To find the equation of the line that passes through the point $(4, 1)$ and is parallel to the line $y = -2x + 7$, we can follow these steps:

1. Identify the slope of the given line: The equation of the given line is in slope-intercept form: $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

   • The slope of the given line is $m = -2$.

2. Parallel lines have the same slope: Since the required line is parallel to the given line, it will have the same slope. Therefore, the slope of our new line will also be $m = -2$.

3. Use the point-slope form of the equation of a line: The point-slope form is given by: $y - y_1 = m(x - x_1)$ where $(x_1, y_1)$ is a point on the line, and $m$ is the slope. Substituting the known values $m = -2$ and $(x_1, y_1) = (4, 1)$, we get: $y - 1 = -2(x - 4)$

4. Simplify the equation: Now, let's simplify this expression: $y - 1 = -2(x - 4)$ $y - 1 = -2x + 8$ $y = -2x + 9$

So, the equation of the line that passes through $(4, 1)$ and is parallel to $y = -2x + 7$ is: $y = -2x + 9$

Would you like a more detailed explanation or have further questions on this topic? Here are some related questions you could explore:

1. How do we determine the slope of a line given two points?
2. What is the general form of the equation of a line?
3. How can we find the equation of a line that is perpendicular to another line?
4. How do we convert between different forms of the equation of a line (point-slope, slope-intercept, standard)?
5. How can we graph the equation $y = -2x + 9$ on a coordinate plane?

Tip: Always remember that parallel lines share the same slope, and only the y-intercept can change between them.