Final answer:
To find the equation of a line that is perpendicular to the given line and passes through the point (5,6), first determine the slope of the given line. Then, find the negative reciprocal of that slope to get the slope of the perpendicular line. Finally, use the point-slope form of a line to find the equation using the given point and the perpendicular slope.
Step-by-step explanation:
To find the equation of a line that is perpendicular to the given line and passes through the point (5,6), we need to follow a few steps:
1. Determine the slope of the given line by rearranging the equation in the form y = mx + b, where m is the slope. In this case, the given line is 6(y+x) - 2(x-y) = -2. Simplifying this equation, we get 8x - 4y = -2. Rearranging, we find that y = 2x + 1. Therefore, the slope of the given line is 2.
2. The slope of a line perpendicular to another line is the negative reciprocal of the slope of the given line. So the slope of the perpendicular line is -1/2.
3. Now, we can use the point-slope form of a line to find the equation. The point-slope form is given by y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope. Substituting the values (5,6) and -1/2 into the equation, we get y - 6 = -1/2(x - 5). Simplifying and rearranging, the equation of the line is y = -1/2x + 11.