Final answer:
After determining the slope of the original line is -3/4, we find the negative reciprocal for the perpendicular line's slope, 4/3. Using the point-slope form with point (2,4), we derive the equation of the new line but the resulting equation does not match any of the answer choices exactly, suggesting an error in the provided options.
Step-by-step explanation:
To find the equation of the line that is perpendicular to another line, you must first determine the slope of the original line. In this case, the slope of the given equation, 9/4x + 3y = -51/4, can be isolated by putting it in y = mx + b form. This results in y = -3/4x - 17/4. The slope of the original line is -3/4, so the slope of the perpendicular line will be the negative reciprocal, which is 4/3.
Now, using the point (2, 4) and the slope 4/3, we can use the point-slope form of the equation, which is y - y1 = m(x - x1). Plugging in the values, we get y - 4 = 4/3(x - 2). Simplifying this will lead to the equation of the line.
Distributing the 4/3 to both x - 2, we get y - 4 = 4/3x - 8/3. To get the equation in the form of y = mx + b, we add 4 to both sides, resulting in y = 4/3x + 4/3. However, our answer choices are not in this form; they are in standard form Ax + By = C. Multiplying everything by 3 to clear the fraction, we get 3y = 4x + 4. Rewriting this in standard form, we get 4x - 3y = -4. We need to find the equivalent equation among the choices that is a multiple of this equation.
By multiplying the equation by -1, we reveal the correct answer: -4x + 3y = 4, which corresponds to 3y - 4x = -4, or rearranged, 4x - 3y = 4. None of the answer choices directly match this, but multiplying by -1 should not change the nature of the equation. It immediately looks like C) x - 3y = -10 is the closest match. Let's check by rearranging our equation to match the structure of choice C: Dividing everything in 4x - 3y = 4 by 4 yields x - 3y = 1, which does not match exactly any of the given choices since they didn't account for fractional coefficients.
Given that the provided options don't exactly match the derived equation, this scenario points towards a potential typo or error in the question or answer choices. However, none of the options given are correct based on standard algebraic manipulations.