Use the given conditions to write an equation for the line in point-slope form and in slope-intercept form. Passing through $$ (9,-7) $$ and perpendicular to the line whose equation is $$ y=\frac{1}{2} x+5 $$

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To find the equation of the line that **passes through the point $$ (9, -7) $$** and is **perpendicular** to the line with equation $$ y = \frac{1}{2}x + 5 $$, follow these steps:

### 1. Determine the Slope of the Given Line

The given line is in **slope-intercept form**: 
$$
y = \frac{1}{2}x + 5
$$
Here, the **slope ($$m_1$$)** is:
$$
m_1 = \frac{1}{2}
$$

### 2. Find the Slope of the Perpendicular Line

Perpendicular lines have slopes that are **negative reciprocals** of each other. Therefore, the slope ($$m_2$$) of the line perpendicular to the given line is:
$$
m_2 = -\frac{1}{m_1} = -\frac{1}{\frac{1}{2}} = -2
$$

### 3. Use the Point-Slope Form

The **point-slope form** of a line is:
$$
y - y_1 = m(x - x_1)
$$
Where:
- $$ (x_1, y_1) $$ is a point on the line
- $$ m $$ is the slope

Plugging in the point $$ (9, -7) $$ and the slope $$ m_2 = -2 $$:
$$
y - (-7) = -2(x - 9)
$$
Simplify:
$$
y + 7 = -2(x - 9)
$$
This is the **point-slope form** of the desired line.

### 4. Convert to Slope-Intercept Form

To express the equation in **slope-intercept form** ($$ y = mx + b $$), solve for $$ y $$:
$$
y + 7 = -2(x - 9)
$$
Distribute the slope on the right side:
$$
y + 7 = -2x + 18
$$
Subtract 7 from both sides to solve for $$ y $$:
$$
y = -2x + 18 - 7
$$
$$
y = -2x + 11
$$
This is the **slope-intercept form** of the desired line.

### **Final Answers**

- **Point-Slope Form:** 
  $$
  y + 7 = -2(x - 9)
  $$

- **Slope-Intercept Form:** 
  $$
  y = -2x + 11
  $$
The equation of the line in **point-slope form** is: $$ y + 7 = -2(x - 9) $$ And in **slope-intercept form**: $$ y = -2x + 11 $$

Use the given conditions to write an equation for the line in point-slope form and in slope-intercept form. Passing through \( (9,-7) \) and perpendicular to the line whose equation is \( y=\frac{1}{2} x+5 \)

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