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Real-World Linear Relationships

A real-world linear relationship is a situation where two quantities change at a constant rate and can be represented by a linear equation such as ax + by = c or y = mx + b.

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Student-friendly explanation

Many daily-life situations can be modelled using linear equations, such as total cost, distance at constant speed, or marks from fixed scoring rules. The coefficient often shows the rate of change, while the constant may show a starting value or fixed charge.

How to write this in exams

  1. 1

    Start with the exact idea

    A real-world linear relationship is a situation where two quantities change at a constant rate and can be represented by a linear equation such as ax + by = c or y = mx + b.

  2. 2

    Then show how to use it

    Identify the changing quantity, identify any fixed amount, assign variables, form the equation, substitute given values, and write the answer with units.

  3. 3

    Add one concrete example

    If a taxi charges Rs 40 fixed and Rs 12 per kilometre, the cost for x kilometres can be written as C = 12x + 40.

  4. 4

    Avoid this incomplete answer

    For a fixed charge of Rs 40 and Rs 12 per km, students may write C = 40x + 12, wrongly multiplying the fixed charge by distance.

Definition

A real-world linear relationship is a situation where two quantities change at a constant rate and can be represented by a linear equation such as ax + by = c or y = mx + b.

Example

If a taxi charges Rs 40 fixed and Rs 12 per kilometre, the cost for x kilometres can be written as C = 12x + 40.

Rule to remember

Common model: total = fixed amount + rate × quantity, written as y = mx + b when one quantity depends on another.

Memory hook

Fixed amount stays once; rate repeats with quantity.

Examples and method

Worked example

A shop charges Rs 20 for packing and Rs 50 per item. For x items, total cost C = 50x + 20. For 4 items, C = 50(4) + 20 = Rs 220.

Method to apply

Identify the changing quantity, identify any fixed amount, assign variables, form the equation, substitute given values, and write the answer with units.

Diagram support

A table or simple line graph can show how the total changes by the same amount each time the quantity increases by 1.

How CBSE asks it

Questions may ask students to form an equation from a word problem, interpret a coefficient, find a total, or read the starting value from a graph.

Avoid common mistakes

Common confusion

Students often write an equation without defining variables, which makes the answer unclear in word problems.

Common wrong answer

For a fixed charge of Rs 40 and Rs 12 per km, students may write C = 40x + 12, wrongly multiplying the fixed charge by distance.

Exam tip

Before forming the equation, write what each variable represents and keep units consistent.

Quick check

A notebook costs Rs 30. How can the cost of x notebooks be written as a linear expression?

The cost of x notebooks can be written as 30x rupees because each notebook adds Rs 30 to the total cost.

Study the real-world linear relationships diagram carefully

Use the labelled diagram to keep real-world linear relationships clear in short answers and revision.

What this diagram makes clear

This diagram keeps the labels and direction of real-world linear relationships in the right order.

Where this helps in exams

Use this for labelled diagram work and short exam answers on real-world linear relationships.

Revision cue

Revise real-world linear relationships through the labels before writing the answer.

Answer writing and exam use

1-mark answer

A real-world linear relationship is a situation where two quantities change at a constant rate and can be represented by a linear equation such as ax + by = c or y = mx + b.

2-mark answer

A real-world linear relationship is a situation where two quantities change at a constant rate and can be represented by a linear equation such as ax + by = c or y = mx + b. Common model: total = fixed amount + rate × quantity, written as y = mx + b when one quantity depends on another. If a taxi charges Rs 40 fixed and Rs 12 per kilometre, the cost for x kilometres can be written as C = 12x + 40.

3-mark answer

Many daily-life situations can be modelled using linear equations, such as total cost, distance at constant speed, or marks from fixed scoring rules. The coefficient often shows the rate of change, while the constant may show a starting value or fixed charge. Common model: total = fixed amount + rate × quantity, written as y = mx + b when one quantity depends on another. A shop charges Rs 20 for packing and Rs 50 per item. For x items, total cost C = 50x + 20. For 4 items, C = 50(4) + 20 = Rs 220. Questions may ask students to form an equation from a word problem, interpret a coefficient, find a total, or read the starting value from a graph. For a fixed charge of Rs 40 and Rs 12 per km, students may write C = 40x + 12, wrongly multiplying the fixed charge by distance.
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