Find an equation of the line that satisfies the given conditions.
Through (−3, −11);  perpendicular to the line passing through (0, 1) and (4, −1)

To find the equation of a line that passes through the point \((-3, -11)\) and is perpendicular to the line passing through the points \((0, 1)\) and \((4, -1)\), follow these steps:

### Step 1: Find the slope of the line passing through (0, 1) and (4, −1).
The slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Substituting the points \((x_1, y_1) = (0, 1)\) and \((x_2, y_2) = (4, -1)\):
\[
m = \frac{-1 - 1}{4 - 0} = \frac{-2}{4} = -\frac{1}{2}
\]
So, the slope of the line passing through these two points is \(m = -\frac{1}{2}\).

### Step 2: Find the slope of the line perpendicular to this line.
The slope of a line perpendicular to a given line is the negative reciprocal of the slope of the original line. Therefore, the slope \(m'\) of the perpendicular line is:
\[
m' = -\left(\frac{1}{-\frac{1}{2}}\right) = 2
\]

### Step 3: Use the point-slope form to find the equation of the line.
The point-slope form of the equation of a line is:
\[
y - y_1 = m'(x - x_1)
\]
where \((x_1, y_1)\) is the point through which the line passes, and \(m'\) is the slope of the line.

Substituting \((x_1, y_1) = (-3, -11)\) and \(m' = 2\):
\[
y - (-11) = 2(x - (-3))
\]
\[
y + 11 = 2(x + 3)
\]
\[
y + 11 = 2x + 6
\]

### Step 4: Simplify the equation.
Subtract 11 from both sides to put the equation in slope-intercept form:
\[
y = 2x - 5
\]

So, the equation of the line that passes through \((-3, -11)\) and is perpendicular to the line passing through \((0, 1)\) and \((4, -1)\) is:
\[
\boxed{y = 2x - 5}
\]

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

Here are some related questions for further practice:
1. Find the equation of a line parallel to the line passing through the points \((2, 3)\) and \((6, 7)\) and passing through \((4, -1)\).
2. Determine the equation of a line passing through \((2, -3)\) and perpendicular to the line \(y = -4x + 9\).
3. What is the slope of a line perpendicular to the line \(y = \frac{1}{3}x + 2\)?
4. Calculate the point of intersection between the lines \(y = 2x - 5\) and \(y = -\frac{1}{2}x + 3\).
5. Find the equation of the line passing through \((1, 1)\) and perpendicular to the line passing through \((3, 2)\) and \((5, 6)\).

**Tip:** When working with perpendicular lines, remember that the product of their slopes is always \(-1\).

Learn how to find the equation of a line passing through (-3, -11) and perpendicular to the line passing through (0, 1) and (4, -1). This problem involves calculating slopes, using the point-slope form, and understanding perpendicular lines. Suitable for Grades 9-12 students.

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