Question

Solve the following simultaneous equations.

`1/(2(3x + 4y)) + 1/(5(2x - 3y)) = 1/4; 5/(3x + 4y) - 2/(2x - 3y) = (-3)/2`.

Answer 1

`1/(2(3"x" + 4"y")) + 1/(5(2"x" - 3"y")) = 1/4`   ...(I)

`5/((3"x" + 4"y")) - 2/((2"x" - 3"y")) = -3/2`      ...(II)

Substituting `1/((3"x" + 4"y"))` = m and `1/((2"x" - 3"y"))` = n in equation (I) and (II),

`1/2"m" + 1/5"n" = 1/4`     ...(III)

5m − 2n = `-3/2`          ...(IV)

Multiply equation (III) by 20,

∴ 10m + 4n = 5       ...(V)

Multiply equation (IV) by 2,

∴ 10m − 4n = − 3     ...(VI)

Adding equation (V) and (VI),

    10m + 4n = 5 
+ 10m − 4n = − 3
∴ 20m = 2

∴ m = `2/20 = 1/10`

Substituting m = `1/10` in equation (V),

∴ `10 xx 1/10 + 4"n" = 5`

∴ 1 + 4n = 5

∴ 4n = 5 − 1 = 4

∴ n = 1

But, `1/((3"x" + 4"y"))` = m and `1/((2"x" - 3"y"))` = n

∴ Substituting the values of m and n in the above equation, we get

`1/((3"x" + 4"y")) = 1/10`

∴ 3x + 4y = 10       ...(VII)

And `1/((2"x" - 3"y")) = 1`

∴ 2x − 3y = 1        ...(VIII)

Multiplying by 3 in equation (VII),

9x + 12y = 30      ...(IX)

Multiplying by 4 in equation (VIII),

8x − 12y = 4       ...(X)

Adding equation (IX) and (X), we get   

9x + 12y = 30
+ 8x − 12y = 4
 17x = 34

∴ x = `34/17`

∴ x = 2

Substituting x = 2 in equation (VIII), we get

2x − 3y = 1        ...(VIII)

∴ 2 × 2 − 3y = 1

∴ 4 − 3y = 1

∴ 4 − 1 = 3y

∴ 3y = 3

∴ y = 1

∴ The solution of the given equations is (x, y) = (2, 1).