Download Topic 1: Algebra • Meaning of terms variable and function • Use of

Topic 1: Algebra  Meaning of terms variable and function  Use of function notation  Rules and techniques for the manipulation, transformation and evaluation of algebraic formulae  Application of algebraic techniques for substitution and factorisation  Solution of linear and quadratic equations by direct methods and simultaneous equations by elimination. Introduction: Algebra is a simple language, used to create mathematical models of real‐world situations and to handle problems that we can't solve using just arithmetic. Rather than using words, algebra uses symbols to make statements about things. In algebra, we often use letters to represent numbers. Algebraic Expressions An algebraic expression is one or more algebraic terms in a phrase. It can include variables, constants, and operating symbols, such as plus and minus signs. It's only a phrase, not the whole sentence, so it doesn't include an equal sign. 5
9
13 E.g.: 7
In an algebraic expression, terms are the elements separated by the plus or minus signs. This example has four terms:7 , 5 ,9 & 13. Terms may consist of variables and coefficients, or constants. Variables In algebraic expressions, letters represent variables. In this expression, the variables are and . We call these letters "variables" because the numbers they represent can vary—that is, we can substitute one or more numbers for the letters in the expression. Coefficients 5
Coefficients are the number part of the terms with variables. In,7
9
13 the coefficient of the first term is 7. The coefficient of the second term is (‐5), and the coefficient of the third term is 9. If a term
m consists of only varriables, its coefficient is 1. E.g.:: Constan
nts Constan
nts are the
e terms in tthe algebraaic expresssion that co
ontain only numberss. That is, they're the terms without variables. We call them constants because their ever changges, since tthere are n
no variablees in the teerm that caan change its value ne
value. In
n the abovve expression the con
nstant term
m is "(‐13).." Functio
ons: A functiion is a relaation (a se
et of ordere
ed pairs) w
where the vvalue of on
ne variablee depends on the vaalue of the
e other varriable. How
wever, in a function, each inputt (x coordin
nate) may b
be paired w
with only O
ONE outpuut (y coordiinate). on is only aa function if each inp
put is only paired witth one output. In oth
her A relatio
words ‐ focus on yyour x coordinate (in
nput). If you have 2 o
or more x ccoordinates e the same
e ‐ they mu
ust all have
e the same output orr it is not a function!
that are
e whether the given relation iss a function
n or not, ju
ust by We can determine
ng the orde
ered pairs.. analyzin
E.g.: L
,3), (2,4), ((3,5), (4, 7))} 1. Let A = {(1
Since each S
input in bo
old letters is paired w
with a diffeerent outp
put (y coordinate
c
e), we can ssay that th
he set A is aa function.. ONE TO ON
NE RELATIO
ON) (O
2. Let B = { (‐
L
3, 2) (‐1, 6) (1,2) } The set B w
T
will be a function beccause the o
outputs forr (‐3) and 1
1 is same b
but the inputs t
ent. are differe
RELATION)
(MANYY TO ONE R
3) (‐1,0) (3
L
3,‐3)} 3. Let C = {(3,
The set C w
T
will not be a function because tthe input 33 is giving ttwo differeent outputs as o
C is not a fu
unction ass for each 3 & (‐3). TThat is the given set C
value of x, v
there is no
o unique vaalue of y.
O MANY RELATIONS)) (ONE TO
The typical notation for a function is f(x). This is read as "f of x" This does NOT mean f times x. This is a special notation used only for functions! However, f(x) is not the only variable used in function notation. We can use g(x), or h(x), or even b (a). We can use any letters, but they must be in the same format ‐ a variable followed by another variable in parenthesis. f(x) is another way of representing the "y" variable in an equation. E.g.: 3
5 can also be written as 3
5 Simplification of algebraic formulae: We can often simplify algebraic expressions by 'collecting like terms'. We can re‐
order the terms in the expression so that the x terms are together and the y terms are together. Example: Simplify 5 3
2
7 4 Solution: 3
2
4
5 7 2
6
2 2
6
2 Exercise: Simplify the following: 17 4
5
10
4
20 1. 3
3
5
9
3 11 2. 3
4
3.
3
5
1
7 15 9
32 4. 3
2
7 5
3 5. 3
6. 2 6
5
3
2 Write the following as algebraic expression and simplify if possible: 1. The sum of 6 times a number and 3, subtracted from 10 times a number. 2. The sum of 9 times a number and 6, subtracted from 3 times a number 3. Five times a number subtracted from 3 times the square of that number. 4. A number divided by 3 gives the same result as adding 2 to it. 5. Four times a number is subtracted from 50 6. Two more than three times a number is 11. Write algebraic expression for the following statements and then solve them to get the answer for the problem given: 1. A number is three more than another number. When added together they give the result 11. What are the numbers? 2. The sum of three consecutive numbers is 48.What are the numbers? 3. Mrs. Prasad made cookies for her family. Sharon ate three times as many as Janit and Shivani ate six times as many as Janit. How many cookies did Janit eat if a total of 30 cookies were made? 4. Arti spends 1 3 of her monthly salary on charity. In December she spent $150 on charity. How much did she earn in December? 5. A rectangle has its length 4 cm more than its width. If its perimeter is 32 cm, what are the dimensions of the rectangle? 6. When 5 is added to a number and the result divided by 2, it gives the same answer as when 3 is added to the same number and the result divided by 3. What is the number? 7. Today Josefa studied one hour less than he did yesterday. Tomorrow he will study 2 hours more than he did today. If the total hours he would study on the three days are 10 hours, for how many hours did he study today? Solve the following equations for : 1.
