If the equation ax + byc of This may be shown as follows: the given type is solved for y, the value of y will involve only the first power of x, and since the degree of equation (2) in x and યુ is 2, the result of substituting this value of y in equation (2) will be a quadratic equation in x, which will give two values of x. To each of these values there will correspond but one value of y, determined by equation (1). 442. TYPE II ( (1) ax2 + bxy + cy3 = d, (2) Ax2 + Bxy+ Cy2=D, When first members of both equations are homogeneous and of the second degree. The equations in Type II are called homogeneous equations of the second degree, because the degree of every term involving x or y is the same and is equal to 2. Such a system of equations can always be solved by a definite plan explained in the following examples. Let yvx, and substitute in both equations; thus, (3) x2 (1— v2) = 7, x2v = 12; Substitute these values in either of equations (3); then The values of y which correspond to x = ±4 are real and are found by substituting x 4 in equation (1); thus Hence, the real system of values of x and y which satisfy equations (1) and (2), are x1 = +4, x=-4, 31 = +3; The values of y which correspond to the values x = ±√-3 are imaginary. Equations (1) and (2) are homogeneous and may be solved by the method indicated in Example 1; but they may be solved also as follows: multiply equation (2) by 2, and add and subtract the resulting equations member by member, to and from equation (1); thus After extracting the square root of both members of equations (3) and (4), To the four possible combinations of signs there correspond the four following systems of equations the solution of which gives the four systems of values, respectively, which satisfy the given equations (1) and (2). Let yvx, and substitute in equations (1) and (2); thus (3) x2(3+8v) 14 and c2 (1 + v + 4v2) = 6; then, by division, = The system of equations y=ve and equations (3) constitute a system of equations equivalent to the given system. On substitut ing successively the values of v in the equations in (3), the result is The artifice here employed may be conveniently used when both equations are homogeneous and of the second degree. In solving examples of this type four pairs of values of x and y, real or imaginary, have been found, which satisfy the given equations. EXAMPLE 4. Solve the equations J (1) 2x2+3 xy + y2 = 70, This system of equations has the general form of Type II. (1) and (2); then Divide the first equation of number (3) by the second, then NOTE. It will always be possible to find four systems of values of x and y, real or imaginary, which will satisfy two equations of the second degree in x and y. An expression is symmetrical with respect to two letters, x and y, when they are involved in the same way, so that the expression is unaltered in form when x and y are interchanged. Thus, Ax2+2 Bxy + Ay2 is symmetrical with respect to x and y, because if x and y are interchanged the result is Ay2+2 Bxy + Ax2, which is identical with the former expression. Similarly, x2 + 4x3y + 5 x2y2+ 4 xy3 + y1 is symmetrical with respect to x and y. Many examples involving symmetrical expressions may be solved by substituting for the unknown quantities the sum and the difference of two new variables. (1) becomes (3) (u + v)2 + (u2 — v2) + (u — v)2 = 84. (2) becomes (4) (u + v) — √ u2 — v2 + (u — v) = 6. 3 u2 + v2 = 84, 3 u2 - 24u + v2 + 36 = 0. |