Let's solve the problem shown in the image:

We are given the line equation $-x + 7y = 3$. We need to find:

1. The equation of the line parallel to this line that passes through the point $(-3, -6)$.
2. The equation of the line perpendicular to this line that passes through the same point.

Step 1: Rewriting the given line in slope-intercept form

The given equation of the line is $-x + 7y = 3$. Let's convert it to slope-intercept form ($y = mx + b$).

$-x + 7y = 3$ Add $x$ to both sides: $7y = x + 3$ Now, divide by 7 to solve for $y$: $y = \frac{1}{7}x + \frac{3}{7}$ Thus, the slope of the given line is $m = \frac{1}{7}$.

Step 2: Finding the parallel line

The slope of a parallel line is the same as the slope of the original line, $m = \frac{1}{7}$. Using the point-slope form of a line equation, $y - y_1 = m(x - x_1)$, where $(x_1, y_1) = (-3, -6)$, we can substitute the values:

$y - (-6) = \frac{1}{7}(x - (-3))$ Simplifying: $y + 6 = \frac{1}{7}(x + 3)$ Distribute the $\frac{1}{7}$: $y + 6 = \frac{1}{7}x + \frac{3}{7}$ Subtract 6 from both sides: $y = \frac{1}{7}x + \frac{3}{7} - 6$ Convert 6 to have a denominator of 7: $y = \frac{1}{7}x + \frac{3}{7} - \frac{42}{7}$ Simplify: $y = \frac{1}{7}x - \frac{39}{7}$ So, the equation of the parallel line is: $y = \frac{1}{7}x - \frac{39}{7}$

Step 3: Finding the perpendicular line

The slope of a perpendicular line is the negative reciprocal of the slope of the original line. Since the slope of the original line is $m = \frac{1}{7}$, the slope of the perpendicular line will be $m = -7$.

Again, using the point-slope form with $(x_1, y_1) = (-3, -6)$:

$y - (-6) = -7(x - (-3))$ Simplifying: $y + 6 = -7(x + 3)$ Distribute the $-7$: $y + 6 = -7x - 21$ Subtract 6 from both sides: $y = -7x - 21 - 6$ Simplify: $y = -7x - 27$ Thus, the equation of the perpendicular line is: $y = -7x - 27$

Final Answers:

1. Equation of the parallel line: $y = \frac{1}{7}x - \frac{39}{7}$
2. Equation of the perpendicular line: $y = -7x - 27$

Would you like further clarification on any of these steps?

5 Follow-up Questions:

1. What is the general form of a line equation, and how can it be converted to slope-intercept form?
2. How do we find the equation of a line given a point and a slope?
3. What is the relationship between the slopes of parallel and perpendicular lines?
4. Can we find the point of intersection between two lines if we have their equations?
5. How do we graphically represent parallel and perpendicular lines?

Tip:

To quickly find the slope of a perpendicular line, just take the negative reciprocal of the original slope.