Simple Equation And Variations

CONTENT

• Change of subject of formulae
• Types of variation such as: direct, inverse, joint and partial
• Application of variations

EQUATIONS

An equation is a statement of two algebraic expressions which are equal in value. For example, 4 – 4x = 9 – 12x is a linear equation with an unknown x. this equation is only true when x has a particular numerical value. To solve an equation means to find the real number value of the unknown that makes the equation true.

Types of Variation such as: Direct, Inverse, Joint and Partial

Direct Variation

If a person buys some packets of sugar, the total cost is proportional to the number of packets bought

The cost of 2 packets atNx per packet is N2x

The cost of 3 packets at Nx per packet is N3x.

The cost of n packets at Nx per packet is Nnx.

Thus, the ratio of total cost to number of packets is the same for any number of packets bought.

These are both examples of direct variation, or direct proportion. In the first example, the cost, C, varies directly with the number of packets, n.

Joint Variation

The mass of a sheet of metal is proportional to both the area and the thickness of the metal. Therefore M ∝ At (where M, A and t are the mass, area and thickness). This is an example of joint variation. The mass varies jointly with the area and thickness.

EVALUATION

1. A rectangle has a constant area, A. its length is l and its breadth is b.
   1. Write a formula for l in terms of A and b
   2. Write a formula for b in terms of A and l.
   3. Does l vary inversely or directly with b?

Indirect Variation

The main idea in inverse variation is that as one variable increases the other variable decreases. That means that if x is increasing y is decreasing, and if x is decreasing y is increasing. The number k is a constant so it’s always the same number throughout the inverse variation problem.

Partial Variation

When a tailor makes a dress, the total cost depends on two things: first the cost of the cloth; secondly the amount of the time it takes to make the dress. The cost of the cloth is constant, but the time taken to make the dress can vary. A simple dress will take a short time to make; a dress with a difficult pattern will take a long time. This is an example of parital variation. The cost is partly constant and partly varies with the amount of time taken. In algebraic from, C= a + kt, where C is the cost, t is the time taken and a and k are constants.

EVALUATION

1. The cost of a car service is partly constant and partly varies with time it takes to do the work. It cost N3, 500 for a 5
    hour service and N2, 900 for a 4-hour service. Find the formula connecting cost, NC with time, T hours Hence find the cost of a 7 hour service. X is partly constant and partly varies as y. when y = 2, x = 30, and when y = 6, x = 50.Find the relationship between x and y.Find x when y = 3

GENERAL EVALUATION

1. If a man cycles 15km in 1 hour, how far will he cycle in two hours if he keeps up the same rate?
   1. A piece of string is cut into n pieces of equal length l.
      1. Does n vary directly or inversely with l?
      1. The mass of rice that each woman gets when sharing a sack varies inversely with the number of women. When there are 20 women, each gets 6kg of rice. If there are nine woman, how much does each get?

READING ASSIGNMENT

New General Mathematics SSS 1 pages 220 Exercise 18a 11 – 15