Find an equation of the line that satisfies the given conditions.
Through 1,2 and -2/3
perpendicular to the line 4x − 8y = 1


We are asked to find the equation of a line that passes through the point \((1, 2)\) and has a slope perpendicular to the line given by \(4x - 8y = 1\).

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

We begin by writing the equation of the line in slope-intercept form, \(y = mx + b\).

Start with the equation \(4x - 8y = 1\):

\[
4x - 8y = 1 \quad \Rightarrow \quad -8y = -4x + 1 \quad \Rightarrow \quad y = \frac{1}{2}x - \frac{1}{8}
\]

From this, the slope \(m_1\) of the given line is \(\frac{1}{2}\).

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

The slope of a line perpendicular to another is the negative reciprocal of the slope. Thus, the slope \(m_2\) of the perpendicular line is:

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

### Step 3: Use the point-slope form to write the equation.

We are given the point \((1, 2)\) and the slope \(m_2 = -2\). We use the point-slope form of a line equation:

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

Substitute \(m = -2\), \(x_1 = 1\), and \(y_1 = 2\):

\[
y - 2 = -2(x - 1)
\]

Simplify the equation:

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

### Final Answer:
The equation of the line is:

\[
y = -2x + 4
\]

Let me know if you need further details or clarifications!

### Related Questions:
1. What is the general formula for finding the slope of a line given two points?
2. How can you derive the slope of a line from the standard form equation \(Ax + By = C\)?
3. How do you convert from point-slope form to slope-intercept form?
4. What is the meaning of the negative reciprocal in terms of slopes of perpendicular lines?
5. How do you calculate the distance between a point and a line?

### Tip:
When dealing with perpendicular lines, always remember that the slopes are negative reciprocals of each other!

Find an equation of the line that satisfies the given conditions. Through (1, 2) and slope -2/3, perpendicular to the line 4x − 8y = 1

Learn how to find the equation of a line that passes through (1, 2) and is perpendicular to 4x - 8y = 1. This problem explores key algebraic concepts, including slope-intercept and point-slope forms, and demonstrates the property of perpendicular slopes. Ideal for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation