To find the equation of the line that passes through the point $(5,7)$ and is parallel to a line passing through the points $(-2,3)$ and $(2,6)$, we can follow these steps:

Step 1: Determine the slope of the given line

First, we need to find the slope of the line passing through the points $(-2,3)$ and $(2,6)$. The formula for the slope $m$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

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

Substituting the given points:

$m = \frac{6 - 3}{2 - (-2)} = \frac{3}{4}$

So, the slope of the line is $\frac{3}{4}$.

Step 2: Use the point-slope form to find the equation of the line

Since parallel lines have the same slope, the slope of the line we want to find is also $\frac{3}{4}$. We can use the point-slope form of the equation of a line, which is:

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

Where $(x_1, y_1)$ is the point $(5,7)$ and $m = \frac{3}{4}$.

Substituting the values:

$y - 7 = \frac{3}{4}(x - 5)$

Step 3: Simplify to the slope-intercept form (optional)

To get the equation in slope-intercept form $y = mx + b$, distribute and simplify:

$y - 7 = \frac{3}{4}x - \frac{15}{4}$

Add 7 to both sides:

$y = \frac{3}{4}x - \frac{15}{4} + 7$

To combine the constant terms, convert 7 to a fraction with a denominator of 4:

$y = \frac{3}{4}x - \frac{15}{4} + \frac{28}{4}$

Simplifying further:

$y = \frac{3}{4}x + \frac{13}{4}$

So, the equation of the line is:

$\boxed{y = \frac{3}{4}x + \frac{13}{4}}$

Would you like more details or have any questions?

Here are some related questions you might find useful:

1. How do you find the equation of a line given two points?
2. What are the properties of parallel and perpendicular lines?
3. How do you convert an equation from point-slope form to slope-intercept form?
4. What is the significance of the slope in the equation of a line?
5. How do you find the intersection of two lines given their equations?

Tip: When working with fractions, converting to a common denominator can help simplify addition or subtraction.