Simplify : `( 3y ( x - y) - 2x ( y -2x))/( 7x ( x - y) - 3 ( x ^(2) - y ^(2)))`

Options

A. `(x -y)/(x + y)`
B. `1`
C. `(x +y)/(x -y)`
D. 0

Answer

To simplify the expression \(( 3y ( x - y) - 2x ( y -2x))/( 7x ( x - y) - 3 ( x ^{2} - y ^{2}))\), we will follow these steps:

### Step 1: Expand the Numerator
Start by expanding the numerator \(3y(x - y) - 2x(y - 2x)\).

\[ 3y(x - y) = 3yx - 3y^2 \]
\[ -2x(y - 2x) = -2xy + 4x^2 \]

Combining these, we get:
\[ 3yx - 3y^2 - 2xy + 4x^2 = (3yx - 2xy) + 4x^2 - 3y^2 = xy + 4x^2 - 3y^2 \]

### Step 2: Expand the Denominator
Now, expand the denominator \(7x(x - y) - 3(x^2 - y^2)\).

\[ 7x(x - y) = 7x^2 - 7xy \]
\[ -3(x^2 - y^2) = -3x^2 + 3y^2 \]

Combining these, we get:
\[ 7x^2 - 7xy - 3x^2 + 3y^2 = (7x^2 - 3x^2) - 7xy + 3y^2 = 4x^2 - 7xy + 3y^2 \]

### Step 3: Write the Simplified Expression
Now, we can write the entire expression as:
\[ \frac{xy + 4x^2 - 3y^2}{4x^2 - 7xy + 3y^2} \]

### Step 4: Factor the Numerator and Denominator
Next, we will factor both the numerator and the denominator.

For the numerator \(xy + 4x^2 - 3y^2\):
We need to find two numbers that multiply to \(4 \times -3 = -12\) and add to \(1\) (the coefficient of \(xy\)). The numbers are \(-4\) and \(3\). Thus, we can rewrite:
\[ 4x^2 - 3y^2 + xy = 4x^2 - 4xy + 3xy - 3y^2 = 4x(x - y) + 3y(x - y) = (4x + 3y)(x - y) \]

For the denominator \(4x^2 - 7xy + 3y^2\):
We need to find two numbers that multiply to \(4 \times 3 = 12\) and add to \(-7\). The numbers are \(-4\) and \(-3\). Thus, we can rewrite:
\[ 4x^2 - 7xy + 3y^2 = 4x^2 - 4xy - 3xy + 3y^2 = 4x(x - y) - 3y(x - y) = (4x - 3y)(x - y) \]

### Step 5: Cancel Common Factors
Now, we can simplify the expression:
\[ \frac{(4x + 3y)(x - y)}{(4x - 3y)(x - y)} \]

Since \(x - y\) is common in both the numerator and the denominator, we can cancel it out (assuming \(x \neq y\)):
\[ \frac{4x + 3y}{4x - 3y} \]

### Final Answer
Thus, the simplified expression is:
\[ \frac{4x + 3y}{4x - 3y} \]