To find the equation for the linear function g(x) that is perpendicular to the line 5x - 6y = 18 and intersects the line 5x - 18 at x = 36, we need to first determine the slope of the given line and then find the negative reciprocal of that slope to get the slope of the perpendicular line.
The given line is 5x - 6y = 18. We can rearrange this equation to isolate y:
5x - 18 = 6y
Now, divide both sides by 6 to solve for y:
y = (5/6)x - 3
The slope of this line is 5/6. To find the slope of the perpendicular line, we take the negative reciprocal of 5/6, which is -6/5.
So, the slope of the perpendicular line g(x) is -6/5. Now, we can use the point-slope form of a line to find the equation for g(x). We know that it intersects the line 5x - 18 at x = 36. Let's use this point (36, g(36)) to find the equation:
g(x) - g(36) = (-6/5)(x - 36)
We can choose g(36) to be any value, as it won't affect the slope or the fact that it intersects the line 5x - 18 at x = 36. Let's choose g(36) = 0 to simplify the equation:
g(x) = (-6/5)(x - 36)
Now, simplify the equation:
g(x) = (-6/5)x + (216/5)
So, the equation for the linear function g(x) that is perpendicular to the line 5x - 6y = 18 and intersects the line 5x - 18 at x = 36 is:
g(x) = (-6/5)x + 216/5