Reference · module 1
Letters instead of numbers
Everything this module explains, on one page.
Variables and substitution
Read an expression with letters and work out its value.
- A letter in algebra stands for a number we do not know yet or that can change. In 3 + 2k, k is the number of kilometres, and the expression works for any trip.
- A number written right before a letter means multiplication: 2k is 2 × k, and 5ab is 5 × a × b. The multiplication sign is left out because x looks like a letter.
- To find the value, replace each letter with its number and calculate in the usual order. If x = 4, then 3x − 5 is 3 × 4 − 5, which is 7.
The mistake you are about to make
x = 4, 3x = 34
x = 4, 3x = 12
Writing 3 next to 4 feels like substituting. The 3 in 3x multiplies, so 3x with x = 4 is 3 × 4, which is 12.
Collecting like terms
Shorten an expression by combining terms with the same letters.
- Like terms have exactly the same letters: 3x and 5x are alike, and so are 2ab and 7ab. They can be added like apples: 3x + 5x is 8x.
- The number in front is the coefficient, and only coefficients are added. x on its own has coefficient 1, so x + 4x is 5x, and 6y − y is 5y.
- Different letters cannot be combined: 3x + 2y stays as it is, and so does x + 5. Collect each kind separately: 4x + 3 + 2x − 1 is 6x + 2.
The mistake you are about to make
3x + 2y = 5xy
3x + 2y
Two terms side by side look unfinished, so it is tempting to merge them. x and y are different unknowns, like apples and pears, and cannot be added.
Expanding brackets
Multiply out brackets without losing a term or a sign.
- A number in front of brackets multiplies everything inside: 3(x + 4) is 3 × x plus 3 × 4, which is 3x + 12. Every term inside gets multiplied, not just the first.
- A minus in front flips every sign inside: −(x − 5) is −x + 5, and −2(a + 3) is −2a − 6. The second term is where people slip.
- After expanding, collect like terms: 2(x + 3) + 4x is 2x + 6 + 4x, which is 6x + 6. Expand first, then simplify.
The mistake you are about to make
3(x + 4) = 3x + 4
3(x + 4) = 3x + 12
The 3 sits right next to x, so it seems to belong to x alone. It multiplies the whole bracket, so the 4 becomes 12 too.