Solve : `{:(x+y=7xy),(2x-3y=-xy):}`

Answer

To solve the system of equations given by:

1. \( x + y = 7xy \)
2. \( 2x - 3y = -xy \)

we will follow these steps:

### Step 1: Rearranging the First Equation
Start with the first equation:

\[ x + y = 7xy \]

Divide both sides by \( xy \):

\[ \frac{x}{xy} + \frac{y}{xy} = 7 \]

This simplifies to:

\[ \frac{1}{y} + \frac{1}{x} = 7 \]

Let’s denote this as Equation (1).

### Step 2: Rearranging the Second Equation
Now take the second equation:

\[ 2x - 3y = -xy \]

Divide both sides by \( xy \):

\[ \frac{2x}{xy} - \frac{3y}{xy} = -1 \]

This simplifies to:

\[ \frac{2}{y} - \frac{3}{x} = -1 \]

Let’s denote this as Equation (2).

### Step 3: Substituting Variables
Let’s substitute:

\[ \frac{1}{x} = X \quad \text{and} \quad \frac{1}{y} = Y \]

Then we can rewrite our equations as:

1. \( X + Y = 7 \) (Equation 3)
2. \( -3X + 2Y = -1 \) (Equation 4)

### Step 4: Solving the New System of Equations
Now we will solve Equations (3) and (4).

From Equation (3):

\[ Y = 7 - X \]

Substituting \( Y \) in Equation (4):

\[ -3X + 2(7 - X) = -1 \]

Expanding this gives:

\[ -3X + 14 - 2X = -1 \]

Combining like terms results in:

\[ -5X + 14 = -1 \]

### Step 5: Isolating \( X \)
Now, isolate \( X \):

\[ -5X = -1 - 14 \]

\[ -5X = -15 \]

\[ X = 3 \]

### Step 6: Finding \( Y \)
Substituting \( X \) back into Equation (3):

\[ 3 + Y = 7 \]

\[ Y = 7 - 3 = 4 \]

### Step 7: Converting Back to \( x \) and \( y \)
Recall that:

\[ X = \frac{1}{x} \quad \text{and} \quad Y = \frac{1}{y} \]

Thus:

\[ \frac{1}{x} = 3 \implies x = \frac{1}{3} \]

\[ \frac{1}{y} = 4 \implies y = \frac{1}{4} \]

### Final Answer
The solutions are:

\[ x = \frac{1}{3}, \quad y = \frac{1}{4} \]

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