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 $-1$.