VCE General Mathematics (Year 11) · Algebra and Number · 11m read

Linear Equations and Modelling

Ever wondered how Netflix predicts how many people will watch a show, or how a plumber quotes you a price based on call-out fees plus hourly rates? That's linear equations at work. You'll learn to solve equations (finding the unknown value) and use them to model real situations—turning messy real-world problems into maths you can actually work with. This skill is essential for VCE maths, physics, and economics, plus it's genuinely useful whenever you need to figure something out.

Learning objective: Solve linear equations and use them to model real situations.

By the end of this lesson you can:

• I solve linear equations confidently.
• I rearrange formulas.
• I build linear models from scenarios.

A linear model is a phone plan. Fixed monthly fee (y-intercept) plus cost per GB (gradient). Predict your bill for any usage.

Linear models in practice

LINEAR EQUATION FORMS:
• y = mx + c — slope-intercept.
• y − y₁ = m(x − x₁) — point-slope.
• ax + by = c — general form.

GRADIENT (slope):
m = (y₂ − y₁) / (x₂ − x₁).

INTERPRETING PARTS:
• m — rate of change per unit increase in x.
• c — y-intercept (starting value).

SOLVING LINEAR EQUATIONS:
• Do the same to both sides.
• Collect x terms on one side.
• Isolate x.

FORMULA REARRANGEMENT:
• Example: v = u + at, solve for t.
• v − u = at → t = (v − u)/a.

WORD PROBLEMS:
• Define variables.
• Translate into equation.
• Solve.
• INTERPRET back in context.

MODELLING REAL SITUATIONS:
• Phone plan: cost = flat fee + per-minute rate.
• Car depreciation: value = initial − loss × years.
• Taxi fare: fare = flag-fall + per-km rate.

SIMULTANEOUS EQUATIONS:
• Two equations, two unknowns.
• Methods: substitution, elimination, graphical.

INEQUALITIES:
• Similar to equations.
• Flip sign when multiplying/dividing by negative.

PIECEWISE LINEAR:
• Different rules for different ranges.
• Example: tax brackets.

LIMITATIONS OF LINEAR MODELS:
• Only valid within a realistic range.
• Extrapolation risky.
• Many real trends are non-linear.

CAS USE:
• Solve linear equations on calculator.
• Graph lines, find intersections.

COMMON ERRORS:
• Distributing wrong.
• Sign errors.
• Forgetting to answer the original question in context.

Linear essentials

• Gradient: Rate of change.
• Y-intercept: Starting value.
• Context: Interpret answers.

Linear modelling

• Slope-intercept: y = mx + c
• Two-point gradient: m = \dfrac{y_2 - y_1}{x_2 - x_1}
• Point-slope: y - y_1 = m(x - x_1)

Quick reference

| Feature | Meaning |
|---|---|
| y = a + bx | General form |
| a | Starting value (fixed cost) |
| b | Rate of change |
| Interpolation | Predict within data range |
| Break-even | Cost = revenue |

Worked example (Medium)

A taxi charges $4.00 flag-fall and $2.50 per km. Write a model for fare F.

1. Variables. Let k = km travelled, F = fare.
2. Equation. F = 4.00 + 2.50k.
3. Interpret. Y-intercept = flag-fall, slope = per-km rate.
4. Check. For 10km: F = 4 + 25 = $29 — reasonable.

Answer: F = 4 + 2.5k; a linear model where each part represents a real cost.

Worked example (Hard)

A plumber charges a $90 call-out fee plus $70 per hour. An electrician charges $50 call-out plus $80 per hour. After how many hours is the plumber cheaper? Model both and solve.

1. Define. Let h = hours worked. Plumber cost P = 90 + 70h. Electrician cost E = 50 + 80h.
2. Set equal. 90 + 70h = 50 + 80h.
3. Solve. 40 = 10h → h = 4 hours.
4. Interpret. For h < 4: electrician cheaper. For h > 4: plumber cheaper. At exactly 4 hours they cost the same ($370).

Answer: After 4 hours the plumber becomes cheaper. Break-even at h = 4 ($370 each).

Common mistakes

• Confusing gradient (rate) with y-intercept (fixed cost) The gradient is the per-unit rate; the y-intercept is the starting value. Fix: Gradient = slope = rate of change. Y-intercept = value when x = 0 (fixed/initial cost).
• Forgetting to interpret the answer in context A bare number like "h = 4" earns fewer marks than a contextual interpretation. Fix: Always write: "After 4 hours of work, the plumber becomes the cheaper option."
• Sign errors when rearranging equations Moving terms across the equals sign requires changing signs. Fix: Check by substituting your answer back into the original equation.

Linear models in the real world — and their limits

Linear models are powerful because they are simple and interpretable, but they have important limitations that VCE examiners love to test.

Strengths of linear models: Every linear model has two easily interpretable parameters — the gradient (rate of change) and the y-intercept (starting value). A taxi fare model F = 4 + 2.5k immediately tells you the flag-fall ($4) and the per-kilometre rate ($2.50). This simplicity makes linear models ideal for short-term predictions within a known range.

Extrapolation danger: Linear models assume the same rate continues forever. A model predicting a baby's weight gain as W = 3.5 + 0.2t (kg, weeks) works for early months but would predict a 13.9 kg baby at one year — possibly reasonable — but a 24.3 kg child at two years, which is too heavy. Real growth slows, following a curve, not a line.

Domain restrictions: Always state the valid range for your model. A depreciation model V = 25000 − 3000t becomes negative after t = 8.33 years — a car cannot have negative value. The model is only valid for 0 ≤ t ≤ 8.33.

Piecewise models address this by using different linear rules for different intervals — like tax brackets, where different rates apply to different income ranges. These are more realistic but harder to work with algebraically.

Practice questions

1. In y = mx + c, the y-intercept is:

   1. m
   2. x
   3. c
   4. y

   (1 marks)

2. A gym charges $50 joining fee plus $20/month. Write a linear model. When does cost reach $250?

   (3 marks)

Summary

• Solve linear equations confidently.
• Rearrange formulas.
• Build linear models from scenarios.