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Write an equation of a line that satisfies the given conditions

We are asked to find the equation of a line that:

1. Passes through the point \( (3, 5) \).
2. Is parallel to the line passing through the points \( (-5, -6) \) and \( (2, 1) \).

### Step 1: Find the slope of the line through points \( (-5, -6) \) and \( (2, 1) \).
To find the slope \( m \), use the formula:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Here, \( (x_1, y_1) = (-5, -6) \) and \( (x_2, y_2) = (2, 1) \):

\[
m = \frac{1 - (-6)}{2 - (-5)} = \frac{1 + 6}{2 + 5} = \frac{7}{7} = 1
\]

So, the slope of the line through these points is \( m = 1 \).

### Step 2: Use the point-slope form to find the equation.
Since the line we need to find is parallel to this line, it will have the same slope, \( m = 1 \).

Now, use the point-slope form of a line:

\[
y - y_1 = m(x - x_1)
\]

Here, \( m = 1 \) and the line passes through the point \( (3, 5) \). Substituting these values into the equation:

\[
y - 5 = 1(x - 3)
\]

### Step 3: Simplify the equation.
Simplify the equation to get the slope-intercept form:

\[
y - 5 = x - 3
\]
\[
y = x + 2
\]

### Final Answer:
The equation of the line that passes through \( (3, 5) \) and is parallel to the line passing through \( (-5, -6) \) and \( (2, 1) \) is:

\[
y = x + 2
\]

Do you have any questions or would you like further details?

### Related Questions:
1. How do you find the equation of a line that is perpendicular instead of parallel?
2. What is the general form of the line equation, and how can it be derived from the slope-intercept form?
3. How would the equation change if the slope were negative?
4. What happens to the equation if the line passes through the origin?
5. Can you convert this line equation into standard form?

**Tip:** When dealing with parallel lines, remember they always have the same slope but different y-intercepts unless they are the same line.

Write an equation of a line that passes through the point (3, 5) and is parallel to the line passing through (-5, -6) and (2, 1).

Learn how to write the equation of a line passing through the point (3, 5) and parallel to the line through (-5, -6) and (2, 1). Understand key concepts in coordinate geometry, including slopes of parallel lines and using the point-slope form.

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