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To find the equation of a line perpendicular to \( y = 2x - 5 \) that passes through the point \( (2, -5) \), let's break down the process.

### Step 1: Find the slope of the given line
The given line is in slope-intercept form \( y = mx + b \), where \( m \) is the slope. From \( y = 2x - 5 \), we see the slope of the given line is \( m = 2 \).

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

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

We are given the point \( (2, -5) \) and the slope \( -\frac{1}{2} \). Substituting these values into the point-slope form:
\[
y - (-5) = -\frac{1}{2}(x - 2)
\]
This simplifies to:
\[
y + 5 = -\frac{1}{2}(x - 2)
\]

### Step 4: Simplify the equation
Distribute the slope on the right-hand side:
\[
y + 5 = -\frac{1}{2}x + 1
\]
Now subtract 5 from both sides to get the equation in slope-intercept form:
\[
y = -\frac{1}{2}x - 4
\]

So, the equation of the line perpendicular to \( y = 2x - 5 \) that passes through the point \( (2, -5) \) is:
\[
y = -\frac{1}{2}x - 4
\]

Would you like a more detailed explanation or have any further questions?

Here are 5 related questions:
1. What is the general process for finding the slope of a line from its equation?
2. How do you find the equation of a line parallel to another line?
3. Can you find the equation of a line perpendicular to \( y = -\frac{3}{4}x + 2 \)?
4. What are the properties of perpendicular lines in geometry?
5. How can you convert the point-slope form of a line into slope-intercept form?

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

Write the equation of a line perpendicular to y = 2x - 5 that passes through the point (2, -5).

Learn how to find the equation of a line perpendicular to y = 2x - 5 that passes through the point (2, -5). This problem demonstrates the use of algebraic concepts such as slope-intercept form and point-slope form, suitable for students in Grades 8-10.

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