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Examples 1 to 3

(1) Solve : 3x – 2y = 1, 5x + 3y = 8.

Soln. First of all,

3x – 2y – 1 = 0

5x + 3y – 8 = 0

$\therefore$ Solution is $\left (\frac{(-2)(-8)-(3)(-1)}{(3)(3)-(5)(-2)},- \frac{(3)(-8)-(5)(-1)}{(3)(3)-(5)(-2)} \right )$

= $\left (\frac{19}{19},\frac{-19}{19} \right )$

= ( 1, 1)

(2) Solve : 53x + 72y = 178, 72x + 53y = 197.

Soln. Here, 53x + 72y = 178 ––– ( 1 ) and 72x + 53y = 197 –– ( 2 )

Adding eq. ( 1 ) and ( 2 ), we get

53x + 72y = 178

+ 72x + 53y = 197
125x + 125y = 375

$\therefore$ x + y = 3

Subtracting eq. ( 1 ) and ( 2 ), we get

– 53x + 72y = 178

72x + 53y = 197
– 19x + 19y = – 19

$\therefore$ x – y = 1

Now,

x + y – 3 = 0

x – y – 1 = 0

$\therefore$ Solution is $\left (\frac{(1)(-1)-(-3)(-1)}{(1)(-1)-(1)(1)},- \frac{(1)(-1)-(1)(-3)}{(1)(-1)-(1)(1)} \right )$

= $\left (\frac{-4}{-2}, - \frac{2}{-2} \right )$

= ( 2, 1)

(3) Solve : $\frac{10}{x}$ + $\frac{3}{y}$ = 3, $\frac{5}{x}$ + $\frac{9}{y}$ = 4.

Soln. Here, $\frac{10}{x}$ + $\frac{3}{y}$ – 3 =0 , $\frac{5}{x}$ + $\frac{9}{y}$ -4 = 0

$\therefore$ Solution is $\left (\frac{(10)(9)-(5)(3)}{(3)(-4)-(9)(-3)},- \frac{(10)(9)-(5)(3)}{(10)(-4)-(5)(-3)} \right )$

= $\left (\frac{75}{15}, - \frac{75}{25} \right )$

= ( 2, 1)