Final answer:
The equation for a line perpendicular to 3x+6y=36 that passes through the point (3,4) is y = 2x - 2. This is found by calculating the slope of the given line, determining the negative reciprocal for the perpendicular slope, and using the point provided to find the y-intercept.
Step-by-step explanation:
To find an equation for a line that is perpendicular to the given line 3x+6y=36 and goes through the point (3,4), we first need to determine the slope of the given line. Remembering that the slope-intercept form of a line is y = mx + b, we start by writing our given equation in this form.
By dividing the entire equation 3x+6y=36 by 6, we get y = -0.5x + 6. This tells us that the slope (m) of the given line is -0.5. The slope of a line perpendicular to this one would then be the negative reciprocal of -0.5, which is 2.
Now that we have the slope of the perpendicular line, we can use the point (3,4) to find its y-intercept (b) using the point-slope form, which is y - y1 = m(x - x1) where (x1, y1) is our point. Plugging in our values, we get 4 = 2(3) + b, simplifying to b = -2.
So the final equation of our perpendicular line in slope-intercept format is y = 2x - 2.