A Review of Algebra

Name: A Review of Algebra
Author: Rivenburg, Romeyn Henry

Rivenburg, Romeyn Henry

Algebra -- Problems, exercises, etc.

Any two of the quantities
             x + y
             x^2 + y^2
             xy
             x - y
             x^3 + y^3
             x^3 - y^3
             x^2 + xy + y^2
             x^2 - xy + y^2
        given.

            x + y = 5,
            x^2 - xy + y^2 = 7.

        METHOD: Solve for x + y and x - y; then add to get x, subtract
                to get y.

    CASE IV.

        Both equations symmetrical or symmetrical except for sign.
        Usually one equation of high degree, the other of the first
        degree.
            x^5 + y^5 = 242,
            x + y = 2.

        METHOD: Let x = u + v and y = u - v, and substitute in both
                equations.

    ~Special Devices~

          I. Consider some compound quantity like xy, [x - y]^(1/2),
             [xy]^(1/2), x/y, etc., as the unknown, at first. Solve
             for the compound unknown, and combine the resulting
             equation with the simpler original equation.

                x^2 y^2 + xy = 6,
                x + 2y = -5.

         II. Divide the equations member by member. Then solve by
             Case I, II, or III.

                x^3 - y^3 = 152,
                x - y = 2.

        III. Eliminate the quadratic terms. Then solve by Case I,
             II, or III.

                xy + x = 15,
                xy + y = 16.




SIMULTANEOUS QUADRATICS


Solve:

  1. x + y = 7,
     x^2 + 4xy = 57.

  2. 2x^2 = 46 + y^2,
     xy + y^2 = 14.

  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.

Public-domain text, read in full here on John Shaqi.

A Review of Algebra

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