Find the equation of a line which passes through (– 3, 2) and perpendicular to the 3x + 4y = 5.

Answer

To find the equation of a line that passes through the point (-3, 2) and is perpendicular to the line given by the equation \(3x + 4y = 5\), we will follow these steps:

### Step 1: Find the slope of the given line
The first step is to convert the equation of the line \(3x + 4y = 5\) into slope-intercept form \(y = mx + b\), where \(m\) is the slope.

Starting with the equation:
\[ 3x + 4y = 5 \]

Rearranging it to solve for \(y\):
\[ 4y = -3x + 5 \]
\[ y = -\frac{3}{4}x + \frac{5}{4} \]

From this, we can see that the slope \(m_1\) of the given line is \(-\frac{3}{4}\).

### 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 \(m_1 = -\frac{3}{4}\), the slope \(m_2\) of the line we want to find is:
\[ m_2 = -\frac{1}{m_1} = -\frac{1}{-\frac{3}{4}} = \frac{4}{3} \]

### Step 3: Use the point-slope form to find the equation of the line
Now that we have the slope \(m_2 = \frac{4}{3}\) and a point \((-3, 2)\) through which the line passes, we can use the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
Substituting in our values:
\[ y - 2 = \frac{4}{3}(x + 3) \]

### Step 4: Simplify the equation
Now we will simplify this equation:
\[ y - 2 = \frac{4}{3}x + \frac{4}{3} \cdot 3 \]
\[ y - 2 = \frac{4}{3}x + 4 \]
Adding 2 to both sides gives:
\[ y = \frac{4}{3}x + 4 + 2 \]
\[ y = \frac{4}{3}x + 6 \]

### Step 5: Convert to standard form
To convert this to standard form \(Ax + By + C = 0\), we can rearrange it:
\[ -\frac{4}{3}x + y - 6 = 0 \]
Multiplying through by 3 to eliminate the fraction:
\[ -4x + 3y - 18 = 0 \]
Rearranging gives:
\[ 4x - 3y + 18 = 0 \]

Thus, the equation of the line is:
\[ \boxed{4x - 3y + 18 = 0} \]