Part 25

3. x^2 + y^2 = 25, x + y = 1.

4. x^4 + y^4 = 2, x - y = 2.

5. x^3 + y^3 = 28, x + y = 4.

6. x^2 y^2 + xy - 12 = 0, x + y = 4.

7. 2xy - x + 2y = 16, 3xy + 2x - 4y = 10.

8. (3x - 2y)(2x - 3y) = 26, x + 1 = 2y.

9. 4x^2 + 3xy + 2y^2 = 18, 3x^2 + 2xy - y^2 = 3.

10. x^5 + y^5 = 242, x + y = 2.

11. x - y + [x - y]^(1/2) = 6, xy = 5.

12. 4x^2 - x + y = 67, 3x^2 - 3y = 27.

13. x - y - [x - y]^(1/2) = 2, x^3 - y^3 = 2044. (Yale.)

14. x^2 + xy + x = 14, y^2 + xy + y = 28. (Princeton.)

15. x^2 + y^2 = 13, y^2 = 4(x - 2). Plot the graph of each equation. (Cornell.)

16. x^2 + y^2 = xy + 37, x + y = xy - 17. (Columbia.)

In grouping the answers, be sure to associate each value of x with the corresponding value of y.

17. The course of a yacht is 30 miles in length and is in the shape of a right triangle one arm of which is 2 miles longer than the other. What is the distance along each side?

~Reference:~ The chapter on Simultaneous Quadratics in any algebra.

RATIO AND PROPORTION

1. Define ratio, proportion, mean proportional, third proportional, fourth proportional.

2. Find a mean proportional between 4 and 16; 18 and 50; 12m^2n and 3mn^2.

3. Find a third proportional to 4 and 7; 5 and 10; a^2 - 9 and a - 3.

4. Find a fourth proportional to 2, 5, and 4; 35, 20, and 14.

5. Write out the proofs for the following, stating the theorem in full in each case:

(a) The product of the extremes equals etc.

(b) If the product of two numbers equals the product of two other numbers, either pair etc.

(c) Alternation.

(d) Inversion.

(e) Composition.