line v has an equation of y=-2x+5. perpendicular to v is w, whcih passes through (6,-5). equation for line w? answer in slope intercept form

The equation of line \( v \) is given as \( y = -2x + 5 \). To find the equation of the line \( w \), which is perpendicular to \( v \) and passes through the point \( (6, -5) \), follow these steps:

### Step 1: Find the slope of line \( w \)
- The slope of line \( v \) is \( -2 \).
- Since line \( w \) is perpendicular to line \( v \), its slope will be the negative reciprocal of \( -2 \). The negative reciprocal of \( -2 \) is:

\[
m_w = \frac{1}{2}
\]

Thus, the slope of line \( w \) is \( \frac{1}{2} \).

### Step 2: Use the point-slope form to find the equation of line \( w \)
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 a point on the line. For line \( w \), the slope \( m_w = \frac{1}{2} \) and the point \( (6, -5) \) are given. Substituting these values into the point-slope form:

\[
y - (-5) = \frac{1}{2}(x - 6)
\]

Simplifying this:

\[
y + 5 = \frac{1}{2}(x - 6)
\]

### Step 3: Solve for \( y \) to get the equation in slope-intercept form
Distribute the \( \frac{1}{2} \) on the right side:

\[
y + 5 = \frac{1}{2}x - 3
\]

Now, subtract 5 from both sides to isolate \( y \):

\[
y = \frac{1}{2}x - 3 - 5
\]

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

### Final Answer
The equation of line \( w \) in slope-intercept form is:

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

Would you like more details or have any questions?

Here are 5 related questions:
1. What is the general method for finding the negative reciprocal of a slope?
2. How do you determine if two lines are perpendicular using their slopes?
3. What is the significance of the point-slope form of a line equation?
4. How does slope-intercept form help in graphing a line?
5. Can two lines with the same slope be perpendicular?

**Tip:** When two lines are perpendicular, the product of their slopes is always \( -1 \).

Learn how to find the equation of a line perpendicular to y = -2x + 5 that passes through the point (6, -5). This problem covers key algebraic concepts including the negative reciprocal of a slope, the point-slope form, and the slope-intercept form. Suitable for students in Grades 8-10.

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