X AND Y INTERCEPTS
Any straight line which is not parallel to one. of the axes has an X intercept and a Y intercept. These are the points at which the line crosses the X and Y axes. At the X intercept, the graph line is touching the X axis, and thus the Y value at that point is 0. At the Y intercept, the graph line is touching the Y axis; the X value at that point is 0.
In order to find the X intercept, we simplylet y = 0 and find the corresponding value of x. The Y intercept is found by letting x = 0 and finding the corresponding value of y. For ex- ample, the line
5x + 3y = 15
crosses the Y axis at (0,5). This may be verlfied by letting x = 0 in the equation. The X intercept is (3,0), since x is 8 when y is 0. Picture 12-3 shows the Line
5x + 3y = 15
graphed by means of the X and Y intercepts.
EQUATIONS IN ONE VARIABLE
An equation containing only one variable itseasily graphed, since the line it represents lies parallel to an axis. For example, in
the value of y is
Figure 12-3.-Graph of 5x + 3y = 15.
The line 2y = 9 lies parallel to the X axis at adistance of 4 1/2 units above it. (See fig. 12-4.) Notice that each small division on the graph paper in figure 12-4 represents one-half unit.
The line 4x + 15 = 0 lies parallel to the Yaxis. The value of x is - 15/4. Since this value is negative, the line lines to the left of the Y axis at a distance of 3/4 units. (See fig. 12-4.)
Figure12-4.-Graphs of 2y = 9 and 4x + I5 = 0.
From the foregoing discussion, we arrive attwo important conclusions:
1. A pair of numbers that satisfy an equation are the coordinates of a point on the graphof the equation.
2. The coordinates of any point on the graphof an equation will satisfy that equation.
SOLVING EQUATIONS INTWO VARIABLES
A solution of a linear equation in two variables consists of a pair of numbers that satisfythe equation. For example, x = 2 and y = 1 constitute a solution of
3x - 5y = 1
When 2 is substituted for x and 1 is substitutedfor y, we have
3(2) - 5(1) = 1
The numbers x = -3 and y = -2 also form asolution. This is true because substituting -3 for x and -2 for y reduces the equation to an identity:
Each pair of numbers (x, y) such as (2, 1) or(-3, -2) locates a point on the line 3x - 5y = 1. Many more solutions could be found. Any two numbers that constitute a solution of the equation are the coordinates of a point on the line represented by the equation.
Suppose we were asked to solve a problemsuch as: Find two numbers such that their sum is 35 and their difference is 5. We could indicate the problem algebraically by letting x represent one number and y the other. Thus, the problem may be indicated by the two equations
Considered separately, each of these equationsrepresents a straight line on a graph. There are many pairs of values for x and y which satisfy the first equation, and many other pairs which satisfy the second equation. Our problem
is to find ONE pair of values that will satisfyBOTH equations. Such a pair of values is said to satisfy both equations at the same time, or simultaneously. Hence, two equations for which we seek a common solution are called SIMULTANEOUS EQUATIONS. The two equations, taken together, comprise a SYSTEM of equations.