Question

22/x+y+15/x-y=5,55/x+y +40/x-y=13​

Answer 1

Question :- Solve :- 22/x+y+15/x-y=5,55/x+y +40/x-y=13

Solution :-

given that,

22/x+y + 15/x-y = 5

55/x+y + 40/x-y = 13

Let ,

• 1/x + y = a
• 1/x - y = b

so,

→ 22a + 15b = 5 -------- Eqn.(1)

→ 55a + 40b = 13 --------- Eqn.(2)

multiply Eqn.(1) by 5 and Eqn.(2) by 2 and then subtracting the result from Eqn.(2) result ,

→ 2(55a + 40b) - 5(22a + 15b) = 2 * 13 - 5 * 5

→ 110a - 110a + 80b - 75b = 26 - 25

→ 5b = 1

→ b = (1/5)

putting value of b in Eqn.(1)

→ 22a + 15(1/5) = 5

→ 22a = 5 - 3

→ a = 2/22

→ a = (1/11)

then,

→ (1/x + y) = a

→ (1/x + y) = 1/11

→ x + y = 11 -------- Eqn.(3)

and,

→ (1/x - y) = b

→ (1/x - y) = 1/5

→ x - y = 5 -------- Eqn.(4)

adding Eqn.(3) and Eqn.(4) now,

→ x + y + x - y = 11 + 5

→ 2x = 16

→ x = 8 (Ans.)

putting value of x in Eqn.(4)

→ 8 - y = 5

→ y = 8 - 5

→ y = 3 (Ans.)

Hence, value of x is 8 and value of y is 3 .

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Answer 2

Step-by-step explanation:

Given:

$\frac{22}{x + y} + \frac{15}{x - y} = 5 \\ \\ \frac{55}{x + y} + \frac{40}{x - y} = 13$

To find: Find the value of x and y

Solution:

Tip: Convert given equations in standard linear equations

Step 1: Let

$\frac{1}{x + y} = a...eq1 \\ \\ \frac{1}{x - y} = b...eq2$

Substitute these in given equations

$22a + 15b = 5 ...eq3\\ \\ 55a + 40b = 13...eq4$

Step 2: Solve eq3 and eq4 for a and b

Multiply eq3 by 5 and eq4 by 2 and subtract both

$110a + 75b = 25 \\ 110a + 80b = 26 \\ (-) \: \: \: \: (-) \: \: \: \: \: \: \: ( -) \\ - - - - - - - \\ - 5b = - 1 \\ \\ \bold{\pink{b = \frac{ 1}{5}}}$

put the value of b in eq3

$22a + 15 \times \frac{1}{5} = 5 \\ \\ 22a + 3 = 5 \\ \\ 22a = 2 \\ \\ \bold{\purple{a = \frac{1}{11}}}$

Step 3: Put the value of a and b in eq 1 and eq2

$\frac{1}{x + y} = \frac{1}{11} \\ \\ or \\ \\ x + y = 11...eq5 \\ \\ \frac{1}{x - y} = \frac{1}{5} \\ \\ or \\ \\ x - y = 5 \: ...eq6$

Step 4: Add eq5 and eq 6 to find value of x and y

$x + y = 11 \\ x - y = 5 \\ - - - - - \\ 2x = 16 \\ \\ x = \frac{16}{2} \\ \\ x = 8$

put the value of x in eq5

$8 + y = 11 \\ \\ y = 11 - 8 \\ \\ y = 3$

Final answer:

$\bold{\red{x = 8}}\\ \\\bold{\green{ y = 3}}$

Verification:

Put value if x and y in eq

$\frac{22}{x + y} + \frac{15}{x - y} = 5 \\ \\\frac{22}{8+3} + \frac{15}{8-3} = 5\\\\\frac{22}{11} + \frac{15}{5} = 5\\\\2+3 = 5\\\\5=5$

Hope it helps you.

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