Final answer:
The line perpendicular to the given equation has a negative reciprocal slope of -3. Using the point (-1, -1) and slope -3, we find the perpendicular line to be y = -3x - 4. None of the answer choices provided perfectly match this equation.
Step-by-step explanation:
To find the equation of the line perpendicular to the given equation 3y = 4x + 6 - 3x + 6, which simplifies to 3y = x + 12 or y = (1/3)x + 4, we first identify the slope of the original line. This line has a slope of 1/3. The slope of a line that is perpendicular to this line would be the negative reciprocal of 1/3, which is -3.The next step is to use the slope-point form y - y1 = m(x - x1), where (x1, y1) is the point through which the new line passes, and m is the slope of the new line. Substituting the point (-1, -1) and the slope -3, we get y - (-1) = -3(x - (-1)), which simplifies to y + 1 = -3x - 3Finding the y-intercept, we set x to 0 in the equation and solve for y: y + 1 = -3(0) - 3; y = -4. The equation of the perpendicular line in slope-intercept form y = mx + b is y = -3x - 4.The correct choice that matches this equation is A) y = -4x - 1, however, there seems to be a discrepancy.
The coefficient before x is correct (-4 matches -3x), but the constant term does not match what we derived (-1 instead of -4). Therefore, none of the choices provided perfectly match the correct equation of the perpendicular line y = -3x - 4.To find the equation of a line perpendicular to the given equation, we need to determine the slope of the given line and then find the negative reciprocal of that slope. The given equation is 3y = 4x + 6 - 3x + 6. We can rewrite this equation as 3y = x + 12. The slope of this line is 1/3. The negative reciprocal of 1/3 is -3.Now, we have the slope of the perpendicular line. To find its equation, we can use the point-slope form of a line, which is y - y1 = m(x - x1), where (x1, y1) is a point on the line.Given the point (-1, -1), we can substitute these values into the equation and the slope -3 to get: y - (-1) = -3(x - (-1)). Simplifying this equation gives y + 1 = -3x - 3. Rearranging the terms gives y = -3x - 4.Therefore, the equation of the line perpendicular to the given equation and passes through the point (-1, -1) is y = -3x - 4.