write equation for line if line contains point (2,-2) and is perpendicular To a line that has slope -1

To write the equation of the line that is perpendicular to a line with a slope of \(-1\) and passes through the point \((2, -2)\), follow these steps:

1. **Find the slope of the new line:**  
   If two lines are perpendicular, the product of their slopes is \(-1\).  
   Given that the slope of the first line is \(-1\), let the slope of the perpendicular line be \(m\). We know that:
   \[
   m \cdot (-1) = -1
   \]
   Solving for \(m\), we get:
   \[
   m = 1
   \]
   So, the slope of the new line is \(1\).

2. **Use the point-slope form of the line equation:**  
   The point-slope form is:
   \[
   y - y_1 = m(x - x_1)
   \]
   where \((x_1, y_1)\) is a point on the line and \(m\) is the slope.  
   Substituting \(m = 1\) and \((x_1, y_1) = (2, -2)\), we get:
   \[
   y - (-2) = 1(x - 2)
   \]
   Simplifying this:
   \[
   y + 2 = x - 2
   \]
   \[
   y = x - 4
   \]

Thus, the equation of the line is:
\[
y = x - 4
\]

Would you like more details on any part of this process?

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Here are 5 related questions to further explore this topic:

1. What is the equation of a line parallel to \(y = -x + 3\) that passes through the point \((1, 2)\)?
2. How do you find the slope of a line passing through two points \((4, 2)\) and \((6, -1)\)?
3. What is the equation of a line perpendicular to \(y = 2x - 5\) that passes through \((3, 1)\)?
4. How do you convert a line equation from point-slope form to slope-intercept form?
5. What is the slope of a line that is horizontal?

**Tip:** When two lines are perpendicular, their slopes are negative reciprocals of each other (i.e., their product is \(-1\)).

Learn how to write the equation of a line that passes through the point (2, -2) and is perpendicular to a line with a slope of -1. This solution involves key algebraic concepts like slopes of perpendicular lines and the point-slope form equation. Suitable for students in Grades 8-10.

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