Expert: Sherry Wallin - 2/8/2009

Question

Question:  Given the slope of the line, find the equation for each line.

a) A line of slope 4 passing through the point (0,2)

b) A line of slope -5 passing through the point (3,4)

c) A line of slope 0 passing through the point (1,2)

I am absolutely lost on these problems


Answer
Hi Chad~
    You must be in a basic elementary algebra course. There are a few well known formulas that can help you out here though. The first I am going to tell you about is your 'best friend'. The equation of the form y = mx + b is an equation that gives you the most information for the least amount of work. It is also well known as the 'slope intercept' method in most textbooks because all you need is the slope m and the y intercept to know where the line is on the graph. But I prefer to think of it as my best friend and you will too when I explain why. In this equation you have the y-intercept, b, (where the line crosses the y-axis) and the slope 'm' of the line. The slope tells you just how much your line grows going uphill left to right or how much it falls left to right on your graph. When m is positive them it grows uphill and when it grows downward m is negative. Armed with these two pieces of information you can easily graph any line. Now I understand you didn't ask to graph this information but it is important to realize and recognize when you can use your 'best friend'. Notice in your first problem that you are given the slope m = 4  so the line is going uphill and the point (0,2) tells you x is 0 and y is 2 which is exactly where the line crosses the y axis,, at y = 4. You simply plug in the x and y value with the slope BECAUSE you have the y-intercept. NOTE this case only works if you have the y intercept.
So y = 4(0) + 4 -> y = 4 a horizontal line. You know it is a horizontal line because m = 0, there is no upward or downward movement, only straight a cross and because x is zero and the only time or place x is 0 is on the y axis. Which brings me to your c) question because you are told that the slope is 0, so guess what? Your y intercept is the y coordinate of the ordered pair (point) (1,2), thus y = 2 is the equation. Picture the equation this way y = 1x + 2 and when x is 0 what do you have? Go ahead and cover x up and see that y has to be 2. When x is 0 it is just like it isn't there. Now part b requires you to be a little bit more creative but actually if you know the slope formula and can find the slope you can just use the point slope formula which it is called that because all you need is one point and the slope to write the equation. Recall to find the slope you want (in English) to find the differences in the y values and divide that result by the difference in the x values in the same order. Symbolically this is (Y_2-Y_1)/(X_2-X_1) where the 1's and 2's are called subscripts (because when you write them they are smaller and near the bottom of x and y sort of like in the basement) and all they are doing is telling you that Y_2 and X_2 comes from point 2 and Y_1 and X_1 comes from point 1. You don't need to find the slope in your problem b but you might and I am sure you will be faced with a similar problem where they don't give you the slope they just give you two point and tell you to write an equation. Also that formula for the slope is where the point slope formula comes from.

   (Y_2-Y_1
m = ---------
   (X_2-X_1)

because with a very small change in it the point slope formula comes right out of it. the first change I am going to make is I am going to remove the Y_2 and X_2 and just put in Y and X because you only need one point and you can just leave those alone, I will show you how.


   (Y-Y_1)
m = ---------
   (X-X_1)

Now multiply both sides by (X-X_1) and get m(X-X_1)=(Y-Y_1) and since if 2 = x and x =2 change their position around the equal sign getting:

(Y-Y_1)= m(X-X_1) and now you have created the point slope form of a line and you have the slope -5 and you have the point (3,4), put them in their rightful spots: Y-4 = -5(x-3) and this is the point slope form of the equation of a line and if your teacher wants you to put it in your best friend form this is how you do it: use the distributive law on the right getting: Y-4 = -5x +15 and simply add 4 to both sides: Y = -5x + 19. Done deal!!

Well as I sign off you may feel you got way more than you bargained for in an answer to your question but I guarantee you, if you study this you will be a pro in no time at all...

Best Wishes,
Math Prof