What is 1/x+1/y =a, 1/x-1/y =b ?

The solution to 1/x+1/y =a, 1/x-1/y =b is

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1/x+1/y =a, 1/x-1/y =b

\frac{1}{x} + \frac{1}{y} = a, \frac{1}{x} - \frac{1}{y} = b

(x = \frac{2}{a + b}, y = - \frac{2}{- a + b})

Solution steps

[\frac{1}{x} + \frac{1}{y} = a \frac{1}{x} - \frac{1}{y} = b]

Add the equations

\frac{1}{x} + \frac{1}{y} = a

\frac{1}{x} + \frac{1}{y} + (\frac{1}{x} - \frac{1}{y}) = a + b

Simplify

\frac{2}{x} = a + b

Solve

\frac{2}{x} = a + b : x = \frac{2}{a + b}; a \neq = - b

Plug the solutions

x = \frac{2}{a + b}

into

\frac{1}{x} + \frac{1}{y} = a

For

\frac{1}{x} + \frac{1}{y} = a

, subsitute

x

with

\frac{2}{a + b} : y = - \frac{2}{- a + b}; a \neq = b

Verify solutions by plugging them into the original equations

Therefore, the final solution for

\frac{1}{x} + \frac{1}{y} = a, \frac{1}{x} - \frac{1}{y} = b

(x = \frac{2}{a + b}, y = - \frac{2}{- a + b})

x + y = 4, x - y = 3

4 x + y = 17, 2 x + y = 11

y = - 2, x = 3

x + 3 y = 6, 2 x + y = - 3

3 x + 2 y = 6, 2 x - 3 y = 17