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Parallel and Perpendicular Lines - Problem 1
Remember that that parallel lines have the same slope. So if you are given the equation of a line and asked to write the equation of a line that is parallel to the given line, you know that the new line will have the same slope as the given line. If given a point where the new line passes through, plug the coordinates of the point along with the slope in the point-slope form of a line. Solve for y to write the equation in slope-intercept form.
Before I can do this problem, I have to think about what I know about parallel lines and slopes. I know that parallel lines mean they have the same steepness, they never cross, they look like this, or like that, or like that, or whatever, they never cross. The way we describe it with their equations is by making sure they have the exact same slope.
So here write the equation of the line parallel to y equals 2x take away -4 through the point (1,3). It didn’t ask me to draw a picture, but I’m a visual person, so I’m just going to draw a really rough sketch to wrap my head around what this question is asking. 2x take away 4, that means I’m going start at the y intercept of -4, go up to over 1. It’s going to look like something like that. There is my given line and I’m working with the point (1,3) something like that.
Now something you want to know about parallel lines is that there is infinitely many parallel lines to this given line. This line is parallel, that one is parallel, that one that I mean all of these lines are parallel because they’re equally as steep. They have the same slope. So this point is super important. I don’t want any parallel line, I want the exact one parallel line that goes right through that dot right there, so I’m not sure exactly what the equation is from my graph because I didn’t graph it on graph paper, I didn’t do it very precisely, but we can do it using Algebra.
Let’s see, in order to find the line that’s parallel, I’m going to use my same slope which is 2, y equals mx plus b tells me that slope is the co-efficient of x, and I also have the point (1,3) so I’m going to use the Point-Slope formula like this and plug in my x1 value and y1 value. Y minus 3 is equal to my slope number times x take away 1. Let’s go ahead and distribute so we can get this into Slope-Intercept form.
Add 3 to both sides to finish up my solving for y and I’ll have y equals 2x plus 1. That’s my final answer, and it makes sense. When I was showing you guys this line right here, I was pretty close to approximating y equals 2x plus 1, so if you wanted too, you could graph to check your solution, you don’t have to, but if you’re a visual learner like me, sometimes it helps to make a graph even if you’re not asked to.