Practice Maths

Solving Linear Equations III

Applying Equations to Real Life

Key Terms

independent variable
the input value that you control or that naturally varies (x)
dependent variable
the output value that results from the independent variable (y)
modelling
using mathematics to represent a real-world situation with a formula or equation
substitution
replacing a variable with a known value to evaluate an expression

Part 1 — Revise Solving

The goal is always to isolate the unknown — either by balancing (same operation to both sides) or backtracking (reversing each operation in order).

Quick Revision

Solve 4x + 2 = 22

Subtract 2 from both sides:   4x = 20

Divide both sides by 4:   x = 5

Part 2 — Modelling Real Situations

In real life, one value often depends on another.

  • Independent Variable (x): the input — the thing that changes naturally (e.g., time, kilometres).
  • Dependent Variable (y): the output — the result that depends on x (e.g., cost, distance).

Example: The Bricklayer

A bricklayer charges a $100 call-out fee, plus $50 per hour.

Step 1 — Write the rule:
y = 50x + 100    (where x = hours, y = total cost)

Step 2a — Find the cost for 3 hours (substitute x):
y = 50(3) + 100 = 150 + 100 = $250

Step 2b — Find hours if the bill is $400 (solve for x):
400 = 50x + 100
300 = 50x    (subtract 100)
x = 6    (divide by 50)
He worked 6 hours.

Hot Tip Always identify what 'x' stands for and what 'y' stands for before starting the maths. If you are asked to find the total (y), just calculate. If you are given the total, you must solve the equation (backtrack) to find x.

Worked Example

Problem: A taxi company charges a $3 flag fall plus $2 per kilometre. The rule is C = 2k + 3. If the total fare was $19, how far was the trip?

Step 1 — Substitute the known value (total fare) and write the equation.
C = 19, so: 2k + 3 = 19

Step 2 — Solve using backtracking or balancing.
Subtract 3 from both sides: 2k = 16
Divide both sides by 2: k = 8

Step 3 — State the answer with units and check.
The trip was 8 km. Check: 2(8) + 3 = 16 + 3 = 19. ✓

Bringing It All Together: The Modelling Process

In this lesson, equations meet real life. The process for any word problem is the same every time:

  1. Define your variables: Name what x and y represent, including units. "Let h = hours worked" is more useful than just "let h = hours."
  2. Write the equation: Translate the words into maths. Fixed fee + rate per unit × variable.
  3. Decide which way you're solving: If you're given x, substitute and calculate y. If you're given y, write the equation and solve for x.
  4. State the answer with units and check.

Two Types of Questions — Know the Difference

The bricklayer example from the Key Ideas tab (y = 50x + 100) shows both directions perfectly:

  • Forward (substitution): "How much for 3 hours?" → substitute x = 3 → y = 50(3) + 100 = $250. Easy!
  • Backward (solving): "If the bill is $400, how many hours?" → substitute y = 400 → solve 400 = 50x + 100 → 300 = 50x → x = 6 hours.

Every real-world problem is one of these two types. The trick is recognising which unknown you need to find.

Remember: If the question gives you the TIME or QUANTITY and asks for the TOTAL, substitute x and calculate. If the question gives you the TOTAL and asks for the TIME or QUANTITY, set up the equation and solve.

Descending Models: When Things Shrink

Some real-life models decrease. A water tank draining: y = 200 − 10x. A candle burning down. A car using up its fuel. These models still work the same way — just remember the subtraction means the output gets smaller as x grows.

For the water tank (y = 200 − 10x), when does it empty? Set y = 0:

  • 0 = 200 − 10x
  • 10x = 200
  • x = 20 minutes

This is a really useful skill — engineers use it to calculate when a chemical reaction will be complete, or when a fuel tank will need refilling.

Table, Equation, Graph: Three Representations

Maths becomes much more powerful when you can move between a table, an equation, and a graph. A table gives you specific values. An equation gives you the rule for any value. A graph gives you a visual picture of the relationship. They all describe the same thing — just in different languages. Getting comfortable with all three is the mark of a confident mathematician.

Common Mistake: Forgetting to state units in your answer. "x = 6" is incomplete — always write "x = 6 hours" or "the trip was 8 km." Without units, the number is meaningless in a real-world context.

Practice Questions

  1. Solving Equations Fluency

    Solve the following equations using any method (show your working):

    1. 2x + 5 = 19
    2. 3x − 4 = 20
    3. 10 = 4m − 2

  2. Equations with Brackets Understanding

    Solve for the variable:

    1. 2(x + 3) = 18
    2. (x ÷ 2) + 5 = 11

  3. The Pancake Recipe Understanding

    A pancake recipe uses a specific amount of milk. The relationship is described by the rule: y = 3x + 2.

    Where x is cups of milk, and y is the number of pancakes made.

    1. How many pancakes can be made with 4 cups of milk?
    2. If you need to make 23 pancakes, how many cups of milk do you need? (Write the equation and solve.)

  4. Hexagonal Tables Understanding

    A school cafeteria puts hexagonal (6-sided) tables in a row.
    1 table seats 6 people.
    2 tables joined together seat 10 people.
    3 tables joined together seat 14 people.

    1. Draw a table of values for 1, 2, and 3 tables.
    2. The rule is y = 4x + 2. Check this rule against your table.
    3. How many people can sit at a row of 10 tables?
    4. If you have 26 students to seat in a single row, how many tables do you need?

  5. The Electrician Understanding

    An electrician charges a $80 call-out fee plus $60 per hour of work.

    1. Write an equation for the total cost (C) based on hours worked (h). (Hint: C = ... h + ...)
    2. Calculate the cost for a 2-hour job.
    3. If the total bill comes to $320, how many hours did the electrician work?

  6. Tree Planting Understanding

    A gardener plants a layout of trees. The rule for total trees (t) based on the number of rows (r) is: t = 8r + 4.

    1. How many trees are there in total if there are 5 rows?
    2. The gardener has exactly 60 trees to plant. How many rows will this fill?

  7. Savings Account Understanding

    Sarah opens a savings account with $50. She deposits $25 every week after that.

    1. Complete the table of values:
      Weeks (x)123
      Total ($) (y)75100
    2. Write the algebraic rule connecting x and y.
    3. How many weeks will it take for Sarah to reach $300?

  8. Car Rental Problem Solving

    A rental company charges a flat fee of $100 plus $40 per day to rent a car.

    1. Write the equation using d for days and C for cost.
    2. How much does it cost to rent the car for a week (7 days)?
    3. The customer has a budget of $500. What is the maximum number of days they can rent the car?

  9. Water Tank Understanding

    A water tank starts with 200 litres of water. A tap is opened, and water drains out at 10 litres per minute.

    The rule is: y = 200 − 10x (where x is minutes, y is litres remaining).

    1. How much water is left after 5 minutes?
    2. How many minutes until there are only 80 litres left?
    3. How many minutes until the tank is completely empty?

  10. Mobile Phone Data Problem Solving

    A mobile plan costs $20 per month plus $10 for every GB of extra data used.

    1. Write an equation for the monthly bill (y) based on extra GB used (x).
    2. If you use 4 GB of extra data, what is the bill?
    3. If your bill was $90, how much extra data did you use?
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