What in the world is y = y1 + b (x - x1) ?

Question

Is it supposed to be a formula or something? :P

The slope is -2, and the points are (2,5) and (4,1).

What does [b] stand for? Is it like the [b] in y = mx + b ?

How about the [x]?

Thankies.... :)

Answer 1

y = y1 + b (x - x1) is a point slope formula. It is also in the form y - y1 = m (x - x1). There is also a 2 point formula where you substitute m = (y2 - y1) / (x2 - x1).

This is consistent with the points you were given as (1-5) / (4-2) = -2. This is the slope you were given. Therefore, (x1,y1) = (2,5) and you need only plug the numbers into the formula, or...

y = 5 - 2 (x - 2) => y = -2x + 9

Answer 2

I guess you know that  y  =  mx + c is the "general" equation for a straight line? (usually "c" is used where you have put "b" because it stands for  "constant"  -  i.e. it is a fixed number, not a variable.  Actually, in the formula with which you begin your question, your "b" corresponds to the "m"  in y  =  mx + c (see below).

x and y stand for any point on the line.  m represents the slope of the line, and  b  is the point where the line crosses the y-axis.

A common problem is to find the equation of the line when you are given two points on it  -  here, you are given  (2, 5) and (4, 1)

To show that these are specific points and not just any points on the line, they are given a subscript, so they will be referred to as  (x1, y1) and (x2, y2).

The formula for finding the slope from the two given points is usually written as

m  =  (y2 - y1) /(x2 - x1)

So the slope of your line is  (1 - 5) / (4 - 2)  =  - 4 / 2  =  -2, as you state.

You will see that the way I wrote the formula above gives you the answer  - 2 for the value of m, so the equation of the line therefore is

y  =  - 2x  +  c

but you still need to find the value of  c, which you do by inserting the values for one point into the equation - say you choose  (2, 5).

Then  5  =  - 2(2) + c,  from which you find that  c  =  5 + 4  =  9

And the full equation  is  y  =  - 2x + 9

--------------------------------------...

Now, say that instead of being given two points on the line, you are told that the slope  is  - 2  and that it goes through the single point (2, 5).  (This is another common way of putting a problem)

You write  y  =  - 2x + c as I did just above, then put in the values (2, 5) for x  and  y to determine what value to use for c.

However, since you already know the value for the slope (m), you can write down an expression for the straight line directly like this :

y  =  5  +  (- 2)(x - 2)

and this is exactly the formula which you quoted to begin with (except that you have used "b" instead of "m" to represent the slope of the line.

(If you multiply out and re-arrange this equation, you get  

y  =  5  -  2x  +  4   =  - 2x + 9, the same as before.)

So you have squashed two forms of the problem into one question, your formula is the quick way to the solution when you are given  the slope and one point on the line.  

If, however, you are given two points on the line and need to find the slope, you use  (y2 - y1) / (x2 - x1).

Finally, if you are given two points and need to find the equation of the line joining them, you can combine the two methods above and write the equation in one fell swoop like this

y - y1 =  {(y2 - y1) / (x2 - x1)} {x - x1}

and this is exactly your formula, except that you have put y1 on the right-hand side instead of the left  ( remembering that  (y2 - y1) / (x2 - x1)  is what you have called "b").

Answer 3

y = y1 + b (x - x1)

Y=5(1)+B(X-2(1))

Y=5-B2-B2

Y=5

I THINK....NOT SURE THOUGH

Answer 4

it is the "point - slope' form of the line..[y - y1] / [x - x1] = b , the slope between the points (x,y) and (x1,y1) ....and no , it is not the y intercept of y = mx + b