CHAPTER 12
LINEAR EQUATIONS IN TWO VARIABLES
Thus far in this course, discussions of equations have been limited to linear
equations in one variable. Linear equations which have
two variables are common, and their solution involves
extending some of the procedures which have already
been introduced.
RECTANGULAR COORDINATES
An outstanding characteristic of equations in two
variables is their adaptability to graphical analysis.
The rectangular coordinate system, which was
introduced in chapter 3 of this course, is used in
analyzing equations graphically. This system of
vertical and horizontal lines, meeting each other at
right angles and thus forming a rectangular grid, is
often called the Cartesian coordinate system. It is
named after the French philosopher and mathematician,
Rene Descartes, who invented it.
COORDINATE AXES
The rectangular coordinate
system is developed on a framework of reference similar to figure
32 in chapter 3 of this course. On a piece of graph
paper, two lines are drawn intersecting each other at right angles, as in
figure 121. The vertical line is usually labeled
with the capital letter Y and called the Y axis. The
horizontal line is usually labeled with the capital
letter X and called the X axis. The point where the X
and Y axes intersect is called the ORIGIN and is
labeled with the letter o. Above the origin, numbers
measured along or parallel to the Y axis are positive;
below the origin they are negative. To the right of
the origin, numbers measured along or parallel to
the X axis are positive; to the left they are negative.
COORDINATES
A point anywhere on the graph may be located by two numbers, one showing the
distance of the point from the Y axis, and the other
showing the distance of the point from the X axis.
Figure 121 Rectangular coordinate system. Point P
(fig. 121) is 6 units to the right of the Y axis and
3 units above the X axis. We call the numbers that
indicate the position of a point COORDINATES. The
number indicating the distance of the point measured
horizontally from the origin is the X coordinate (6 in
this example), and the number indicating the distance
of the point measured vertically from the origin (3 in
this example) is the Y coordinate.
In describing the location of a point by means of
rectangular coordinates, it is customary to place the
coordinates within parentheses and separate them with
a comma. The X coordinate is always written first. The
coordinates of point P (fig. 121) are written (6, 3).
The coordinates for point Q are (4, 5); for point R, they
are (5, 2); and for point S, they are (8, 5).
Usually when we indicate a point on a graph, we
write a letter and the coordinates of the point. Thus,
in figure 121, for point S, we write S(8, 5).
The other points would ordinarily be
written, P(6, 3), Q(4, 5), and R(5, 2). The Y
coordinate of a point is often called its ORDINATE and the X coordinate is often
called its ABSCISSA.
QUADRANTS
The X and Y axes divide the graph into four parts
called QUADRANTS. In figure 121, point P is in
quadrant I, point S is in quadrant II, R is in
quadrant III, and Q is in quadrant IV. In the first
and fourth quadrants, the X coordinate is positive,
because it is to the right of the origin. In the
second and third quadrant it is negative, because it
is to the left of the origin. Likewise, the Y
coordinate is positive in the first and second
quadrants, being above the origin; it is negative in
the third and fourth quadrants, being below the
origin. Thus, we know in advance the signs of the
coordinates of a point by knowing the quadrant in
which the point appears. The signs of the coordinates
in the four quadrants are shown in figure 121.
Locating points with respect to axes is called PLOTTING.
As shown with point P (fig. 12l), plotting a point is
equivalent to completing a rectangle that has segments
of the axes as two of its sides with lines dropped
perpendicularly to the axes forming the other two
sides. This is the reason for the name
"rectangular coordinates."
PLOTTING A LINEAR EQUATION
A linear equation in two variables may have many
solutions. For example, in solving the equation 2x  y = 5, we can find an
unlimited number of values of x for which there will
be a corresponding value of y. When x is 4, y is 3,
since (2 x 4)  3 = 5. When
x is 3, y is 1, and when x is 6, y is 7. When we graph
an equation, these pairs of values are considered coordinates of points on the
graph. The graph of an equation is nothing more than a
line joining the points located by the various pairs
of numbers that satisfy the equation.
To picture an equation, we first find several pairs
of values that satisfy the equation. For example, for
the equation 2x  y = 5, we assign several values to x
and solve for y. A convenient way to find values is to first solve the equation
for either variable, as follows:
Once this is accomplished, the value of y is readily
apparent when values are substituted for x. The
information derived may be recorded in a table such as table 121. We then
lay off X and Y axes on graph paper, select some
convenient unit distance for measurement along the
axes, and then plot the pairs of values found for x
and y as coordinates of points on the graph. Thus, we
locate the pairs of values shown in table 121 on a
graph, as shown in figure 122 (A).
Table 121.Values of x and y in the equation
Figure 122.Graph of 2x  y = 5.
Finally, we draw a line joining these points, as in
figure 122 (B). It is seen that this is a straight
line; hence the name "linear equation." Once
the graph is drawn, it is customary to write the
equation it represents along the line, as shown in
figure 122 (B).
It can be shown that the graph of an equation is
the geometric representation of all the points whose
coordinates satisfy the conditions of the equation.
The line represents an infinite number of pairs of coordinates for this
equation. For example, selecting at random the point
on the line where x is 2: and y is 0 and substituting
these values in the equation, we find that they
satisfy it. Thus,
If two points that lie on a straight line can be
located, the position of the line is known. The
mathematical language for this is "Two points
DETERMINE a straight line." We now know that the
graph of a linear equation in two variables is a
straight line. Since two points are sufficient to
determine a straight line, a linear equation can be
graphed by plotting two points and drawing a straight
line through these points. Very often pairs of whole
numbers which satisfy the equation can be found by
inspection. Such points are easily plotted.
After the line is drawn through two points, it is
well to plot a third point as a check. If this third
point whose coordinates satisfy the equation lies on the line the graph is
accurately drawn.
