If `(1)/(x) + (1)/(y) = 2 ` and `(3)/(x) + (4)/(y) = 7` , what is the value of (x + y) ?

Answer

To solve the equations given in the problem, we will follow these steps:

### Step 1: Write down the equations
We have the following two equations:
1. \(\frac{1}{x} + \frac{1}{y} = 2\)
2. \(\frac{3}{x} + \frac{4}{y} = 7\)

### Step 2: Express \(\frac{1}{x}\) in terms of \(\frac{1}{y}\)
From the first equation, we can express \(\frac{1}{x}\):
\[ \frac{1}{x} = 2 - \frac{1}{y} \]

### Step 3: Substitute \(\frac{1}{x}\) into the second equation
Now, we substitute \(\frac{1}{x}\) into the second equation:
\[ \frac{3}{x} + \frac{4}{y} = 7 \]
Substituting \(\frac{1}{x}\):
\[ 3(2 - \frac{1}{y}) + \frac{4}{y} = 7 \]

### Step 4: Simplify the equation
Now, simplify the equation:
\[ 6 - \frac{3}{y} + \frac{4}{y} = 7 \]
Combine the terms involving \(\frac{1}{y}\):
\[ 6 + \frac{1}{y} = 7 \]

### Step 5: Solve for \(\frac{1}{y}\)
Now, isolate \(\frac{1}{y}\):
\[ \frac{1}{y} = 7 - 6 \]
\[ \frac{1}{y} = 1 \]

### Step 6: Solve for \(y\)
Taking the reciprocal gives us:
\[ y = 1 \]

### Step 7: Substitute \(y\) back to find \(x\)
Now substitute \(y = 1\) back into the first equation:
\[ \frac{1}{x} + \frac{1}{1} = 2 \]
\[ \frac{1}{x} + 1 = 2 \]
Isolate \(\frac{1}{x}\):
\[ \frac{1}{x} = 2 - 1 \]
\[ \frac{1}{x} = 1 \]

### Step 8: Solve for \(x\)
Taking the reciprocal gives us:
\[ x = 1 \]

### Step 9: Calculate \(x + y\)
Now we can find \(x + y\):
\[ x + y = 1 + 1 = 2 \]

### Final Answer
The value of \(x + y\) is \(2\).

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