https://file.aiquickdraw.com/mathbot/file/1727247251602th1kkotf.png

The problem asks for the equation of a line that passes through the point \( (2, -1) \) and is perpendicular to the line \( 2x - y = 4 \).

### Step 1: Determine the slope of the given line.

We start by rewriting the equation of the line \( 2x - y = 4 \) in slope-intercept form \( y = mx + b \), where \( m \) is the slope.

\[
2x - y = 4
\]
Solve for \( y \):

\[
y = 2x - 4
\]

The slope of the given line is \( m_1 = 2 \).

### Step 2: Find the slope of the perpendicular line.

For two lines to be perpendicular, the product of their slopes must equal \(-1\). Let \( m_2 \) represent the slope of the perpendicular line. Then:

\[
m_1 \cdot m_2 = -1
\]
Substitute \( m_1 = 2 \):

\[
2 \cdot m_2 = -1
\]
Solve for \( m_2 \):

\[
m_2 = -\frac{1}{2}
\]

### Step 3: Use the point-slope form to find the equation of the perpendicular line.

The point-slope form of a line is:

\[
y - y_1 = m(x - x_1)
\]

Where \( m \) is the slope and \( (x_1, y_1) \) is the point through which the line passes. Here, \( m = -\frac{1}{2} \) and \( (x_1, y_1) = (2, -1) \). Substituting these values:

\[
y - (-1) = -\frac{1}{2}(x - 2)
\]

Simplify:

\[
y + 1 = -\frac{1}{2}(x - 2)
\]

Distribute the slope:

\[
y + 1 = -\frac{1}{2}x + 1
\]

Solve for \( y \):

\[
y = -\frac{1}{2}x + 1 - 1
\]
\[
y = -\frac{1}{2}x
\]

### Final Answer:
The equation of the line that passes through \( (2, -1) \) and is perpendicular to the line \( 2x - y = 4 \) is:

\[
y = -\frac{1}{2}x
\]

Would you like further details or have any questions?

Here are 5 related questions to explore:
1. How do you find the slope of a line given its equation?
2. What does it mean for two lines to be perpendicular in terms of slopes?
3. How can you convert from standard form to slope-intercept form?
4. How would you graph the line \( y = -\frac{1}{2}x \)?
5. Can two lines be both parallel and perpendicular to another line?

**Tip:** When two lines are perpendicular, the slopes of the lines are negative reciprocals of each other.

What is an equation of the line that passes through the point (2, -1) and is perpendicular to the line 2x - y = 4?

Learn how to find the equation of a line that passes through the point (2, -1) and is perpendicular to the line 2x - y = 4. This problem covers key geometry concepts, such as finding slopes and using point-slope form.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation