Q. 74.0( 116 Votes )

# Solve the following pair of linear equations.

(i) *px* + *qy* = *p* - *q*

*qx* - *py* = *p* + *q*

(ii) *ax* + *by* = *c*

*bx* + *ay* = 1 + *c*

(iii)

*ax* + *by* = *a*^{2} + *b*^{2}

(iv) (*a* - *b*) *x* + (*a* + *b*) *y* = *a*^{2} - 2*ab* - *b*^{2}

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

(v) 152 *x* - 378 *y* = - 74

-378 *x* + 152 *y* = - 604

^{2}

Answer :

**(i)** *p x* + *q y* = *p* - *q* … (1) * q x* - *p y* = *p* + *q* … (2)

Multiplying equation (1) by *p* and equation (2) by *q*,

we obtain *p*^{2 }*x* + *pq y* = *p*^{2} - *pq* … (3)

*q*^{2 }*x* - *pq y* = *pq* + *q*^{2} … (4)

Adding equations (3) and (4),

we obtain *p*^{2}*x* + *q*^{2} *x* = *p*^{2} + *q*^{2}

(*p*^{2} + *q*^{2}) *x* = *p*^{2} + *q*^{2}

From equation putting the value of x (1),

we obtain *p* (1) + *qy* = *p* - *q*

*qy* = - *q *

*y* = - 1

**(ii)** *ax* + *by* = *c* … (1) * bx* + *ay* = 1 + *c* … (2)

Multiplying equation (1) by *a* and equation (2) by *b*,

we obtain *a*^{2 }*x* + *ab y* = *ac* … (3)

*b*^{2 }*x* + *ab y* = *b* + *bc* … (4)

Subtracting equation (4) from equation (3),

(*a*^{2} - *b*^{2}) *x* = *ac* - *bc* – *b*

From equation (1), we obtain *ax* + *by* = *c, now putting the value of x in the equation*

**(iii)**

Or, *bx* - *ay* = 0 … (1)

*ax* + *by* = *a*^{2} + *b*^{2} … (2)

Multiplying equation (1) and (2) by *b* and *a* respectively, we obtain *b*^{2 }*x* - *ab y* = 0 … (3)

*a*^{2 }*x* + *ab y* = *a*^{3} + *ab*^{2} … (4)

Adding equations (3) and (4), we obtain *b*^{2}*x* + *a*^{2}*x* = *a*^{3} + *ab ^{2}*

*x* (*b*^{2} + *a*^{2}) = *a* (*a*^{2} + *b*^{2}) *x*

Thus, x = a

By using (1), and putting the value of x in the equation we obtain *b* (*a*) - *ay* = 0

*ab* - *ay* = 0

*ay* = *ab*

*y* = *b*

**(iv)** (*a* - *b*) *x* + (*a* + *b*) *y* = *a ^{2}* - 2

*ab*-

*b*

^{2}… (1)

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

(*a* + *b*) *x* + (*a* + *b*) *y* = *a*^{2} + *b*^{2} … (2)

Subtracting equation (2) from (1),

we obtain

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

- 2*bx* = - 2*b* (*a* + *b*) * x* = *a* + *b*

Using equation (1), and putting the value of x in the equation we obtain

(*a* - *b*) (*a* + *b*) + (*a* + *b*) *y* = *a*^{2} - 2*ab* - *b*^{2}*a*^{2} - *b*^{2} + (*a* + *b*) *y* = *a*^{2} - 2*ab* - *b*^{2}

(*a* + *b*) *y* = - 2*ab*

**(v)** 152 x - 378 y = - 74 ------------(1)

-378 x + 152 y = - 604 -------- (2)

Multiply eq (2) by 152 and equation (1) by 378

378 × 152x – 378^{2}y = -74 × 378

-378 × 152x + 152^{2}y = -604 × 152

Adding both the questions we get

(152^{2} – 378^{2})y = -119780

-119780 y = -119780

y = 1

put the value in eq 1,

152 x - 378 x 1 = - 74

152 x = 378 - 74

152 x = 304

x = 2

we get x = 2.

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