2}(a+b)}=\frac{1}{a^{2}b^{2}(a+b)}\)

=y= \(\frac{1}{b^{2}}\)

The solution of the given pair of equations are \(\frac{1}{a^{2}}\) and  \(\frac{1}{b^{2}}\) respectively.

Q26

mx-my=m²+n²

x+y=2m

Soln:

mx-my=m²+n²………………………………… (i)

x+y=2m…………………………………………….. (ii)

Here, a₁= m , b₁ = -n , c₁=-(m²+n²)

Here, a₂= 1 , b₂ = 1 , c₂=-(2m)

By cross multiplication method

\(\frac{x}{(m+n)^{2}}=\frac{-y}{-m^{2}+n^{2}}=\frac{1}{m+n}\)

Now, \(\frac{x}{(m+n)^{2}}=\frac{1}{m+n}\)

=x= m+n

And, \(\frac{-y}{-m^{2}+n^{2}}=\frac{1}{m+n}\)

=y=m-n

The solutions of the given pair of equations are m+n and m-n respectively.

Q27

\(\frac{ax}{b}-\frac{by}{a}=a+b\)

ax-by=2ab

Soln:

The given pair of equations are:

\(\frac{ax}{b}-\frac{by}{a}=a+b\) ……………………….. (i)

ax-by=2ab …………………………….. (ii)

Here, a₁= \(\frac{a}{b}\) , b₁ = – \(\frac{b}{a}\) , c₁=-(a+b)

Here, a₂= a , b₂ = – b, c₂=-(2ab)

By cross multiplication method

= \(\frac{x}{b(b-a)}=\frac{-y}{a(-a+b)}=\frac{1}{b-a}\)

Now, \(\frac{x}{b(b-a)}=\frac{1}{b-a}\)

=x=b

And , \(\frac{-y}{a(-a+b)}=\frac{1}{b-a}\)

=y=-a

The solution of the given pair of equations are b and –a respectively.

Q28

\(\frac{b}{a}x+\frac{a}{b}y-(a^{2}+b^{2})=0\)

X+y-2ab=0

Soln:

\(\frac{b}{a}x+\frac{a}{b}y-(a^{2}+b^{2})=0\)…………………….. (i)

X+y-2ab=0…………………………………………. (ii)

Here, a₁= \(\frac{b}{a}\) , b₁ =  \(\frac{a}{b}\) , c₁=-(a²+b²)

Here, a₂= 1 , b₂ = -1, c₂=-(2ab)

By cross multiplication method

= \(\frac{x}{b^{2}-a^{2}}=\frac{-y}{-b^{2}+a^{2}}=\frac{1}{\frac{b^{2}-a^{2}}{ab}}\)

Now, \(\frac{x}{b^{2}-a^{2}}=\frac{1}{\frac{b^{2}-a^{2}}{ab}}\)

=x=ab

And, \(\frac{-y}{-b^{2}+a^{2}}=\frac{1}{\frac{b^{2}-a^{2}}{ab}}\)

=y=ab

The solutions of the given pair of equations are ab and ab respectively.