. CHAPTER XVIII SIMULTANEOUS QUADRATIC EQUATIONS 145. Two simultaneous quadratic equations, involving two unknown numbers, are solved by various methods, depending upon the form of the equations. I. When one of the equations is a simple equation. These values must be properly grouped, showing the corresponding values of x and y. The student should note that if the proper pair of values is substituted in the original equations, the equations will be satisfied. II. When the left sides of the two equations are homogeneous and of the second degree. 146. Equations are homogeneous when all their terms are of the same degree with respect to the unknown numbers. Thus, 2x2, 3xy, y2 are terms of the second degree and the equation 2x2 + 3y2 2xy= 0, is homogeneous. The equation 2x2 - 3xy + y2 = 12, is homogeneous on one side only, or it may be said to be homogeneous in its unknown terms. Such equations are most easily solved by the following method. III. When the two equations are symmetrical with respect to the two unknown numbers. 147. Algebraic expressions are said to be symmetrical with respect to two unknowns, when the two unknowns are similarly involved. In all symmetrical expressions, the two unknowns may be interchanged without altering the form of the expression. As, x2 - xy + y2, 2x2 + 3xy + 2y2, x3 + 3x2y + 3xy2 + y3. x2 The direct object of this method of solving symmetrical equations is to obtain the values of x + y and x y, whence, by combining these two simple equations, we obtain the values of x and y. 98 2x4 + 4x3y + 6x2y2 + 4xy3 + 2y1 Divide by 2, x1 + 2x3y + 3x2y2 + 2xy3 + Extract the square root, = y1 = 49 5 = ±7 As the value of x + y is given in (2), we must find the value of x When the signs are used in their proper order, the corresponding values of x and y are found in their proper order, as above 2 -55 |