2.
2
3.
4
4.
6 5.
3
2
1
2 2
3
18 6.
2
7.
8.
9.
5
2
6 45 Note: 1.
2
2
2.
3.
Solving Quadratic Equations using Formula: 0 where , and are just A quadratic equation is of the form numerical coefficients, then the quadratic formula for finding the value of x is given by √
Nature of roots of a quadratic equation: 1. If
4
0, then the equation has only one root
, called the double root or repeated root. 4
0, then the equation has two distinct real roots . 2. If
3. If
4
0, then the equation has two distinct complex roots that are conjugates of each other. 3
4 0 Example: (1) Solve Solution: Here 1, 3 and 4 √
√
√
1, 4 Thus, the solution is 1 and 4 . 10
34 0 (2) Solve Solution: Here 1, 10 and 34 √
√
√
5 3 Thus, the solution is 5 3 and
5 3 . Factorise the following algebraic expressions : 36 1. 2. 16
3. 25 4. 6
5. 14
49 6. 10
16 7. 2
8. 17
9. 9 24 18 30 10.3
9
11.5
30
40 12.4
12
40 6 13.4
13
14.
3 20
15.5
100 11
16.
4
2 4 17.4
20
18.2
9
10 19.2
5
3 20.
16
64 16 Solve for x: 21.2
22.9
23.
24.3
25.2
26.
27.
28.
4
3
12
2
0 4
0 4
0 4
2
0 4
3
0 2
3
0 6
9
0 3
3
0 Expand the following: 1.
11
2.
5
12 5 3.
3
4 4.
2
1
5.
3
4
2 6 6. 2
11 3
3
7.
1
6
8.
7 2
4
9. 4
3
11. 4
1 5
4
3 2
11 2
14. 4
3 4
15.
10
16.
2 17.
3 4 3 2 18. 2
1 19. 3
5 20. 2
3
21. 7
9
22. 2
7 12.
13.
5 1 2
10. 2
3 6
Simplify the following algebraic expression: 1.
2.
3.
4.
5. 6. 7. 8. 9. Solve the following by transformation of formulae: 1. If 2. If 2
3. If 2
then make g as the subject of formula. 2
4. If 5. If then make t as the subject of the formula. 3
6 then make r as the subject of the formula. then make u as the subject of formula. then make a as the subject of formula. Evaluate the following algebraic expressions: 4
1. Evaluate3
3
3. Evaluate 5. Evaluate 2 and
5 . for a = –2, b = 3, c = –4, and d = 4. 2. Evaluate 4. If 5 , given that √25
3
6 for – 3. then evaluate 1 ,
2 when 1 and 3 and 3 2 Solution of two simultaneous linear equations with two unknowns: Here we will be learning two methods of solving two simultaneous linear equations with two variables.  Method of Substitution  Method of Elimination Example: Solve for & 10 ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐> (Equation 1) – 2 ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐> (Equation 2) Solution: Method 1: Method of Substitution From equation (1) we have 10– ‐‐‐‐‐‐‐‐‐‐‐‐> (Equation 3) Substitute the value of equation (3) in equation (2), 10
– 10
2 2
2 12
2 2 10
6 Again, substituting the value of in equation (3) we have, 10– 6
4 Hence, 6 and 4 Method 2: Method of Elimination Adding equation (1) & (2) gives, 10 – 2 _+_ _ +____+_________ 2 ∴
12 6
Substituting the value of in equation (1) gives, 6
10 10– 6 4 Exercise: 1. Solve the following systems of linear equations by method of substitution and elimination: 2x  y  4
a)
x  y 1
x y2
b
2 x  y  10
c
x  y  24
2 x  y  6
d
2x  y  7
3x  y  8
2. 1000 tickets were sold at a theatre last Saturday. If the adult tickets cost $8.50, children's cost $4.50, and a total of $7300 was collected, then how many tickets of each kind were sold? 3. The sum of two numbers is 15. The difference of the same two numbers is 7. What are the two numbers? 4. A woman is now 30 years older than her son. 15 years ago, she was twice as old. What are the present ages of the woman and her son?
5. Mr. B. has $20,000 to invest. He invests part at 6%, the rest at 7%, and he earns $1,280 interest. How much did he invest at each rate? 6. A landscaping company placed two orders with a nursery. The first order was for 13 bushes and 4 trees, and totaled $487. The second order was for 6 bushes and 2 trees, and totaled $232. The bills do not list the per‐item price. What were the costs of one bush and of one tree? 7. The sum of twice one number and three times another number is 23 and the product of the numbers is 20. Find the numbers. _____________________________________________________________________________________