Locating the points tabulated and drawing the graphs we have a curve for equation (1) and a straight line for equation (2). The intersections are points whose coördinates satisfy both equations and therefore give the roots of the system.

The roots are approxi

mately

x = 2.1, 2.7.

y = .4, 2.8.

If solved by the usual

method, we find

x = 2.1+, - 2.6+.

y = .45+, 2.8+.

The student is not to understand that the graphical method of solving a system of simultaneous equations is to replace the algebraic method. The algebraic method is generally much shorter than the graphical method. However, the graphical method of representing equations plays a very important part in higher

mathematics and in the applications of mathematics to problems of physics and engineering. It may also be noted that the algebraic methods do not

always furnish the solutions of simultaneous quadratics (§ 551). The graphical method can generally be depended upon to give good approximations to the real roots in such cases.

The algebraic solution gives

x = √10, -√10 or 3.16+, 3.16+, −3.16+, −3.16+

y = ±√õ, ±√6 or 2.44+, - 2.44+, 2.44+, - 2.44+.

The curve for equation (2) is a hyperbola. The curve for a second degree equation in two unknown numbers is, in general, a circle, a parabola, an ellipse, or a hyperbola.

579. Imaginary roots cannot be found by this method. The presence of imaginary roots is indicated by a failure of the graphs to intersect. Thus, if we attempt to solve the system

x2-y=4, (1), x − y = 5, (2), we shall find that the graphs have no common points. The graph of the first equation is shown in § 578. The second gives the line (2) as shown in the figure. The algebraic solution of this system gives

y=

:

XXV. THE PROGRESSIONS

ARITHMETICAL PROGRESSION

581. Series. A succession of terms formed according to some definite law is a series.

Thus,,, and a, a2, a3

each?

...

are series. What is the fourth term of

582. Arithmetical Progression. A series in which each term after the first is found by adding a constant quantity to the preceding term is an arithmetical progression (A.P.).

.

583. Common Difference. common difference.

The constant number added is the The common difference is found by sub

tracting any term from the term immediately following it.

Thus, 1, 3, 5, 7

and 12, 8, 4, 0, -4, - 8 ... are arithmetical progressions. In the first, 2 is the common difference and is added to each term to form the next; in the second, - 4 is the common difference and

is added to each term to form the next.

ORAL EXERCISE

584. What is the common difference in each of the following series?

9. Form the next two terms in each of the series in examples

1 to 6.