Using complete sentences, explain how to find the equation of the line, in standard form and slope-intercept form, passing through (-1, 5) and (1, 9). Also comp

Question

Using complete sentences, explain how to find the equation of the line, in standard form and slope-intercept form, passing through (-1, 5) and (1, 9). Also compare the benefits of writing an equation in standard form to the benefits of writing an equation in slope-intercept form.

Answer 1

To find the equation, in slope-intercept form, of the line that passes through the points (-1,5) and (1,9) you first have to find the slope of the line. Slope is the change in the y coordinates, Δy, divided by the change in the x coordinates, Δx. In general, given two points (x₀,y₀) and (x₁,y₁), the slope of a line is m = Δy/Δx = (y₁-y₀)/(x₁-x₀). Once you have found the slope, in the case given it is m = (9-5)/(1-(-1)) = 4/2 = 2, you must use the general slope-intercept equation to plug in the slope and one of the two given points. For example y = mx+b becomes 9 = 2*1+b. Then you solve for b. In our case you get b = 7. Now you have the slope intercept form of our line: y = 2x+7. To get standard form from here is simple. You subtract 2x from both sides to obtain the equation -2x + y = 7. If you like the first number to be positive you can simply multiply every term by -1. This yields 2x - y = -7. I encourage the reader to prove to themselves that we would have gotten the same line if we had used the other point (-1,5) to plug in to the general slope-intercept equation and solved for b. I also encourage the reader to plot all these lines to see they are all the same.

 

The benefit of writing the equation of a line in slope-intercept form is that you can see the x-intercept, the point where x=0, by covering the mx term with you finger. Because when you plug in 0 for x that term just disappears. After you have the x-intercept you can count up and over according to the slope, m, to get one more point. It only takes the two to draw a line, so you would be done. 

 

The benefit of writing the equation of a line in standard form is that it is easy to solve for both x and y intercepts. Since these are two points they are all you need to draw the line. If drawing the line is all you want, then slope-intercept form is easier. However, there are times when you are interested in both intercepts. These are the most important parts of the line in some application problems, such as maximizing profit of a company that produces multiple different products. In this case, and others like it, standard form would make your life a little easier.

 

All that being said: since you can pretty simply turn one into another, as was shown above, it doesn't make a big difference which you use, as long as you use it correctly. So if you are trying to decide which you should use for important problems, test questions for example, then use the one you are most comfortable with.

 

Good Luck