Find an equation for the line with the given properties. Express your answer using either the general form or the​ slope-intercept form of the equation of a line.

Perpendicular to the line 

y equals one half x plus 4y=12x+4​; 

containing the point 

​(11​,negative 2−2​)

To find the equation of the line that is perpendicular to the given line \( y = \frac{1}{2}x + 4 \) and passes through the point \( (11, -2) \), we'll follow these steps:

### Step 1: Find the slope of the given line
The equation \( y = \frac{1}{2}x + 4 \) is already in slope-intercept form \( y = mx + b \), where \( m \) represents the slope.

From this equation, we see that the slope of the given line is:
\[
m_{\text{given}} = \frac{1}{2}
\]

### Step 2: Find the slope of the perpendicular line
The slope of the line perpendicular to another line is the negative reciprocal of the original slope. So, the slope of the perpendicular line is:
\[
m_{\text{perpendicular}} = -\frac{1}{\frac{1}{2}} = -2
\]

### Step 3: Use the point-slope form of the line equation
We now know the slope \( m_{\text{perpendicular}} = -2 \), and the line passes through the point \( (11, -2) \). We can use the point-slope form of a line equation, which is:
\[
y - y_1 = m(x - x_1)
\]
Substitute \( m = -2 \), \( x_1 = 11 \), and \( y_1 = -2 \):
\[
y - (-2) = -2(x - 11)
\]
Simplifying:
\[
y + 2 = -2(x - 11)
\]
\[
y + 2 = -2x + 22
\]
\[
y = -2x + 20
\]

### Step 4: Final equation
Thus, the equation of the line in slope-intercept form is:
\[
y = -2x + 20
\]

If you prefer the general form, rearrange this equation:
\[
2x + y - 20 = 0
\]

### Final Answer:
- Slope-intercept form: \( y = -2x + 20 \)
- General form: \( 2x + y - 20 = 0 \)

Would you like more details or have any questions?

Here are 5 related questions to explore further:
1. How do you find the slope of a line given two points?
2. What is the geometric interpretation of perpendicular slopes?
3. How would the equation change if the point were different but still perpendicular?
4. How can you rewrite an equation from slope-intercept form to general form?
5. What happens to the graph if the slope of the line is zero?

**Tip:** Always remember that perpendicular slopes are negative reciprocals of each other.

Find an equation for the line with the given properties. Express your answer using either the general form or the slope-intercept form of the equation of a line. Perpendicular to the line y = (1/2)x + 4; containing the point (11, -2).

Learn how to find the equation of a line that is perpendicular to y = (1/2)x + 4 and passes through the point (11, -2). This step-by-step solution covers concepts such as slope-intercept form, point-slope form, and negative reciprocal slopes. Suitable for students in Grades 8-10.

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