We are asked to find equations of lines passing through the point $(-5, 4)$ that are:

1. Parallel to the line $x + y = 7$
2. Perpendicular to the line $x + y = 7$

Let's start by analyzing the given line $x + y = 7$.

Step 1: Determine the slope of the given line

To find the slope, we can rewrite the equation $x + y = 7$ in slope-intercept form $y = mx + b$, where $m$ is the slope:

$x + y = 7 \implies y = -x + 7$

So, the slope of the line is $m = -1$.

Part (a): Equation of the line parallel to the given line

A line parallel to $x + y = 7$ will have the same slope, which is $m = -1$. Using the point-slope form of the equation of a line, $(y - y_1) = m(x - x_1)$, where $(x_1, y_1)$ is $(-5, 4)$, we substitute the values:

$y - 4 = -1(x + 5)$

Simplifying this:

$y - 4 = -x - 5 \implies y = -x - 1$

Thus, the equation of the line parallel to $x + y = 7$ and passing through $(-5, 4)$ is:

$y = -x - 1$

Part (b): Equation of the line perpendicular to the given line

For a line to be perpendicular to the given line, its slope will be the negative reciprocal of $-1$. The negative reciprocal of $-1$ is $1$.

Using the point-slope form again with slope $m = 1$ and point $(-5, 4)$:

$y - 4 = 1(x + 5)$

Simplifying this:

$y - 4 = x + 5 \implies y = x + 9$

Thus, the equation of the line perpendicular to $x + y = 7$ and passing through $(-5, 4)$ is:

$y = x + 9$

Final Answer:

• (a) The equation of the parallel line: $y = -x - 1$
• (b) The equation of the perpendicular line: $y = x + 9$

Would you like more details or have any questions?

Here are 5 related questions to expand on this topic:

1. How do you derive the general form of a line equation from slope-intercept form?
2. How does the distance between two parallel lines relate to their equations?
3. Can you explain how to find the point of intersection of two lines?
4. What is the significance of slopes in determining the angle between two lines?
5. How would you find the shortest distance from a point to a line?

Tip: When determining whether two lines are parallel or perpendicular, always remember that parallel lines share the same slope, while perpendicular lines have slopes that are negative reciprocals of each other.