242 Graphs of Linear Relations
Solution
Table 15-1 is an S/R derivation of a SI equation from the standard form of a two-variable linear equation.
Here, the familiar m for slope is replaced by −a/b, and the familiar b for slope is replaced by −c/b. The
result is in the correct form; that’s the important thing! The coefficient of y cannot equal 0 in the original
equation; that would cause both the slope and the y-intercept to be undefined. The graph in such a case
would exist, but it would be a vertical line, so it would not represent a function of x.Point-Slope Form
Another common way to express a linear function is known as the point-slope form. As the
name suggests, we can draw the graph of a function if we know the coordinates of any single
point on the line, and if we also know the slope of the line. Let’s call this the PS form.The form
Here is the standard PS form for a linear function. Later in this chapter, we’ll figure out how
this form is derived:y − y 0 =m(x − x 0 )where x is the independent variable, y is the dependent variable, m is the slope, and (x 0 ,y 0 ) are
the coordinates of a known point on the graph.An example
Suppose we’re told that there’s a linear function whose graph contains the point (−1, 2). We
are also told that the slope of the graph is 2. The independent variable is x, and the dependent
variable is y. Our task is to draw a graph of the function.
Let’s begin by assigning x 0 =−1 and y 0 = 2. When we plug these numbers into the stan-Table 15-1. Conversion of a general two variable linear equation to SI form. This
only works if the constant b (the coefficient of y) is not equal to 0. In this result, the
slope is equal to −a/b, and the y-intercept is equal to −c/b.
Statements Reasons
ax+by+c= 0 This is the equation we are given
ax+by=−c Subtract c from each side
by=−ax−c Subtract ax from each side
Require that b≠ 0 We’re about to divide through by b
y= (−ax−c)/b Divide through by b
y=−ax/b−c/b Right-hand distributive law for division over subtraction
y= (−a/b)x−c/b Rearrange to define the coefficient for x
y= (−a/b)x+ (−c/b) Change subtraction to negative addition, putting the
equation into strict SI form