What is the equation of the line perpendicular to the line ` 2 x + 3y = - 6` and having y - intercept 3 ?

Options

A. `3 x - 2y = 6`
B. ` 3 x - 2y = - 6`
C. ` 2 x - 3y = - 6 `
D. ` 2 x - 3y = 6 `

Answer

To find the equation of the line that is perpendicular to the line given by \(2x + 3y = -6\) and has a y-intercept of 3, we can follow these steps:

### Step 1: Convert the given line to slope-intercept form
The first step is to rewrite the equation \(2x + 3y = -6\) in the slope-intercept form, which is \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept.

1. Start with the original equation:
\[ 2x + 3y = -6 \]
2. Isolate \(y\):
\[ 3y = -2x - 6 \]
3. Divide by 3:
\[ y = -\frac{2}{3}x - 2 \]

### Step 2: Identify the slope of the given line
From the slope-intercept form \(y = -\frac{2}{3}x - 2\), we can see that the slope \(m_1\) of the given line is:
\[ m_1 = -\frac{2}{3} \]

### Step 3: 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, we can find the slope \(m_2\) of the perpendicular line:
\[ m_2 = -\frac{1}{m_1} = -\frac{1}{-\frac{2}{3}} = \frac{3}{2} \]

### Step 4: Use the y-intercept to write the equation of the new line
We know that the y-intercept of the new line is 3. In slope-intercept form, the equation of the line can be written as:
\[ y = m_2x + c \]
Substituting \(m_2\) and \(c\):
\[ y = \frac{3}{2}x + 3 \]

### Step 5: Convert to standard form (optional)
To convert the equation \(y = \frac{3}{2}x + 3\) into standard form \(Ax + By = C\):
1. Multiply through by 2 to eliminate the fraction:
\[ 2y = 3x + 6 \]
2. Rearrange to standard form:
\[ -3x + 2y = 6 \]
3. Multiply by -1 to make \(A\) positive:
\[ 3x - 2y = -6 \]

Thus, the equation of the line perpendicular to \(2x + 3y = -6\) and having a y-intercept of 3 is:
\[ 3x - 2y = -6 \]