Fixed part
US$3 booking fee, even when distance is zero.
Learning Objectives
- explain partial variation;
- recognise a fixed part and a varying part;
- use equations of the form y = kx + c;
- find k and c from two pairs of values;
- interpret the fixed amount in context.
Introduction
This lesson develops Partial Variation from meaning and patterns before moving to calculations. Look for what the quantities, shapes or symbols represent, then connect that idea to the method.
Key Notes
Read the definitions and key facts below carefully. Each rule is most useful when you can explain why it fits the situation, not simply recall it.
Visual Exploration
Notice→Connect→Explain
Use the diagrams, tables or examples in this lesson to identify what changes, what stays fixed and which relationship connects the values.
What Is Partial Variation?
Partial variation occurs when a quantity has one fixed part and another part that varies directly.
y = kx + c
Key idea: c is the fixed amount, while kx is the part
that changes.
Everyday Example
A delivery company charges a fixed booking fee of US$3 plus US$1.50 per kilometre.
Cost = 1.5 × distance + 3
Varying part
US$1.50 for every kilometre travelled.
Worked Example 1
y varies partly as x and partly as a constant. When x = 2, y = 11. When x = 5, y = 20. Find the relationship.
1
Write y = kx + c.
2
11 = 2k + c.
3
20 = 5k + c.
4
Subtract: 9 = 3k, so k = 3.
5
11 = 6 + c, so c = 5.
6
Therefore y = 3x + 5.
The Graph of Partial Variation
The graph is a straight line, but unlike direct variation, it usually does not pass through the origin.
gradient = k and y-intercept = c
Do not confuse a linear relationship with direct variation. Direct
variation requires c = 0.
🔍 How Do I Recognise Partial Variation?
- “varies partly as x and partly as a constant”
- a fixed charge plus a charge per unit;
- a straight-line graph not passing through the origin;
- an equation of the form y = kx + c.
✅ Quick Check
In y = 4x + 7, which part is fixed?
Does y = 5x + 2 represent direct variation?
What does c represent on the graph?
🧠 Let’s Think Together
Before calculating, identify the information given, the result required and the mathematical relationship that connects them. Predict whether the answer should be larger, smaller or unchanged, then use that prediction to check the result.
Practice Questions
Foundation- Identify k and c in:
- y = 6x + 4;
- C = 2.5d + 10.
- A taxi charges US$2 fixed plus US$0.80 per kilometre. Write a formula for cost C after d kilometres.
- y = kx + c. When x = 1, y = 8. When x = 4, y = 17. Find k and c.
- A printing company charges a fixed setup fee and a charge per page. Printing 20 pages costs US$9 and printing 50 pages costs US$15. Find the cost formula.
- A straight-line relationship passes through (3, 14) and (8, 29). Find its equation and explain why it is not direct variation.
✅ Check Your Work
Show practice answers
Foundation
-
- k = 6, c = 4
- k = 2.5, c = 10
- C = 0.8d + 2
Developing
- k = 3 and c = 5, so y = 3x + 5.
-
C = 0.2p + 5.
9 = 20k + c
15 = 50k + c
6 = 30k, so k = 0.2.
9 = 4 + c, so c = 5.
Challenge
- y = 3x + 5. It is not direct variation because the y-intercept is 5 rather than 0.
Common Mistakes
- Choosing a rule before identifying what the question describes.
- Skipping working or changing notation part-way through a solution.
- Accepting an answer without checking its size, sign or units.
Exam Focus
Show the mathematical relationship you are using, keep notation consistent and give a clear final answer. Use a quick estimate, diagram or inverse operation to check that the result is reasonable.
Summary
- Partial variation uses y = kx + c.
- c is the fixed part.
- kx is the varying part.
- The graph is straight but may not pass through the origin.