Recall that the general form for an equationin the first degree in one variable is ax + b = 0. The general form for first-degree equations in two variables is
ax + by + c =0
It is interesting and often useful to note whathappens graphically when equations differ, in certain ways, from the general form. With this information, we know in advance certain facts concerning the equation in question.
LINES PARALLEL TO THE AXES
If in a linear equation the y term is missing, as in
2x - 15 = 0
the equation represents a line parallel to the Yaxis and 7 1/2 units from it. Similarly, an equation such as
4y - 9 = 0
which has no x term, represents a line parallel to the X axis and 2 1/4 units from it. (Seefig. 12-a.)
The fact that one of the two variables doesnot appear in an equation means that there are no limitations on the values the missing variable can assume. When a variable does not appear, it can assume any value from zero to plus or minus infinity. This can happen only if the line represented by the equation lies parallel to the axis of the missing variable.
Lines Passing Through the Origin
A linear equation, such as
that has no constant term, represents a linepassing through the origin. This fact is obvious since x = 0, y = 0 satisfies any equation not having a constant term. (See fig. 12-a.)
Lines Parallel to Each Other
An equation such as
has all possible terms present. It representsa line that is not parallel to an axis and does not pass through the origin.
Equations that are exactly alike, except forthe constant terms, represent parallel lines. As shown in figure 12-8, the lines represented by the equations
3x - 2y = -18 and 3x - 2y = 6
Parallel lines have the same slope. Changing the constant term moves a line away fromor toward the origin while its various positions remain parallel to one another. Notice in figure 12-8 that the line 3x - 2y = 6 lies closer to the origin than 3x - 2y = -18. This is revealed at sight for any pair of lines by comparing their constant terms. That one which has the constant term of greater absolute value will lie farther from the origin. In this case 3x - 2y = -18 will be farther from the origin since |-18| > |16|..
The fact that lines are parallel is indicatedby the result when we try to solve two equations such as 3x - 2y = -18 and 3x - 2y = 6 simultaneously. Subtraction eliminates both x and y immediately. If both variables disappear, we cannot find values for them such that both equations are satisfied at the same time. This means that there is no solution. No solution implies that there is no point of intersection for the straight lines represented by the equations. Lines that do not intersect in the finite plane are parallel.