To find the equation of a line that is perpendicular to line \( g \) and passes through the point \( (4, 2) \), we will follow these steps:
- Find the slope of line \( g \).
- Determine the slope of the perpendicular line.
- Use the slope-point form to write the equation.
- Convert to slope-intercept form \( y = mx + b \).
Step 1: Find the slope of line \( g \)
The points that line \( g \) passes through are \( (-3, 2) \) and \( (0, 5) \). We can calculate the slope \( m \) using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Assigning the coordinates, we have:
- \( (x_1, y_1) = (-3, 2) \)
- \( (x_2, y_2) = (0, 5) \)
Now substituting the values into the formula:
\[ m = \frac{5 - 2}{0 - (-3)} = \frac{3}{3} = 1 \]
So, the slope of line \( g \) is \( 1 \).
Step 2: Determine 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. Therefore, if the slope of line \( g \) is \( 1 \), the slope of the perpendicular line is:
\[ m_{perpendicular} = -\frac{1}{1} = -1 \]
Step 3: Use the slope-point form to write the equation
Now we will use the point-slope form of the equation of a line, which is given by:
\[ y - y_1 = m(x - x_1) \]
Substituting \( m = -1 \) and the point \( (4, 2) \) (where \( x_1 = 4 \) and \( y_1 = 2 \)):
\[ y - 2 = -1(x - 4) \]
Step 4: Simplify to slope-intercept form
Now simplifying the equation:
\[ y - 2 = -1x + 4 \]
Adding \( 2 \) to both sides:
\[ y = -x + 6 \]
Thus, the equation of the line that is perpendicular to line \( g \) and passes through the point \( (4, 2) \) in slope-intercept form is:
\[ \boxed{y = -x + 6} \]