6. At what time between 8 and 9 o'clock are the hands of a watch (_a_) opposite each other? (_b_) at right angles? (_c_) together? Work out (_a_) and state the equations for (_b_) and (_c_). 7. The formula for converting a temperature of F degrees Fahrenheit into its equivalent temperature of C degrees Centigrade is C = (5/9)(F - 32). Express F in terms of C, and compute F for the values C = 30 and C = 28. (_College Entrance Exam. Board._) 8. What is the price of eggs when 2 less for 24 cents raises the price 2 cents a dozen? (_Yale._) 9. Solve 2/(x - 2) + 2/(4 - x^2) = 5/(x + 2). ~Reference:~ The Chapter on Fractional Equations in any algebra. Note particularly the special _short_ methods, usually given about the middle of the chapter. SIMULTANEOUS EQUATIONS NOTE. Up to this point each topic presented has reviewed to some extent the preceding topics. For example, factoring reviews the special rules of multiplication and division; H. C. F. and L. C. M. review factoring; addition and subtraction of fractions and fractional equations review H. C. F. and L. C. M., etc. From this point on, however, the interdependence is not so marked, and miscellaneous examples illustrating the work already covered will be given very frequently in order to keep the whole subject fresh in mind. 1. Solve by three methods--addition and subtraction, substitution, and comparison: { 5x + y = 11, { 3x + 2y = 1. Solve and check: 2. { 12R_1 - 11R_2 = b + 12c, { R_1 + R_2 = 2b + c. 3. { (r - s)/2 = 25/6 - (r + s)/3, { (r + s - 9)/2 - (s - r - 6)/3 = 0. 4. One half of A's marbles exceeds one half of B's and C's together by 2; twice B's marbles falls short of A's and C's together by 16; if C had four more marbles, he would have one fourth as many as A and B together. How many has each? (_College Entrance Board._) 5. The sides of a triangle are a, b, c. Calculate the radii of the three circles having the vertices as centers, each being tangent externally to the other two. (_Harvard._) 6. Solve { 2x + 3y = 7, x - y = 1 } graphically; then solve algebraically and compare results. (Use cooerdinate or squared paper.) Factor: 7. x^4 + 4. 8. 2d^(10) - 1024d. 9. 2(x^3 - 1) - 7(x^2 - 1). ~References:~ The chapters on Simultaneous Equations and Graphs in any algebra. SIMULTANEOUS EQUATIONS AND INVOLUTION 1. Solve (3/4)x - (5/3)y = 11-1/2, (5/8)x - (3/2)y = 10-1/4. Look up the method of solving when the unknowns are in the denominator. Should you clear of fractions? 2. Solve 1/x - 1/y - 1/z = 1/a, 1/y - 1/z - 1/x = 1/b, 1/z - 1/x - 1/y = 1/c. 3. Solve graphically and algebraically 2x - y = 4, 2x + 3y = 12.