Mr. and Mrs. Ahuja weigh x kg and y kg respectively. They both take a dieting course, at the end of which Mr. Ahuja loses 5 kg and weighs as much as his wife weighed before the course.
Mrs. Ahuja loses 4 kg and weighs -7/8 th of what her husband weighed before the course. Form two equations in x and y to find their weights before taking the dieting course.

Answer

To solve the problem, we need to form two equations based on the information provided about Mr. and Mrs. Ahuja's weights before and after their dieting course.

### Step 1: Define the Variables
Let:
- \( x \) = weight of Mr. Ahuja before the diet (in kg)
- \( y \) = weight of Mrs. Ahuja before the diet (in kg)

### Step 2: Form the First Equation
According to the problem, Mr. Ahuja loses 5 kg and ends up weighing as much as Mrs. Ahuja did before the diet. This can be expressed as:
\[ x - 5 = y \]
This is our **first equation**.

### Step 3: Form the Second Equation
Mrs. Ahuja loses 4 kg and weighs \( \frac{7}{8} \) of what Mr. Ahuja weighed before the diet. This can be expressed as:
\[ y - 4 = \frac{7}{8}x \]
This is our **second equation**.

### Step 4: Rewrite the Equations
We can rewrite the equations for easier manipulation:
1. From the first equation:
\[ x - y = 5 \quad \text{(Equation 1)} \]
2. From the second equation, we can multiply through by 8 to eliminate the fraction:
\[ 8(y - 4) = 7x \]
Simplifying this gives:
\[ 8y - 32 = 7x \]
Rearranging gives:
\[ 7x - 8y = -32 \quad \text{(Equation 2)} \]

### Step 5: Solve the Equations
Now we have the two equations:
1. \( x - y = 5 \)
2. \( 7x - 8y = -32 \)

We can solve these equations simultaneously. From Equation 1, we can express \( x \) in terms of \( y \):
\[ x = y + 5 \]

Substituting this into Equation 2:
\[ 7(y + 5) - 8y = -32 \]
Expanding this gives:
\[ 7y + 35 - 8y = -32 \]
Combining like terms:
\[ -y + 35 = -32 \]
Subtracting 35 from both sides:
\[ -y = -32 - 35 \]
\[ -y = -67 \]
Thus:
\[ y = 67 \]

### Step 6: Find \( x \)
Now that we have \( y \), we can find \( x \) using Equation 1:
\[ x = y + 5 = 67 + 5 = 72 \]

### Final Weights
- Mr. Ahuja's weight before the diet: \( x = 72 \) kg
- Mrs. Ahuja's weight before the diet: \( y = 67 \) kg

### Summary
Mr. Ahuja weighs 72 kg and Mrs. Ahuja weighs 67 kg before the dieting course.

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