121. Equivalence of pairs of equations. In the theorems of this section the capital letters represent polynomials in x and y, and the small letters represent numbers not equal to zero.

If (x1, y1) be a pair of values that satisfy (1), then when x and y in B2 are replaced by x1 and the equation B2 : Yı

= 62 is a numerical identity. These values (x1, 71) must then satisfy one of the equations Bb, for if they did not, but only satisfied the equation say Bc when c±b, then the hypothesis that B2 = b2 is satisfied by (x1, 1) would be contradicted.

Conversely, any pair of values that satisfy B = ±b evidently satisfy B2 = b2.

This theorem is used, for instance, in exercise 2, p. 114, and justifies the assumption that

If A

A =
= α

AB

=

=a and B

ab

}(1) and

A:

B = b } (2)

= b are satisfied by a pair of numbers (x1, yı), we multiply the identities and obtain AB

=

= ab.

Conversely, if Aa, AB ab are identically satisfied by a pair (x1, 1), since a 0 we can divide the second identity by the first and obtain B = b. Thus if (1, 1) satisfy one pair of equations they satisfy the other pair.

This theorem is assumed in exercise 3, p. 115, to show that

are equivalent where a, b, c, and d are numbers such that

ad - be 0.

If (1, 1) satisfy (1), evidently it also satisfies (2). Thus all solutions of (1) are among those of (2).

Conversely, if (x1, 71) satisfy (2), then

This theorem has been assumed in exercises 1, 2, 3, 6, p. 114. In 1, for example, it is necessary to show that

== 40.

are equivalent. In this case a = c = 1, b = - d= 2. Thus ad bc=

122. Incompatible equations. When a pair of simultaneous equations can be proven equivalent to a pair of equations which contradict each other or are absurd, they are incompatible and have no finite solution.

123. Graphical representation of simultaneous quadratic equations. Every equation that we have considered may be rep

resented graphically by plotting

in accordance with the method already given (p. 93).

The solution of simultaneous equations is represented by the points of intersection of the corresponding graphs.

Thus the equations

x2 + y2 = 25,

These equations have as their graph the preceding figure.
The equations

Simultaneous equations which have

imaginary solutions also lead to non-inter

secting graphs (p. 101).

Thus the equations

x2 + y2 = 4,

4x+5y=20

have the adjacent figure as their graph.

EXERCISES

1. Interpret the graphical meaning of equivalent pairs of equations.

What general statement concerning the graphical meaning of a single solution of quadratic and linear equations does this example suggest? 3. Plot and solve the following:

What general statement concerning the graphical interpretation of four, three, or two real solutions of equations do these examples suggest?

4. State the algebraical condition under which two quadratic equations have four, three, two, or one real solutions (see p. 113).

124. Graphical meaning of homogeneous equations. Consider for example

Y

4y-x=0 and 3x + 2y = 0.

These equations represent two straight lines through the origin which taken together form the graph of equation (1). This example may obviously be generalized: Any homogeneous equation of the form ax2 + bxy + cy2 = 0 with positive discriminant represents two straight lines through the origin. Such an equation is equivalent to two linear equations.

In an example like 5, p. 115, we obtain in place of the given pair of equations a pair of equivalent equations one of which is homogeneous and the other of which is factorable. We can learn the graphical meaning of this method of solution by studying a particular case. Consider for example the equations:

By eliminating the constant terms we obtain the product of the two equations x + y: O and x= -2y= 0. Thus the problem of solving (1) and (2) is replaced by that of solving the two following pairs of equations:

finding the intersection of the graph of (1) with a pair of straight lines. This appears in the figure where the curves and lines are numbered as above. The closed curve represents (1).