Perpendicular lines have slopes that are negative inverses of each others. If the slope of a given line is -4, then the slope of a line perpendicular to this line is 1/4. To find a line perpendicular to a given line through a given point, use the point-slope form and plus in the perpendicular slope (negative reciprocal of the given line) and the given point. Solve for y to write the equation of the line in slope-intercept form.
Here I’m asked to write the equation of a line that’s perpendicular to a given line that goes through a given point. So if you want to, you don’t have to, if you want to, you could draw a little picture of this to help this make sense in your brain. 2x take away 4 looks something like this, well that’s not perfect, but you get the idea. Way to go. Okay and then the point (1,3) looks something like this. Again not exact, but pretty close.
And when you think about perpendicular lines, there are lots of lines that are perpendicular to this given line. Perpendicular meaning they cross at a right angle, but I don’t want any perpendicular line, I want that guy. The one that crosses at a right angle and goes through that point, that’s why this point is really important.
In order to solve this problem, I’m going to be using the Point-Slope form of the equation which looks like that. I’m going to need two pieces of information, the slope and a point. I’m in good shape because they already gave me a point, so I’m halfway there, but I do need to think about the slope of my new line. The slope of this line was 2 and I know perpendicular lines have opposite sign reciprocal slopes, so the slope of my new line is going to be instead of positive, negative and then 2 reciprocalized becomes 1/2. Reciprocalized isn’t a real word my friends, I’m just throwing it out there, you can use it if you want to.
Okay so now I’m ready to boogie. Y minus 3 equals my slope number times x minus 1, I use those numbers. Let’s continue to get this in the Slope-Intercept form, -1/2x plus 1/2 because I distributed. Let’s add 3 to both sides and I’ll have y equals -1/2x, oh I have to add 1/2+3, if I wanted to turn this into a common denominator, I would write 3 as 6/2, so when I add them together,1/2 plus 6/2 is 7/2 that would be my y intercept.
So that’s the equation in Slope-Intercept form that describes this exact line right there, that has slope -1/2 and the y intercept +7/2 that’s enough. So in order to do this kind of problem, a couple of things I want you guys to remember; the first thing is that the slope is the opposite sign reciprocal for perpendicular lines, and then you’re also going to want to start by using the Point-Slope form of the equation.
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