Question

solve this question ​

Answer 1

Ans:- 1/3x+y + 1/3x-y = 3/4

1/2(3x+y) - 1/2(3x-y) = -1/8

Let take 1/3x+y = A and 1/3x - y = B

Now,

A + B = 3/4

=> 4A + 4B = 3 ----------------–--- (1)

again,

A/2 - B/2 = -1/8

=> A - B = -1/4

=> 4A - 4B = -1 ---------——--------- (2)

now, add both these equation we get,

=> 4A + 4B + 4A - 4B = 3 + (-1)

=> 8A = 2

=> A = 1/4

=> 1/3x + y = 1/4 (since A is equal to 1/3x + y)

=> 3x + y = 4 ------------ (3)

put the value of A in (1)

A + B = 3/4

1/4 + B = 3/4

B = 3/4 - 1/4

B = 1/2

1/3x - y = 1/2 (since B is equal to 1/3x - y = 1/2)

3x - y = 2 ------------- (4)

Now, add (3) and (4) we get,

3x + y + 3x - y = 4 + 2

6x = 6

x = 1

now put the value of x in (4) we get,

3x - y = 2

3× 1 - y = 2

y = 1.

here is your answer

hope it will help you

Answer 2

Answer

$\sf \dashrightarrow \dfrac{1}{3x + y} + \dfrac{1}{3x - y} = \dfrac{3}{4} \: \: --- (i)$

$\sf \dashrightarrow \dfrac{1}{2(3x + y)} - \dfrac{1}{2(3x - y)} = \dfrac{-1}{8} \: \: --- (ii)$

Let $\sf \dfrac{1}{3x + y}$ be u.

Let $\sf \dfrac{1}{3x - y}$ be v.

So, the equations become

$\sf \dashrightarrow 1u + 1v = \dfrac{3}{4}$

$\sf \dashrightarrow \dfrac{u}{2} - \dfrac{v}{2} = \dfrac{-1}{8}$

Now, by first equation,

$\sf \dashrightarrow 1u + 1v = \dfrac{3}{4}$

$\sf \dashrightarrow 1u = \dfrac{3}{4} - 1v$

$\sf \dashrightarrow u = \dfrac{3}{4} - v$

$\sf \dashrightarrow u = \dfrac{3 - 4v}{4}$

Now, let's find the value of v by second equation.

$\sf \dashrightarrow \dfrac{u}{2} - \dfrac{v}{2} = \dfrac{-1}{8}$

$\sf \dashrightarrow \dfrac{\dfrac{3 - 4v}{4}}{2} - \dfrac{v}{2} = \dfrac{-1}{8}$

$\sf \dashrightarrow \bigg( \dfrac{3 - 4v}{4} \times \dfrac{1}{2} \bigg) - \dfrac{v}{2} = \dfrac{-1}{8}$

$\sf \dashrightarrow \dfrac{3 - 4v}{8} - \dfrac{v}{2} = \dfrac{-1}{8}$

$\sf \dashrightarrow \dfrac{3 - 4v - 2v}{8} = \dfrac{-1}{8}$

$\sf \dashrightarrow \dfrac{3 - 6v}{8} = \dfrac{-1}{8}$

$\sf \dashrightarrow 8(3 - 6v) = 8(-1)$

$\sf \dashrightarrow 24 - 48v = -8$

$\sf \dashrightarrow -48v = -8 - 24$

$\sf \dashrightarrow -48v = -32$

$\sf \dashrightarrow v = \dfrac{-32}{-48}$

$\sf \dashrightarrow v = \dfrac{2}{3}$

Now, let's find the value of u by first equation.

$\sf \dashrightarrow 1u + 1v = \dfrac{3}{4}$

$\sf \dashrightarrow u + \dfrac{2}{3} = \dfrac{3}{4}$

$\sf \dashrightarrow \dfrac{3u + 2}{3} = \dfrac{3}{4}$

$\sf \dashrightarrow 3u + 2 = \dfrac{3}{4} \times 3$

$\sf \dashrightarrow 3u + 2 = \dfrac{9}{4}$

$\sf \dashrightarrow 3u = \dfrac{9}{4} - 2$

$\sf \dashrightarrow 3u = \dfrac{9 - 4}{4}$

$\sf \dashrightarrow 3u = \dfrac{5}{4}$

$\sf \dashrightarrow u = \dfrac{\dfrac{5}{4}}{3}$

$\sf \dashrightarrow u = \dfrac{5}{4} \times \dfrac{1}{3}$

$\sf \dashrightarrow u = \dfrac{5}{12}$

We know that,

$\sf \dashrightarrow \dfrac{1}{3x + y} = u$

$\sf \dashrightarrow \dfrac{1}{3x + y} = \dfrac{5}{12}$

$\sf \dashrightarrow 5(3x + y) = 12$

$\sf \dashrightarrow 15x + 5y = 12 \: \: --- (iii)$

We also know that,

$\sf \dashrightarrow \dfrac{1}{3x - y} = v$

$\sf \dashrightarrow \dfrac{1}{3x - y} = \dfrac{2}{3}$

$\sf \dashrightarrow 2(3x - y) = 3$

$\sf \dashrightarrow 6x - 2y = 3 \: \: --- (iv)$

Now, by third equation,

$\sf \dashrightarrow 15x + 5y = 12$

$\sf \dashrightarrow 15x = 12 - 5y$

$\sf \dashrightarrow x = \dfrac{12 - 5y}{15}$

Now, let's find the value of y by fourth equation.

$\sf \dashrightarrow 6x - 2y = 3$

$\sf \dashrightarrow 6 \bigg( \dfrac{12 - 5y}{15} \bigg) - 2y = 3$

$\sf \dashrightarrow \dfrac{72 - 30y}{15} - 2y = 3$

$\sf \dashrightarrow \dfrac{72 - 30y - 30y}{15} = 3$

$\sf \dashrightarrow \dfrac{72 - 60y}{15} = 3$

$\sf \dashrightarrow 72 - 60y = 15 \times 3$

$\sf \dashrightarrow 72 - 60y = 45$

$\sf \dashrightarrow -60y = 45 - 72$

$\sf \dashrightarrow -60y = -27$

$\sf \dashrightarrow y = \dfrac{-27}{-60}$

$\sf \dashrightarrow y = \dfrac{9}{20}$

Now, we can find the value of x by third equation.

$\sf \dashrightarrow 15x + 5y = 12$

$\sf \dashrightarrow 15x + 5 \bigg( \dfrac{9}{20} \bigg) = 12$

$\sf \dashrightarrow 15x + \dfrac{45}{20} = 12$

$\sf \dashrightarrow 15x + \dfrac{9}{4} = 12$

$\sf \dashrightarrow \dfrac{60x + 9}{4} = 12$

$\sf \dashrightarrow 60x + 9 = 12 \times 4$

$\sf \dashrightarrow 60x + 9 = 48$

$\sf \dashrightarrow 60x = 48 - 9$

$\sf \dashrightarrow 60x = 39$

$\sf \dashrightarrow x = \dfrac{39}{60}$

$\sf \dashrightarrow x = \dfrac{13}{20}$

Hence, the values of x and y are 13/20 and 9/20 respectively.