The problem asks to find the equation of a line that passes through the point $(3, 7)$ and is parallel to a line passing through the points $(4, 5)$ and $(-5, -2)$.

Step 1: Find the slope of the given line

To find the slope of the line passing through $(4, 5)$ and $(-5, -2)$, use the slope formula:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Substitute the given points $(4, 5)$ and $(-5, -2)$:

$m = \frac{-2 - 5}{-5 - 4} = \frac{-7}{-9} = \frac{7}{9}$

The slope of the line passing through $(4, 5)$ and $(-5, -2)$ is $m = \frac{7}{9}$.

Step 2: Equation of the parallel line

Since the two lines are parallel, they will have the same slope. Therefore, the line passing through $(3, 7)$ will also have a slope of $m = \frac{7}{9}$.

The equation of a line can be written in point-slope form:

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

Substitute $m = \frac{7}{9}$, and the point $(3, 7)$:

$y - 7 = \frac{7}{9}(x - 3)$

Step 3: Simplify the equation

Expand and simplify the equation:

$y - 7 = \frac{7}{9}x - \frac{21}{9}$

$y - 7 = \frac{7}{9}x - \frac{7}{3}$

Add 7 to both sides:

$y = \frac{7}{9}x - \frac{7}{3} + 7$

Convert 7 to a fraction with denominator 3:

$y = \frac{7}{9}x - \frac{7}{3} + \frac{21}{3}$

$y = \frac{7}{9}x + \frac{14}{3}$

Final Equation:

The equation of the line passing through $(3, 7)$ and parallel to the line passing through $(4, 5)$ and $(-5, -2)$ is:

$y = \frac{7}{9}x + \frac{14}{3}$

Would you like further details or have any questions?

Here are some related questions for further practice:

1. How do you find the slope of a line given two points?
2. What is the difference between point-slope form and slope-intercept form?
3. How do you find the equation of a perpendicular line?
4. Can two lines with different slopes be parallel?
5. How can you convert a point-slope equation to slope-intercept form?

Tip: To quickly check if two lines are parallel, just compare their slopes! If the slopes are equal, the lines are parallel.