Week 10 – SS1 First Term General Mathematics Notes

 WEEK 10

SIMPLE EQUATION AND VARIATION

SIMPLE EQUATION: is any algebraic equation with one unknown.
EXAMPLE
Solve for p in the equation p – 7 = 24
SOLUTION
.IF P – 7 = 24, then add 7 to both sides of the equation
P -7 +7 = 24 + 7
P = 31
Solve the equation 5(c +2) – 3(3c -5) = 1
Solution
5c+10-9c+15 =1. First open the bracket, collect like terms and simplify.
5c-9c+10+15 = 1
-4c+25 = 1, subtract 25 from both sides of the equation
-4c = 1-25,
-4c = -24, divide -4 by both sides
C = 6.
CHANGE OF SUBJECT OF FORMULAE
A formula is an equation consisting of letters which represent quantities.
EXAMPLE
Make each of the following letters giving the subject of formula:

1. A= ax + b, x (b) T = a + (n-1)d, a (c) T = 2, g

SOLUTION

1. A=ax + b. make x the subject of formula

   Subtract b from both sides
   A – b = ax divide both sides by a
   = x

2. T = a + (n-1) d, a. subtract (n-1)d from both sides

   T – (n-1) d = a or a =T – nd + d

©T = 2, g. Divide both sides by 2
= , cross multiply
Tg =2l, divide both sides by T
g =
VARIATION: is a change or difference in condition or amount or level etc. within certain limits.
TYPES OF VARIATION    
Direct variation, indirect or inverse variation, joint variation and partial variation
Direct variation is the proportional increase in x with a corresponding increase in y or a decrease in x with a corresponding decrease in y when considering two quantities X and Y. that is X Y, where is sigh of proportionality and the equation becomes X = kY where k is constant.
EXAMPLE
The number of bottles of wine drinks is directly proportional to the cost of the bottles of wine drinks. If 10 bottles of wine drink cost ₦400

1. What is the cost of 18 bottles?
2. How many bottles can₦200 buy?

SOLUTION
Let N = numbers of wine bottles and C = cost of wine drinks
N C. then N = Ck, N = 10 ,C = ₦400
10 = 400x k . k = =
Therefore the equation connecting N and C is N =

1. N = = 18 =

C = 18 x 40 = 720. The cost of 18 bottles of wine drinks is ₦720

1. N = = 5. The numbers of bottles ₦200 can buy is 5 bottles.

INDIRECT OR INVERSE VARIATION
Given two quantities X and Y such that Y increases with a corresponding decrease in X or a
Decrease inY with a corresponding increase in x then Y varies inversely as X. Y then,
the equation becomes Y
EXAMPLE
Y is inversely proportional to x. if y = 9 when x = 4, find the equation connecting x and y
SOLUTION
Y then y =
9 = , then k = 9×4 = 36
The equation connecting x and y is y = .
JOINT VARIATION
This involves three or more variables or quantities in a relationship which occur in many forms. It involves the combination of two direct variations or the combination of one direct and one inverse.
EXAMPLE
Z and z y that is two direct variables. Which is z xy. Equation is z = kxy
V varies directly as T and inversely as P can be written as V
EXAMPLE
Y varies jointly as x and y. W x= 2 and z = 3, y = 30. Find the equation connecting the relationship xyz
SOLUTION
Y xz y = kxz
30 = k x 2 x3 30 = 5k
K = = 6
Equation of the relationship is y = 6xz

PARTIAL VARIATION

Partial or part variation consists of two or more parts of quantities added together. One part
may be constant while the others can vary either directly, indirectly or jointly.
S is partly constant and partly varies directly as T
This statement can be written as :
S k + T. Then the equation is S = k + aT where k and a are constants.
EXAMPLE
X is partly constant and partly varies as y. When y = 5, x = 7 and when y = 7, x = 8. Find

1. The law of the variation. (b) x when y = 11.

SOLUTION
x k + ay x = k + ay where a and k are constants.
When y = 5, x = 7 : 7 = k + 5a …………(1)
When y = 7, x = 8: 8 = k + 7a …………..(2)
Solving the equation simultaneously, subtract (1) from (2)
2k = 1, then a = .
Substitute for a in (2), 8 =k + 7 x
16 = 2k + 7, 16 – 7 = 2k
K = or 4.5
The law of variation is x = or 2x =9 + y.

1. When y = 11, 2x = 9 + 11

2x = 20, x = 10.
ASSESSMENT
Evaluate the following questions,

1. The speed s km/h of a car is partly constant and partly varies as the time t the brake is applied. When t = 0, s = 40 and when t = 8, s =30, find s when t = 10 and t when s = 24.
2. A quantity Q is the sum of two quantities, one of which is constant while the other varies inversely as the square of R. when R =1, Q =-1 and when R =2, Q = 2. Find the positive value of R when Q = 2.
3. Y , y = 27 when x =9 and z = 2. Find
   1. The relation between x, y and z.
   2. Find y when x =14 and z = 12.
4. The price of a material in the market varies indirectly with the number of people demanding the material. When there are 80 people, the price of the material is ₦3.50. what is the price when there are 56 people?
5. Make the given letters the subject of the formula of the following equations:
   1. = , Q (b) A = ) , h,d (c) A = r, r
6. Solve the following equations:
   1. 8y -19 = 5 +3y (b) 12 – 3t – 9 = 3 – 5t (c) 2 = 5(5w – 2) – 9 (3w – 2) (d) + = 6 (e) – =

 MORAL OBJECTIVES: JAMES 1:17 Every good gift and every perfect gift is from above, and cometh down from the Father of lights, with whom is no variableness, neither shadow of turning.