Write an equation of a line passing through (6,-7) that is perpendicular to y=-2x-5


To write the equation of a line that passes through the point \((6, -7)\) and is perpendicular to the line \(y = -2x - 5\), follow these steps:

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

Here, the slope \(m\) of the given line is \(-2\).

### Step 2: Determine the slope of the perpendicular line.
The slope of a line that is perpendicular to another line is the negative reciprocal of the original line's slope.

So, the slope of the perpendicular line is:

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

### Step 3: Use the point-slope form to write the equation.
Now, use the point-slope form of the equation, which is:

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

Substitute the point \((6, -7)\) and the slope \(\frac{1}{2}\):

\[
y - (-7) = \frac{1}{2}(x - 6)
\]

This simplifies to:

\[
y + 7 = \frac{1}{2}(x - 6)
\]

### Step 4: Simplify the equation.
To express it more clearly:

\[
y + 7 = \frac{1}{2}x - 3
\]

Subtract 7 from both sides:

\[
y = \frac{1}{2}x - 10
\]

So, the equation of the line passing through \((6, -7)\) and perpendicular to \(y = -2x - 5\) is:

\[
y = \frac{1}{2}x - 10
\]

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:
1. How would you find the equation of a line parallel to the given line?
2. Can you rewrite the equation \(y = \frac{1}{2}x - 10\) in standard form?
3. What is the y-intercept of the new line?
4. How would the equation change if the line passed through a different point?
5. What is the distance between the point \((6, -7)\) and the original line \(y = -2x - 5\)?

**Tip:** The slope of a line perpendicular to another is always the negative reciprocal. This is a key concept in understanding the relationships between lines!

Learn how to find the equation of a line passing through (6, -7) that is perpendicular to y = -2x - 5. This problem covers key geometric concepts, including perpendicular slopes and point-slope form. Suitable for high school students.

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