Find the equation of a line perpendicular toy, equals, minus, 2, x, minus, 6*y*=−2*x*−6that passes through the point left parenthesis, 2, comma, 9, right parenthesis(2,9).

To find the equation of a line that is perpendicular to the line \( y = -2x - 6 \) and passes through the point \( (2, 9) \), follow these steps:

### Step 1: Determine the slope of the given line
The given line is in the slope-intercept form \( y = mx + b \), where \( m \) is the slope. 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 line is the negative reciprocal of the original slope. If the slope of the given line is \( m_1 = -2 \), then the slope of the perpendicular line \( m_2 \) is:

\[
m_2 = \frac{-1}{m_1} = \frac{-1}{-2} = \frac{1}{2}
\]

### Step 3: Use the point-slope form to find the equation
Now that we have the slope of the perpendicular line \( m_2 = \frac{1}{2} \) and the point \( (2, 9) \), we can use the point-slope form of the equation:

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

Substituting the point \( (2, 9) \) and the slope \( \frac{1}{2} \):

\[
y - 9 = \frac{1}{2}(x - 2)
\]

### Step 4: Simplify the equation
Distribute the slope \( \frac{1}{2} \) on the right side:

\[
y - 9 = \frac{1}{2}x - 1
\]

Next, add 9 to both sides to solve for \( y \):

\[
y = \frac{1}{2}x + 8
\]

### Final Answer:
The equation of the line that is perpendicular to \( y = -2x - 6 \) and passes through the point \( (2, 9) \) is:

\[
y = \frac{1}{2}x + 8
\]

Would you like more details or have any questions? Here are 5 related questions to consider:

1. How do you find the equation of a line parallel to a given line through a specific point?
2. What are the steps to convert an equation from point-slope form to standard form?
3. How can you verify if two lines are perpendicular by comparing their slopes?
4. What is the significance of the y-intercept in the equation of a line?
5. How do you determine the point where two lines intersect?

**Tip:** When dealing with perpendicular lines, always remember to use the negative reciprocal of the slope of the original line.

Learn how to find the equation of a line perpendicular to y = -2x - 6 that passes through the point (2, 9). Detailed step-by-step solution using concepts of linear equations, perpendicular lines, and point-slope form. Suitable for Grades 7-9.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation