Formulae and Variations

Syllabus tag: KCSE | Mathematics | Form 3 | Topic 9 Formulae and Variations

Lesson objectives

By the end of this topic, you should be able to:

• Rewrite a given formula by changing its subject.
• Define direct, inverse, partial and joint variation.
• Determine constants of proportionality.
• Form and use equations of variation to solve problems.

Formulae and Variations

A formula relates quantities. Variation describes how one quantity changes when another does.

a) Changing the subject

The subject is the letter standing alone on one side.

To change it, undo the operations surrounding the letter you want. Do the same to both sides each time.

Work in reverse order. Undo addition and subtraction before multiplication and division. Undo powers last, by taking roots.

Check by substituting numbers into both forms. They must agree.

b) When the letter appears twice

Collect all terms containing it on one side, then factorise it out.

Take x = (a + bx)/c. Multiply by c to get cx = a + bx. Subtract bx to get cx − bx = a. Factorise to x(c − b) = a, so x = a/(c − b).

Factorising is the step people miss.

c) The four kinds of variation

d) Direct variation

y varies directly as x means y = kx, where k is the constant of proportionality.

Doubling x doubles y. The graph is a straight line through the origin.

The quotient y/x stays constant.

y may also vary directly as a power: y = kx² or y = kx³.

e) Inverse variation

y varies inversely as x means y = k/x.

Doubling x halves y. The product xy stays constant.

The graph is a curve approaching but never touching the axes.

f) Joint variation

y varies jointly as x and z means y = kxz.

y varies directly with both at once.

A common case combines them: y = kx/z, where y varies directly as x and inversely as z.

g) Partial variation

y is partly constant and partly varies as x: y = a + kx.

Here there are two unknowns, a and k, so two pairs of values are needed to find them. That gives simultaneous equations.

This models real costs well. A taxi fare has a fixed charge plus a rate per kilometre. A phone bill has line rental plus usage.

h) Finding the constant

Write the correct equation for the type of variation.

Substitute one given pair of values and solve for k.

Rewrite the equation with k known, then use it to answer the question.

For partial variation, substitute both pairs and solve the two equations together.

i) Where this is used

Physical laws, such as pressure varying inversely with volume. Costing with fixed and variable components. Scaling recipes and mixtures.

Words to know

• Subject of a formula -- the letter standing alone on one side.
• Constant of proportionality -- the fixed multiplier k in a variation relation.
• Direct variation -- where the ratio of two quantities is constant.
• Inverse variation -- where the product of two quantities is constant.
• Partial variation -- where a quantity has a fixed part and a varying part.

:::checkpoint Check yourself

1. Make h the subject of V = πr²h.
2. y varies directly as x, and y = 12 when x = 3. Find y when x = 7.
3. y varies inversely as x, and y = 8 when x = 5. Find y when x = 10.
4. Why does partial variation need two pairs of values? :::

Bridge to practice

The exercises begin with changing the subject, including the repeated-subject case, and move through the four types of variation to applications. For every variation question, write the relation with k before substituting any numbers.