CHAPTER XXIII

SIMULTANEOUS QUADRATIC EQUATIONS. PROBLEMS

337. In the solution of simultaneous quadratic equations we have particular methods for dealing with three common types; but no general method for all possible cases can be given.

SOLUTION BY SUBSTITUTION

338. When one equation is of the first degree and the other of the second degree.

Substituting in (1), 2x2-3x (3 – 5 x) — (3 – 5 x)=1.

(1)

(2)

(3)

(4)

Substituting in (3),

If x, y 3-5 (†) = − }.

=

If x = 2, y = 3 − 5 (2) — — 7.

Hence, the corresponding values of x and y are

Corresponding values must be clearly understood as to meaning. From (4) in the above solution two values of x result. Each value of x is substituted in (3), and two values for y result. Therefore, we associate a value of x with that value of y resulting from its use in substitution.

SOLUTION BY COMPARISON AND FACTORING

339. When both equations are of the second degree and both are homogeneous.

2x2 - xy=10,

Solve

3x2-y2=11.

(1)

(2)

A factorable expression in x and y results if the constant terms are eliminated from (1) and (2) by comparison.

NOTE. The sign

is the result of a subtraction of a quantity having the sign. Thus, − (± a) = − (+ a) or − (− a) = − a or + a, = F a.

6.

2xy=-8,

SOLUTION OF SYMMETRICAL TYPES

340. When the given equations are symmetrical with respect to x and y; that is, when x and y may be interchanged without changing the equations.

Subtract (3) from (2),
Extract square root of (4),
Extract square root of (5),

Four pairs of equations result in (6) and (7), viz. :

x2 + 2xy + y2 = 49.
x2-2xy + y2 = 1.

(4)

(5)

x + y = ± 7.

(6)

x − y = ±1.

(7)

Hence, the corresponding values for x and y are

5. x2+xy + y2 = 7, x+y=3.

6. x+y-5=0,

x2 - xy + y2 = 7.

7. x2+ y2=5,

(x + y)2 = 9.

8. x2+3 xy + y2=59,
x2 + xy + y2 = 39.

SOLUTIONS OF MISCELLANEOUS TYPES

341. Systems of simultaneous quadratic equations not conforming to the three types already considered are readily recognized, and the student will gradually gain the experience necessary to properly solve such systems. No general method for these types can be given.

It is frequently possible to obtain solutions of those systems in which a given equation is of a degree higher than the second, derived equations of the second degree resulting from divisions and multiplications.

Combining each of these results with (2), we have two systems:

The two systems (a) and (b) may be readily solved by the principle of Art. 340.