If `9x ^(2) + 16 y ^(2) = 60 and 3x + 4y = 6,` then the value of xy is

Options

A. `-1`
B. 1
C. `-2`
D. 2

Answer

To solve the equations \(9x^2 + 16y^2 = 60\) and \(3x + 4y = 6\) to find the value of \(xy\), follow these steps:

### Step 1: Square the second equation
We start with the equation:
\[ 3x + 4y = 6 \]
Now, we square both sides:
\[ (3x + 4y)^2 = 6^2 \]
This expands to:
\[ 9x^2 + 24xy + 16y^2 = 36 \]

### Step 2: Substitute the first equation into the squared equation
From the first equation, we know:
\[ 9x^2 + 16y^2 = 60 \]
We can substitute this into the equation we obtained from squaring:
\[ 60 + 24xy = 36 \]

### Step 3: Solve for \(24xy\)
Now, we rearrange the equation to isolate \(24xy\):
\[ 24xy = 36 - 60 \]
This simplifies to:
\[ 24xy = -24 \]

### Step 4: Solve for \(xy\)
Now, we divide both sides by 24 to find \(xy\):
\[ xy = \frac{-24}{24} = -1 \]

Thus, the value of \(xy\) is \(-1\).

### Final Answer
Therefore, the value of \(xy\) is:
\[ \boxed{-1} \]

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