To find the equation of a line in slope-intercept form that passes through a given point and is parallel to another line, we will use the slope-intercept form:

$y = mx + b$

where:

• $m$ is the slope of the line.
• $b$ is the y-intercept.

Step 1: Determine the slope of the parallel line

The given line is:

$y = 10x - 5$

The slope of this line is the coefficient of $x$, which is $10$. Since parallel lines have the same slope, the slope of the desired line will also be $10$.

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

The point given is $(-7, -5)$, and the slope $m = 10$. The point-slope form of a line is:

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

Substitute the given point $(-7, -5)$ and the slope $10$:

$y - (-5) = 10(x - (-7))$

Simplify the equation:

$y + 5 = 10(x + 7)$

Step 3: Convert to slope-intercept form

Now, distribute and simplify:

$y + 5 = 10x + 70$

Subtract $5$ from both sides to solve for $y$:

$y = 10x + 65$

Final Equation

The equation of the line in slope-intercept form that passes through the point $(-7, -5)$ and is parallel to the line $y = 10x - 5$ is:

$\boxed{y = 10x + 65}$

Would you like more details on any step or have any questions?

Here are some related questions you might find interesting:

1. How do you find the equation of a line perpendicular to a given line?
2. What are the different forms of linear equations?
3. How do you find the distance between a point and a line?
4. What is the significance of the slope in the context of linear equations?
5. How can you determine if two lines are parallel or perpendicular by their equations?

Tip: When finding equations of lines, remember that parallel lines always share the same slope, while perpendicular lines have slopes that are negative reciprocals.