The image provided shows the equation of a line and asks for the equations of a line parallel to it and a line perpendicular to it, passing through a given point.

The given equation of the line is:

$y = -\frac{7}{4}x - 1$

1. Equation of the Parallel Line

A line parallel to the given line will have the same slope. The slope of the given line is $-\frac{7}{4}$. We need to find the equation of the line parallel to this one that passes through the point $(2, -5)$.

Using the point-slope form of the equation of a line:

$y - y_1 = m(x - x_1)$

where $m$ is the slope and $(x_1, y_1)$ is the point $(2, -5)$,

$y - (-5) = -\frac{7}{4}(x - 2)$

Simplifying:

$y + 5 = -\frac{7}{4}x + \frac{7}{2}$

Subtract 5 from both sides:

$y = -\frac{7}{4}x + \frac{7}{2} - 5$

Convert $-5$ to a fraction with a denominator of 2:

$y = -\frac{7}{4}x + \frac{7}{2} - \frac{10}{2}$

$y = -\frac{7}{4}x - \frac{3}{2}$

Thus, the equation of the line parallel to the given line is:

$y = -\frac{7}{4}x - \frac{3}{2}$

2. Equation of the Perpendicular Line

A line perpendicular to the given line will have a slope that is the negative reciprocal of the slope of the original line. The slope of the given line is $-\frac{7}{4}$, so the slope of the perpendicular line will be:

$m = \frac{4}{7}$

Using the point-slope form again with the point $(2, -5)$:

$y - (-5) = \frac{4}{7}(x - 2)$

Simplifying:

$y + 5 = \frac{4}{7}x - \frac{8}{7}$

Subtract 5 from both sides:

$y = \frac{4}{7}x - \frac{8}{7} - 5$

Convert $-5$ to a fraction with a denominator of 7:

$y = \frac{4}{7}x - \frac{8}{7} - \frac{35}{7}$

$y = \frac{4}{7}x - \frac{43}{7}$

Thus, the equation of the line perpendicular to the given line is:

$y = \frac{4}{7}x - \frac{43}{7}$

Summary of Answers:

• Equation of the parallel line: $y = -\frac{7}{4}x - \frac{3}{2}$
• Equation of the perpendicular line: $y = \frac{4}{7}x - \frac{43}{7}$

Would you like further details on this solution or have any questions?

Related Questions:

1. How do you find the slope of a line given two points?
2. What is the point-slope form of a line, and when is it used?
3. How can you determine if two lines are parallel or perpendicular?
4. How do you convert a linear equation from standard form to slope-intercept form?
5. What are the different methods to find the equation of a line?

Tip:

Always double-check the slope when writing equations for parallel or perpendicular lines to avoid small calculation mistakes.