Algebraic notation

In algebra, we often use letters like x and y to represent unknown numbers. When a number and a letter are right next to each other, such as 8x, it means 8 \times x.

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Note: You might encounter terms with exponents in algebraic expressions. Keep in mind that only like terms with matching exponents can be added or subtracted.

Writing Algebraic Expressions

You can turn everyday sentences into algebraic expressions. Let’s consider a simple example: Tom borrows three pens and two calculators at school. If we use x for pens and y for calculators, then the algebraic expression becomes 3 \times x + 2 \times y.

Simplifying Like Terms

Terms with identical variables are known as “like terms.” You can simplify an expression by adding or subtracting these like terms.

Example:

Collect the like terms in the expression 2 \times y + 3 \times x - 4 \times y + 6 \times x.

1. Rearrange the expression to group like terms: 2 \times y - 4 \times y + 3 \times x + 6 \times x.

2. Perform the addition or subtraction of like terms: -2 \times y + 9 \times x.

It’s important to remember that you can only combine ‘like’ terms that have the same base and exponent. For example, x^{2} + x^{2} = 2x^{2}, but x^{2} + x^{3} cannot be combined into a single term.

Multiplying Algebraic Terms

The rules for multiplying algebraic terms and expressions are similar to those for numbers. For example, y \times y equals y to the power of 2 (y^2), and 2 \times a equals 2a.

Let’s look at some more examples:

  • Multiplying terms with different bases: 3 \times x \times 4 \times y = 12 \times xy
  • Multiplying terms with the same base but different exponents: x^2 \times x^3 = x^{(2+3)} = x^5
  • Multiplying terms with coefficients and the same base: 3 \times x^2 \times 2 \times x^3 = 6 \times x^{(2+3)} = 6 \times x^5

Practice Questions

Question 1:

Translate the following sentence into an algebraic expression: “Sarah buys four books and three pencils.”

Solution

4 \times x + 3 \times y, where x is the number of books and y is the number of pencils.

Question 2:

Simplify 5 \times x + 2 \times y - 3 \times x + 4 \times y

Solution

2 \times x + 6 \times y

Question 3:

Simplify 3 \times x^2 + 4 \times y - 2 \times x^2 + 2 \times y

Solution

x^2 + 6 \times y

Question 4:

Multiply (x + 2) \times (x - 3)

Solution

x^2 - x - 6

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