Question

Line g is shown on the graph.

coordinate plane with line g that passes through the points negative 3 comma 2 and 0 comma 5

What is the equation of the line in slope-intercept form that is perpendicular to line g and passes through the point (4, 2)? Show all necessary steps.

Answer 1

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:

1. **Find the slope of line \( g \)**.
2. **Determine the slope of the perpendicular line**.
3. **Use the slope-point form to write the equation**.
4. **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:

\[
\boxed{y = -x + 6}
\]