Question

Parallel/Perpendicular Through Point
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What is an equation of the line that passes through the point left parenthesis, minus, 5, comma, 2, right parenthesis(−5,2) and is perpendicular to the line x, minus, 2, y, equals, 2x−2y=2?

Answer 1

To find the equation of the line that is perpendicular to the line given by \(2x - 2y = 2\) and passes through the point \((-5, 2)\), we'll follow these steps:

1. **Convert the equation to slope-intercept form (y = mx + b)**:

Start with:
\[
2x - 2y = 2
\]
Rearranging it gives:
\[
-2y = -2x + 2
\]
Dividing through by \(-2\):
\[
y = x - 1
\]

From this, we see that the slope (\(m\)) of the line is \(1\).

2. **Determine the slope of the perpendicular line**:

The slope of a line that is perpendicular to another line is the negative reciprocal of the original line's slope.
- Original slope \(m = 1\)
- Perpendicular slope \(m_{perpendicular} = -\frac{1}{1} = -1\)

3. **Use the point-slope form**:

Now, we can use the point-slope form of a line equation, which is:
\[
y - y_1 = m(x - x_1)
\]
Here, \((x_1, y_1) = (-5, 2)\) and \(m = -1\). Plugging these values in gives:
\[
y - 2 = -1(x + 5)
\]

4. **Simplify the equation**:

Distributing the \(-1\):
\[
y - 2 = -x - 5
\]

Adding \(2\) to both sides:
\[
y = -x - 3
\]

5. **Final result**:

The equation of the line that passes through the point \((-5, 2)\) and is perpendicular to the line \(2x - 2y = 2\) is:
\[
y = -x - 3
\]