Find the equation for the line with the given properties express your answer in slope intercept form of the equation of a line perpendiculer to the line y=1/3x-1 containing the point 3,-9

To find the equation of a line that is **perpendicular** to the line \( y = \frac{1}{3}x - 1 \) and passes through the point \( (3, -9) \), follow these steps:

### Step 1: Determine the slope of the perpendicular line.
The slope of the given line \( y = \frac{1}{3}x - 1 \) is \( \frac{1}{3} \).

- The slope of a perpendicular line is the **negative reciprocal** of the slope of the original line. 
  - The negative reciprocal of \( \frac{1}{3} \) is \( -3 \).

So, the slope \( m \) of the perpendicular line is \( -3 \).

### Step 2: Use the point-slope form of the equation.
The point-slope form of a line is given by:
\[
y - y_1 = m(x - x_1)
\]
Where:
- \( m \) is the slope,
- \( (x_1, y_1) \) is the point on the line (in this case, \( (3, -9) \)).

Substitute the slope \( m = -3 \) and the point \( (3, -9) \) into the equation:
\[
y - (-9) = -3(x - 3)
\]
This simplifies to:
\[
y + 9 = -3(x - 3)
\]

### Step 3: Simplify to slope-intercept form.
Now, simplify the equation to the slope-intercept form \( y = mx + b \).

Expand the right-hand side:
\[
y + 9 = -3x + 9
\]

Subtract 9 from both sides:
\[
y = -3x + 9 - 9
\]
\[
y = -3x
\]

### Final Answer:
The equation of the line perpendicular to \( y = \frac{1}{3}x - 1 \) and passing through \( (3, -9) \) is:
\[
y = -3x
\]

Would you like any more details on this solution, or have further questions?

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Here are 5 related questions to further explore:

1. How do you find the equation of a line parallel to a given line?
2. What is the slope-intercept form and how do you convert other forms of linear equations into this form?
3. Can you find the equation of a line perpendicular to \( y = -2x + 4 \) passing through \( (1, 2) \)?
4. How do you determine if two lines are perpendicular by comparing their slopes?
5. What is the point-slope form and when is it used in finding the equation of a line?

**Tip:** The slope of perpendicular lines is always the negative reciprocal of each other.

Find the equation for the line with the given properties. Express your answer in slope-intercept form for the equation of a line perpendicular to the line y = 1/3x - 1, containing the point (3, -9).

Discover how to find the equation of a line perpendicular to y = 1/3x - 1 and passing through the point (3, -9). This problem covers key linear equation concepts, including slope-intercept form, point-slope form, and the perpendicular line theorem. Perfect for students in Grades 8-10.

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