Algebra Know-It-ALL

248 Graphs of Linear Relations

which is the same as

y= 3 x − 2

When we work with the PS form, we come across another “sign-rigid” situation, but with minus signs

instead of a plus sign! The general form of the equation is always

y − y 0 =m(x − x 0 )

It contains two minus signs, and pays no heed to whether y 0 or x 0 happen to be positive or negative. For

example, if m= 5, x 0 =−4, and y 0 =−8, then

y − (−8)= 5[x − (−4)]

which is the same as

y+ 8 = 5(x+ 4)

We must always pay close attention to signs when working with the standard forms of linear equations. It’s

easy to get them wrong! If we see, for example,

y= 2 x − 4

then the y-intercept is b=−4. If we see

y − 3 =−4(x+ 5)

then the graph contains a point whose coordinates are (x 0 ,y 0 )= (−5, 3).

Here’s a challenge!

Imagine a straight line that passes through two points whose Cartesian coordinates are (x 1 ,y 1 ) and (x 2 ,y 2 ).

Derive an equation for this line in terms of the independent variable x and the dependent variable y. Call

this the two-point form of a linear equation. Consider x 1 ,x 2 ,y 1 , and y 2 to be constants.

Solution

Figure 15-8 shows a generic example of this situation. The line has a negative slope, but that doesn’t make

any difference in the way things will turn out. Let’s start by calculating the slope of the line. It is Δy/Δx.

Let’s move to the right, from the point (x 1 ,y 1 ) to the point (x 2 ,y 2 ). Then

Δy=y 2 − y 1

and

Δx=x 2 − x 1

The slope is

m=Δy/Δx

= (y 2 − y 1 )/(x 2 − x 1 )