To solve this problem, we need to find the equation of lines that are perpendicular to the given lines and pass through the specific points provided.

Key steps:

1. Find the slope of each given line. The slope $m_1$ of a line in slope-intercept form $y = mx + b$ is the coefficient of $x$.
2. Calculate the slope of the perpendicular line. For a line to be perpendicular, its slope $m_2$ must satisfy $m_1 \cdot m_2 = -1$, or $m_2 = -\frac{1}{m_1}$.
3. Use the point-slope form $y - y_1 = m(x - x_1)$ to find the equation of the line passing through the given point $(x_1, y_1)$.
4. Convert to slope-intercept form $y = mx + b$.

Let’s go through each equation:

1. $y = 3x + 6$ ; $(6,7)$

• Slope of given line: $m_1 = 3$
• Slope of perpendicular line: $m_2 = -\frac{1}{3}$
• Use point $(6, 7)$: $y - 7 = -\frac{1}{3}(x - 6)$ $y - 7 = -\frac{1}{3}x + 2$ $y = -\frac{1}{3}x + 9$

2. $y = x - 4$ ; $(-2, 3)$

• Slope of given line: $m_1 = 1$
• Slope of perpendicular line: $m_2 = -1$
• Use point $(-2, 3)$: $y - 3 = -1(x + 2)$ $y - 3 = -x - 2$ $y = -x + 1$

3. $y = \frac{1}{2}x + 5$ ; $(4, -5)$

• Slope of given line: $m_1 = \frac{1}{2}$
• Slope of perpendicular line: $m_2 = -2$
• Use point $(4, -5)$: $y + 5 = -2(x - 4)$ $y + 5 = -2x + 8$ $y = -2x + 3$

4. $y = -2x + 4$ ; $(-4, 2)$

• Slope of given line: $m_1 = -2$
• Slope of perpendicular line: $m_2 = \frac{1}{2}$
• Use point $(-4, 2)$: $y - 2 = \frac{1}{2}(x + 4)$ $y - 2 = \frac{1}{2}x + 2$ $y = \frac{1}{2}x + 4$

5. $3x - 4y = 16$ ; $(3,5)$

• Rewrite the equation in slope-intercept form: $-4y = -3x + 16 \quad \Rightarrow \quad y = \frac{3}{4}x - 4$
  • Slope of given line: $m_1 = \frac{3}{4}$
  • Slope of perpendicular line: $m_2 = -\frac{4}{3}$
  • Use point $(3, 5)$: $y - 5 = -\frac{4}{3}(x - 3)$ $y - 5 = -\frac{4}{3}x + 4$ $y = -\frac{4}{3}x + 9$

Summary of Perpendicular Lines:

1. $y = -\frac{1}{3}x + 9$ for $y = 3x + 6$
2. $y = -x + 1$ for $y = x - 4$
3. $y = -2x + 3$ for $y = \frac{1}{2}x + 5$
4. $y = \frac{1}{2}x + 4$ for $y = -2x + 4$
5. $y = -\frac{4}{3}x + 9$ for $3x - 4y = 16$

Do you want more details or have any questions?

Related questions:

1. How do you find the slope of a line from its equation?
2. What does it mean for two lines to be perpendicular?
3. How do you find the equation of a line given its slope and a point on the line?
4. What is the slope-intercept form of a linear equation?
5. How do you convert a linear equation from standard form to slope-intercept form?

Tip: The slopes of perpendicular lines are negative reciprocals of each other.