Hi,
I think the question is finding the equation of the line which passes through the point (3,4) with y-intercept=- -2
First note that a line has a general equation of the form y=mx+b where m is the slope, and why is the y-intercept.
y-intercept is the value of y when x is equal to 0. Therefore as a coordinate (point), y intercept is denoted as (0,b) and it means that when x=0, y has the value of b which is what we call the y-intercept.
Having said that, the y-intercept is clearly -2 (since when x=0, y=-2). We do not yet know what the slope is, so let's go ahead and plug in the value of b in the equation for y=mx+b for now.
y= mx -2
alright, so all we have to do is to find the slope which is denoted by m. Fortunately we can do this using the other point (3,4). When x=3, we know that y should be equal to 4. Let's plug in those value into the equation:
4 = 3m -2
so 3m = 6 and so m=2
There you go! we found the value of m. Let's plug it back into the equation and we are done!
y = 2x -2
now just a few things worth mentioning:
in order to find m, I could have used a different approach. m or slope is defined as rise over run and if you have two points, you can find the slope by taking the difference of y values and dividing by the difference of x values. So we have (3,4) and (0,2)
change in y= -2-4 = -6 change in x =0-3=-3
so slope would be: -6/-3 = +2
and remember that this is the same as we got earlier.
One last thing! our final answer was y = 2x-2
we can verify this is correct by plugging in the two points we have for x.
if x= 0 then y = 2(0) -2 =-2 which it has to be based on the information we had for y-intercept.
if x = 3 then y= 2(3)-2 = 6-2 =4 and this is also true based on the information we had.
So that means we have found the right equation. And remember that for the equation you just need to find the slope (m) and the y intercept (b) using the values of x and y that are provided, but the final form of the equation should have the variable x in it since x can take infinite number of points. For instance if x is equal to 10, then from our equation y=-2 (10)-2 = -20-2 = -22
so we find the equation and that equation helps us find the y values for any values of x and not just the two that we already know.
Hope that helps.
Kaveh
