Final answer:
The equation of the line parallel to the given line y = -4(x - 4) + 3 and passing through the point (2, -4) is y = -4x + 4.
Step-by-step explanation:
To write the equation of a line that is parallel to a given line and goes through a specific point, we need to identify the slope of the given line and use the same slope for our new line because parallel lines have equal slopes. The equation of the given line is y = -4(x - 4) + 3. This equation is in the slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept.
First, we simplify the given equation.
y = -4(x - 4) + 3
becomes
y = -4x + 16 + 3
which simplifies further to
y = -4x + 19.
The slope of the given line is -4. Since we want a line parallel to this one, our new line will have the same slope, -4. Next, we use the point-slope form of a line, which is y - y1 = m(x - x1), where (x1, y1) is the point the line passes through and m is the slope.
For the point (2, -4), we substitute 2 for x1 and -4 for y1, giving us the new equation:
y - (-4) = -4(x - 2)
or
y + 4 = -4x + 8.
To write the equation in slope-intercept form (y = mx + b), we subtract 4 from both sides:
y = -4x + 8 - 4
which simplifies to
y = -4x + 4.
This is the equation of the line that is parallel to y = -4(x - 4) + 3 and goes through the point (2, -4).