Unit 2
Basic Algebra

This unit is a review of algebra that you did in junior high and high school. Some students will find this unit exactly that—a review of concepts they recall. However if you have been out of school for a while or are rusty on mathematics, then you may find it will take some time and persistence before you begin to recall these concepts. Be diligent and the concepts of algebra will come back to you.

Beaming maiden with beaming eyes, pray tell me, since you understand the right methods, what is that number which when multiplied by two, then increased by one, and then divided by five, gives the number three?

The question is one of the simpler problems found in the works of Hindu mathematicians ca. 500 AD. The Hindus solved such problems by the somewhat complicated method of “inversion”: for each stated operation, they listed the inverse operation, and then performed the inverse operations in reverse order to obtain the solution. The solution to the problem quoted above is obtained as follows.

  1. Operations
    1. Multiply by 2
    2. Add 1
    3. Divide by 5
  2. Inverse operations
    1. Divide by 2
    2. Subtract 1
    3. Multiply by 5
  3. Reversed order of the inverse operations
    1. Multiply by 5
    2. Subtract 1
    3. Divide by 2
  4. Calculations
    1. 3 × 5 = 15
    2. 15 − 1 = 14
    3. 14 + 2 = 7

The solution is 7.

This method looks unfamiliar to most of us, including modern-day Hindus.

Using the modern method of solving the above problem, we would begin by saying that we will let the unknown number be x. When we multiply this number by 2, we get 2x, and when we increase this product by 1, we get (2x + 1). We are told further that when we divide (2x + 1) by 5, we should get 3, as shown below.

(2x + 1) ÷ 5 = 3

The above expression is an “equation in one unknown,” which can be solved using techniques that are the subject of this unit.

Objectives

After completing this unit, you should be able to perform the following tasks.

  1. Do fundamental operations of addition, subtraction, multiplication, and division on algebraic expressions.
  2. Work with exponents.
  3. Perform factoring operations.
  4. Work with algebraic fractions.
  5. Solve systems of linear equations in one, two, or three unknowns.
  6. Express word problems in equation form, and solve word problems using equations.
  7. Work with fractional equations.
  8. Solve quadratic equations using either the quadratic formula or the techniques of factoring.
  9. Describe the properties of an arithmetic progression, compute the nth term and the sum of n terms of such a progression, and calculate arithmetic means.
  10. Describe the properties of a geometric progression, compute the nth term and the sum of n terms of such a progression, and calculate geometric means.
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