Algebraic expressions (grade 7) — introduction

Algebraic expressions (grade 7) — introduction

Algebraic expressions

In grade 6 you started with a variable like `x` and solved simple one-step

equations. In grade 7 we go further — we manipulate expressions even

when there is no equation to solve. Five tools you will use again and again:

  1. Combining like terms. `3a + 5 − a + 2` → `2a + 7`.
  2. Distributive property (expanding). `3·(2x − 4)` → `6x − 12`.
  3. Factoring a common factor. `6x + 9` → `3·(2x + 3)`.
  4. Substitution. Evaluate `a² + 2b` for `a = 3`, `b = 5` → `9 + 10 = 19`.
  5. Power laws (same base). `a^3 · a^2 = a^5`; `(a^2)^3 = a^6`; `a^5 ÷ a^2 = a^3`.

Why bother?

An expression is a piece of mathematics that stands for some number — but

the number depends on the variable. By rewriting the expression in a

simpler or factored form, you understand its structure better, and

later you can solve harder equations by recognising patterns.

Example: `3(x − 2) + 5x = 8x − 6`. Inside the expression there is

structure (a "scale by 3" plus another "scale by 5"). Simplifying makes

it visible.

A worked tour

Combining like terms

The variable `a` is a placeholder. Terms with the same variable

combine like ordinary numbers; constants combine separately. 
3a + 5 − a + 2
= (3a − a) + (5 + 2)
= 2a + 7

Distributive property

The pattern `a · (b + c) = a·b + a·c` lets you eliminate parentheses.

3(2x − 4) = 3·2x + 3·(−4) = 6x − 12

Read backwards, it lets you factor:

6x + 9 = 3·(2x + 3)

Power laws

For the same base `a`:

operationrule
product`a^m · a^n = a^(m+n)`
quotient`a^m ÷ a^n = a^(m−n)` (m ≥ n)
power of power`(a^m)^n = a^(m·n)`

Try it yourself