how do solve this equation? write an equation that is perpendicular to y = 1/2x -3 that passes through the point (4,-2)

Question

y = 1/2x -3 that passes through the point (4,-2)

Answer 1

The equation y = (1/2)x - 3 has a graph that is a line with a slope of 1/2 (m = 1/2) and a y-intercept at (0,-3), The equation is in slope-intercept form.

Any line perpendicular to that line will have a slope of -2, i.e. perpendicular lines have slopes that are negative reciprocals. In other words to find the slope of a perpendicular line find the reciprocal, then the opposite of that.

The slope 1/2 had a reciprocal of 2/1, and the opposite of that is -2/1 = -2.

Now you have a choice for finding the equation of the perpendicular passing through (4,-2). You can use the slope-intercept form (y = mx +b), substituting -2 for y, -2 for m, and 4 for x, then solve for b.

Or (my preference) you can use point-slope form (y-y₁) = m(x-x₁).

y-(-2) = -2(x - 4)

y+2 = -2(x-4) (leave is like this if you want, or continue to slope-intercept form):

y+2 = -2x +8

y = -2x+6

Visit the following graph, use the slider on b to see different lines that are perpendicular to the original.

desmos.com/calculator/1ghgnl2zxb

The above does not work if the original line is horizontal for vertical, because vertical lines do not have a slope and horizontal lines have a slope of zero.

So any perpendicular to x = 5 (for example) will have an equation y = c, where is any number.