To determine the equation of the line passing through $A(1, -4)$ and parallel to the given line $3x + y - 4 = 0$, follow these steps:

Step 1: Identify the slope of the given line

Rewrite the equation of the given line in slope-intercept form $y = mx + c$: $3x + y - 4 = 0 \quad \Rightarrow \quad y = -3x + 4$ Here, the slope ($m$) of the given line is $-3$.

Since the required line is parallel to this line, its slope will also be $-3$.

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

The point-slope form of a line is given by: $y - y_1 = m(x - x_1)$ where $m$ is the slope and $(x_1, y_1)$ is a point on the line. Substituting $m = -3$, $x_1 = 1$, and $y_1 = -4$: $y - (-4) = -3(x - 1)$ $y + 4 = -3x + 3$

Step 3: Simplify the equation

$y = -3x + 3 - 4$ $y = -3x - 1$

Final Answer:

The equation of the line passing through $A(1, -4)$ and parallel to $3x + y - 4 = 0$ is: $y = -3x - 1$

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Would you like to see details on transformations or have any other questions?
Here are 5 related questions for practice:

1. Find the equation of a line perpendicular to $3x + y - 4 = 0$ and passing through $A(1, -4)$.
2. What is the slope of a line perpendicular to $y = -3x + 4$?
3. Determine the equation of a line passing through $B(2, 5)$ and parallel to $y = -3x + 4$.
4. How do you determine if two lines are parallel or perpendicular based on their equations?
5. Verify if the point $(0, -1)$ lies on the line $y = -3x - 1$.

Tip: Remember, parallel lines always have the same slope, and perpendicular lines have slopes that multiply to $-1$.