To write the equation of the line that is parallel to the given line $y = -2x - 1$ and passes through the point $(4, 3)$, we need to follow these steps:

Step 1: Identify the slope of the given line

The given line is in slope-intercept form $y = mx + b$, where $m$ is the slope. From $y = -2x - 1$, we see that the slope $m = -2$.

Step 2: Use the same slope for the parallel line

Since parallel lines have the same slope, the slope of the required line is also $m = -2$.

Step 3: Use the point-slope form of the equation

The equation of a line passing through a point $(x_1, y_1)$ with slope $m$ is given by the point-slope formula: $y - y_1 = m(x - x_1)$ Substitute $m = -2$, $x_1 = 4$, and $y_1 = 3$ into the formula: $y - 3 = -2(x - 4)$

Step 4: Simplify the equation

Now, simplify the equation: $y - 3 = -2x + 8$ Add 3 to both sides: $y = -2x + 8 + 3$ $y = -2x + 11$

Final Answer:

The equation of the line parallel to $y = -2x - 1$ and passing through the point $(4, 3)$ is: $y = -2x + 11$

Do you want more details or have any questions?

Here are 5 related questions to extend your understanding:

1. How do you find the equation of a line that is perpendicular to a given line?
2. What would the equation be if the line passed through a different point, say $(1, 2)$?
3. How do you convert a line from point-slope form to slope-intercept form?
4. What is the geometric significance of two lines being parallel?
5. How does the slope change if the lines are neither parallel nor perpendicular?

Tip: When writing the equation of a line, always start by identifying the slope; this is key for parallel and perpendicular line problems!