Linear equations
Calculating with letters
When calculating with letters, we write the terms in alphabetical order:
Correct order
ab, 2xy, 3abc
Alternative order
ba, 2yx, 3cba
Numbers are written before the letters, and we leave out the multiplication sign:
a + a + a = 3a a · b = ab 2 · x · y = 2xy
Algebraic expressions
An algebraic expression contains numbers, letters and operations. We simplify by collecting like terms:
3x + 2x − 5 + 7 = 5x + 2
Collect like terms
3x + 2x = 5x
Like terms have the same letter part
Distributive law
a(b + c) = ab + ac
3(2x − 1) = 6x − 3
Brackets preceded by a minus sign
−(a + b) = −a − b
−(2x − 5) = −2x + 5
Multiplying brackets
(a+b)(c+d) = ac+ad+bc+bd
(x+2)(x+3) = x²+5x+6
An equation has an equals sign and states that two expressions are equal. We solve the equation to find the value of the unknown.
The balance method (balance principle)
An equation is like balanced scales. Whatever we do to one side, we must also do to the other:
We can add, subtract, multiply or divide by the same number on both sides.
Example 1: Solve 3x + 5 = 20
Subtract 5 from both sides:
3x + 5 − 5 = 20 − 5 ⇒ 3x = 15
Divide by 3 from both sides:
3x / 3 = 15 / 3 ⇒ x = 5
Check: Substitute x = 5:
3 · 5 + 5 = 15 + 5 = 20 ✓
Example 2: Solve 2(x − 3) = 4x + 2
Expand the left-hand side:
2x − 6 = 4x + 2
Subtract 2x from both sides:
−6 = 2x + 2
Subtract 2 from both sides:
−8 = 2x
Divide by 2:
x = −4
Check: LHS = 2(−4 − 3) = 2(−7) = −14. RHS = 4(−4) + 2 = −14 ✓
Example 3: Solve 5x − 3 = 2x + 9
Collect the x terms: Subtract 2x from both sides:
3x − 3 = 9
Collect the number terms: Add 3 to both sides:
3x = 12
Divide by 3:
x = 4
Check: LHS = 5(4) − 3 = 17. RHS = 2(4) + 9 = 17 ✓
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