https://file.aiquickdraw.com/mathbot/file/17272473460667ucpd13r.png

We are asked to find the equation of a line that passes through the point \((2, -1)\) and is perpendicular to the line \(2x - y = 4\).

### Step 1: Find the slope of the given line
We start by rewriting the equation of the given line \(2x - y = 4\) in slope-intercept form \(y = mx + b\).

\[
2x - y = 4 \quad \Rightarrow \quad -y = -2x + 4 \quad \Rightarrow \quad y = 2x - 4
\]

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. Therefore, if the slope of the given line is \(2\), the slope of the perpendicular line will be:

\[
m_{\text{perpendicular}} = -\frac{1}{2}
\]

### Step 3: Use the point-slope form
We now use the point-slope form of the equation of a line, which is:

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

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

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

Simplifying:

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

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

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

Would you like further details or clarification on any of these steps?

Here are 5 related questions:
1. How do we calculate the equation of a line parallel to another given line?
2. What is the relationship between the slopes of perpendicular lines?
3. How do we convert a line from general form to slope-intercept form?
4. How do we find the equation of a line given two points?
5. What is the point-slope form, and when is it useful?

**Tip:** The slope-intercept form \(y = mx + b\) is great for quickly identifying the slope and intercept of a line from an equation!

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 algebraic concepts such as slopes of perpendicular lines and the point-slope form. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation