Algebra Know-It-ALL

238 Graphs of Linear Relations

Switching the order

We can switch the points (x 1 ,y 1 ) and (x 2 ,y 2 ) and still get the same slope when we calculate it as

above. That’s because both the numerator and the denominator end up being additive inverses

(exact negatives) of what they were before. Let’s take

(x 1 ,y 1 )= (3, 5)

and

(x 2 ,y 2 )= (−2, 4)

The slope is again equal to Δy/Δx. Calculating, we get

m=Δy/Δx

= (y 2 − y 1 )/(x 2 − x 1 )

= (4 − 5)/(− 2 − 3)

=− 1 /(−5)

= 1/5

When we know the coordinates of two points on a line, we can figure the slope going from the

first point to the second, or going from the second point to the first; it doesn’t matter. But we

must be careful not to confuse the coordinates. We can reverse the external sequence in which

we work with the points, but we can’t reverse the internal sequence of either of the ordered

pairs defining those points!

What is the intercept?

When we talk about the SI form of a straight line in the Cartesian plane, the term intercept

refers to the value of a variable at the point where the line crosses the axis for that variable.

If y is the dependent variable, then we often talk about the y-intercept. That’s what is usually

meant when we work with the SI form of an equation when graphing it in the xy-plane. Two

examples are shown in Fig. 15-2.

An intercept can be thought of as an ordered pair where one of the values (the one on the

axis not being intercepted) is 0. We can plug 0 into a linear equation for one of the variables,

and solve for the remaining variable to get its intercept. This method can be more convenient

than rearranging everything into SI form or drawing a graph, but all by itself it doesn’t give us

any visual reinforcement of the situation.

Putting it together

In Chap. 12, you learned the standard form for a first-degree equation in one variable. If the

variable is x, the standard form is

ax+b= 0