Final answer:
To find the equation of a line parallel to y - 4 = x(y - 4) that passes through the point (7, -4), we need to find the equation of the given line and then use its slope to find the equation of a parallel line. None of the given options is the equation of the line parallel to the given line.
Step-by-step explanation:
To find the equation of a line parallel to y - 4 = x(y - 4) that passes through the point (7, -4), we need to find the equation of the given line and then use its slope to find the equation of a parallel line. The given equation can be simplified to y - 4 = xy - 4x. Rearranging the terms, we get y - xy = 4 - 4x. Combining like terms, we have (1 - x)y = 4 - 4x. Dividing both sides by (1 - x), we get y = 4 - 4x / (1 - x). Now that we have the equation of the given line, we can find its slope. The slope of a line is given by the coefficient of x, so in this case, the slope is -4 / (1 - x).
A line parallel to the given line will have the same slope. Therefore, the equation of the parallel line passing through (7, -4) can be found by substituting the coordinates into the point-slope form equation: y - y1 = m(x - x1), where (x1, y1) is the point and m is the slope. Substituting the given values, we get y - (-4) = -4 / (1 - x)(x - 7). Simplifying further, we obtain y + 4 = -4 / (1 - x)(x - 7). Therefore, none of the given options (Option 1: y = x - 4, Option 2: y = -x + 3, Option 3: y = 2x + 4, Option 4: y = 3x - 2) is the equation of the line parallel to the given line.