Find the equation of the straight line which passes through the point (2, -3) and is perpendicular to the line 2 x + 3y + 5 = 0

Options

A. 3 x - 2y - 12 = 0
B. 3x + 2y - 12 = 0
C. 2 x + 3y = 12
D. None of these

Answer

To find the equation of the straight line that passes through the point (2, -3) and is perpendicular to the line given by the equation \(2x + 3y + 5 = 0\), we can follow these steps:

### Step 1: Find the slope of the given line
The equation of the line can be rewritten in slope-intercept form \(y = mx + b\), where \(m\) is the slope.

Starting with:
\[ 2x + 3y + 5 = 0 \]
we can isolate \(y\):
\[ 3y = -2x - 5 \]
\[ y = -\frac{2}{3}x - \frac{5}{3} \]
From this, we can see that the slope \(m_1\) of the given line is \(-\frac{2}{3}\).

### Step 2: Find the slope of the perpendicular line
The slope of a line that is perpendicular to another line is the negative reciprocal of the original line's slope. Therefore, if the slope of the given line is \(-\frac{2}{3}\), the slope \(m_2\) of the perpendicular line is:
\[ m_2 = -\frac{1}{m_1} = -\frac{1}{-\frac{2}{3}} = \frac{3}{2} \]

### Step 3: Use the point-slope form of the equation of a line
The point-slope form of a line's equation is given by:
\[ y - y_1 = m(x - x_1) \]
where \((x_1, y_1)\) is a point on the line and \(m\) is the slope.

Substituting the point \((2, -3)\) and the slope \(\frac{3}{2}\):
\[ y - (-3) = \frac{3}{2}(x - 2) \]
This simplifies to:
\[ y + 3 = \frac{3}{2}(x - 2) \]

### Step 4: Simplify the equation
Now, we can distribute and simplify:
\[ y + 3 = \frac{3}{2}x - 3 \]
Subtracting 3 from both sides gives:
\[ y = \frac{3}{2}x - 6 \]

### Step 5: Write the final equation in standard form
To convert this to standard form \(Ax + By + C = 0\), we can rearrange it:
\[ -\frac{3}{2}x + y + 6 = 0 \]
Multiplying through by 2 to eliminate the fraction:
\[ -3x + 2y + 12 = 0 \]
or rearranging gives:
\[ 3x - 2y - 12 = 0 \]

Thus, the equation of the straight line that passes through the point (2, -3) and is perpendicular to the line \(2x + 3y + 5 = 0\) is:
\[ 3x - 2y - 12 = 0 \]