The equation of the lines represented by `3x^(2)-7xy+4y^(2)=0` are

Options

A. `x+2y=0 and 3x+2y=0`
B. `3x-y=0 and x-4y=0`
C. `x-2y=0 and 3x-2y=0`
D. `x-y=0 and 3x-4y=0`

Answer

To find the equations of the lines represented by the quadratic equation \(3x^2 - 7xy + 4y^2 = 0\), we can follow these steps:

### Step 1: Write the equation
We start with the given equation:
\[ 3x^2 - 7xy + 4y^2 = 0 \]

### Step 2: Rearranging the equation
We can rearrange the equation to factor it. We will look for two terms that can be grouped together. We can rewrite the middle term \(-7xy\) as \(-3xy - 4xy\):
\[ 3x^2 - 3xy - 4xy + 4y^2 = 0 \]

### Step 3: Factor by grouping
Next, we can factor by grouping. We take \(3x\) common from the first two terms and \(-4y\) from the last two terms:
\[ 3x(x - y) - 4y(x - y) = 0 \]

### Step 4: Factor out the common term
Now, we can factor out the common factor \((x - y)\):
\[ (3x - 4y)(x - y) = 0 \]

### Step 5: Set each factor to zero
This gives us two equations:
1. \(3x - 4y = 0\)
2. \(x - y = 0\)

### Step 6: Solve for the lines
Now, we can express these equations in slope-intercept form or simply state them as the equations of lines:
1. From \(3x - 4y = 0\), we can rearrange it to \(y = \frac{3}{4}x\).
2. From \(x - y = 0\), we can rearrange it to \(y = x\).

### Final Result
Thus, the equations of the lines represented by the given quadratic equation are:
1. \(y = \frac{3}{4}x\)
2. \(y = x\)

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