The problem provides the line equation $y = 6x - 7$ and asks for the following:

1. The equation of a line parallel to $y = 6x - 7$ and passing through the point $(4, -5)$.
2. The equation of a line perpendicular to $y = 6x - 7$ and passing through the point $(4, -5)$.

Step 1: Parallel Line Equation

For a line parallel to the given one, the slope will remain the same. Since the equation of the line $y = 6x - 7$ has a slope of 6, the parallel line will also have a slope of 6.

We now use the point-slope form of a line: $y - y_1 = m(x - x_1)$ Where $m = 6$, $(x_1, y_1) = (4, -5)$.

Substitute the values: $y - (-5) = 6(x - 4)$ Simplify: $y + 5 = 6(x - 4)$ $y + 5 = 6x - 24$ $y = 6x - 29$

Thus, the equation of the parallel line is: $y = 6x - 29$

Step 2: Perpendicular Line Equation

For the perpendicular line, the slope will be the negative reciprocal of the original slope. The slope of $y = 6x - 7$ is 6, so the perpendicular slope will be: $m = -\frac{1}{6}$

Again, using the point-slope form with the point $(4, -5)$: $y - (-5) = -\frac{1}{6}(x - 4)$ Simplify: $y + 5 = -\frac{1}{6}(x - 4)$ Distribute: $y + 5 = -\frac{1}{6}x + \frac{4}{6}$ $y + 5 = -\frac{1}{6}x + \frac{2}{3}$ Subtract 5 from both sides: $y = -\frac{1}{6}x + \frac{2}{3} - 5$ Convert 5 to a fraction: $y = -\frac{1}{6}x + \frac{2}{3} - \frac{15}{3}$ Simplify: $y = -\frac{1}{6}x - \frac{13}{3}$

Thus, the equation of the perpendicular line is: $y = -\frac{1}{6}x - \frac{13}{3}$

Final Answers:

• Equation of the parallel line: $y = 6x - 29$
• Equation of the perpendicular line: $y = -\frac{1}{6}x - \frac{13}{3}$

Would you like more details on any of these steps or need further clarification?

Here are 5 related questions you might find helpful:

1. How do you find the slope of a line from its equation?
2. How do you find the equation of a line using two points?
3. What is the significance of perpendicular slopes in geometry?
4. How do you convert between different forms of linear equations?
5. How do you graph a line from its slope and intercept?

Tip: When writing equations of lines, always pay close attention to the slope, especially when working with parallel or perpendicular lines, as this property will guide you to the correct form of the line.