Find the combined equation of the lines 2x + 3y = 0 and x - 2y = 0

Answer

To find the combined equation of the lines \(2x + 3y = 0\) and \(x - 2y = 0\), we will follow these steps:

### Step 1: Identify the coefficients
The equations of the lines can be written in the standard form \(A_1x + B_1y + C_1 = 0\) and \(A_2x + B_2y + C_2 = 0\).

For the first line \(2x + 3y = 0\):
- \(A_1 = 2\)
- \(B_1 = 3\)
- \(C_1 = 0\)

For the second line \(x - 2y = 0\):
- \(A_2 = 1\)
- \(B_2 = -2\)
- \(C_2 = 0\)

### Step 2: Write the combined equation
The combined equation of the two lines is given by the product of their equations:
\[ (A_1x + B_1y + C_1)(A_2x + B_2y + C_2) = 0 \]
Substituting the values we identified:
\[ (2x + 3y)(x - 2y) = 0 \]

### Step 3: Expand the product
Now we will expand the product:
\[ 2x \cdot x + 2x \cdot (-2y) + 3y \cdot x + 3y \cdot (-2y) \]
This simplifies to:
\[ 2x^2 - 4xy + 3xy - 6y^2 \]

### Step 4: Combine like terms
Now, we combine the like terms:
\[ 2x^2 + (-4xy + 3xy) - 6y^2 = 2x^2 - xy - 6y^2 \]

### Step 5: Write the final combined equation
Thus, the combined equation of the lines is:
\[ 2x^2 - xy - 6y^2 = 0 \]

### Final Answer
The combined equation of the lines \(2x + 3y = 0\) and \(x - 2y = 0\) is:
\[ 2x^2 - xy - 6y^2 = 0 \]

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