Algebra: Formulas, Equations and Relationships
What you must understand
Algebra in Standard 1 is about using formulas and simple relationships, not heavy symbol manipulation. You need to substitute values into formulas (including formulas you are given on the reference sheet, such as the distance, speed and time relationship), rearrange a formula to make a different variable the subject, solve linear equations, and read and interpret linear graphs. You should also recognise direct linear relationships in tables and graphs and understand that gradient is a rate of change.
Traps that appear in questions
- Units before substitution. A time given in minutes must be converted to hours before using speed = distance ÷ time, or the answer will be out by a factor of 60.
- Rearranging with negatives. When solving 5 − 2x = 11, subtract 5 first to get −2x = 6, then divide by −2 to get x = −3. Dividing only one term is a common slip.
- Order of operations on substitution. In v = u + at with u = 3, a = −2 and t = 4, compute at = −8 before adding.
- Gradient sign and meaning. A falling line has negative gradient; a gradient of 0.5 means the quantity rises 0.5 for every 1 unit across, not that it doubles.
- Reading intercepts. The vertical intercept is the starting value when the horizontal variable is 0.
Worked example
A plumber charges a call-out fee of $80 plus $65 per hour. The cost is C = 80 + 65h. To find the hours for a $310 bill, solve 80 + 65h = 310, so 65h = 230 and h = 3.54 hours (about 3 hours 32 minutes). Check: 80 + 65 × 3.54 ≈ 310. The gradient 65 is the hourly rate; the intercept 80 is the call-out fee.
Rule of thumb
Substitute with units attached, solve by undoing operations in reverse order, and always state what the gradient and intercept mean in the context of the question.