Find the equation of the line which passes through the point (4, -5) and is
perpendicular to the line `3x+4y+5=0`

Answer

To find the equation of the line that passes through the point (4, -5) and is perpendicular to the line given by the equation \(3x + 4y + 5 = 0\), we can follow these steps:

### Step 1: Find the slope of the given line
The equation of the line is given in the standard form \(Ax + By + C = 0\). We can rewrite the equation \(3x + 4y + 5 = 0\) in slope-intercept form \(y = mx + c\).

1. Rearranging the equation:
\[ 4y = -3x - 5 \]
\[ y = -\frac{3}{4}x - \frac{5}{4} \]

2. From this equation, we can see that the slope \(m_2\) 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 slope of the original line.

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

### Step 3: Use the point-slope form of the equation of a line
We can use the point-slope form of the equation of a line, which 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.

1. Substituting the point \((4, -5)\) and the slope \(\frac{4}{3}\):
\[ y - (-5) = \frac{4}{3}(x - 4) \]
\[ y + 5 = \frac{4}{3}(x - 4) \]

### Step 4: Simplify the equation
1. Distributing the slope on the right side:
\[ y + 5 = \frac{4}{3}x - \frac{16}{3} \]

2. Subtracting 5 from both sides:
\[ y = \frac{4}{3}x - \frac{16}{3} - 5 \]
\[ y = \frac{4}{3}x - \frac{16}{3} - \frac{15}{3} \]
\[ y = \frac{4}{3}x - \frac{31}{3} \]

### Step 5: Convert to standard form
To convert the equation \(y = \frac{4}{3}x - \frac{31}{3}\) into standard form \(Ax + By + C = 0\):

1. Multiply through by 3 to eliminate the fraction:
\[ 3y = 4x - 31 \]

2. Rearranging gives:
\[ 4x - 3y - 31 = 0 \]

Thus, the equation of the line is:
\[ 4x - 3y - 31 = 0 \]