Download Algebraic Modelling (AM1) Basic Algebraic Skills

 Algebraic Modelling (AM1) Basic Algebraic Skills Name .......................................................................... G. Georgiou 1 General Mathematics (Preliminary Course) | Basic Algebraic Skills (AM1) Gener
• Add, Subtract, Multiply and Divide Algebraic Terms Consider the following expression: 3x + 4y In this expression we have two terms: 3x and 4y. The co-­‐efficient of x is 3 and the co-­‐efficient of y is 4. The ‘letters’ x and y are actually called pronumerals or variables. Define “like terms”: .................................................................................................................... .................................................................................................................................................... Only add or subtract the coefficients of like terms Example 1 Circle the like terms from the lists below. (a) 4x, 5y, 3x2, -­‐x, 2x, 4xy (b) 5xy, 3x2y, 4yx, -­‐3xy, 4y, -­‐7x (c) 4x2y, 4xy2, 3x2, 5x2y, -­‐yx2 (d) 3fgh, -­‐fg2h, 7fhg2 Example 2 Simplify the following algebraic expressions. (a) 3a + 5a (b) 4a + 7b – 3a + 2b .................................................................. ........................................................................... (c) 5a2 – 4a + 3a2 (d) 5a – a .................................................................. ........................................................................... (e) 6ab + 4a – 10ab + a – b (f) 3xy – 5yx + 4x – 2yx .................................................................. ........................................................................... !
(g) 4x2y – 5yx2 + 4y2x (h) 5ab2 + !b2a .................................................................. ........................................................................... .................................................................. ........................................................................... 2 General Mathematics (Preliminary Course) | Basic Algebraic Skills (AM1) Gener
Multiply the numerals Add t
he powers of like variables Example 3 Simplify the following expressions. (a) 3x 4y (b) 5a 4 .................................................................. ........................................................................... (c) 3a 4a (d) 5ab 7a .................................................................. ........................................................................... !
(e) 4x2y -­‐5y2 (f) 4abc3 !abc .................................................................. ........................................................................... Divide the numerals Subtract the powers of like variables (NOTE: be careful to distinguish between the numerator and the denominator) Example 4 Evaluate the following expressions. !" !"
(a) 9L3 3L2 (b) "!
.................................................................. ........................................................................... .................................................................. ........................................................................... (c) !" !" #
$% " !
.................................................................. .................................................................. (e) ! ! ÷"! ! " .................................................................. .................................................................. (d) !"! ""
#" $
........................................................................... ........................................................................... (f) 24xy 6y2 ........................................................................... ...........................................................................
3 General Mathematics (Preliminary Course) | Basic Algebraic Skills (AM1) Gener
• Simplify Algebraic Expressions involving Multiplication and Division of Fractions When multiplying algebraic fractions: 1. Multiply the numerators (top terms) 2. Multiply the denominators (bottom terms) 3. Simplify Example 5 !! #!
!!" ##
! (a) !
(b) "
$
" "!
.................................................................. ........................................................................... .................................................................. ........................................................................... .................................................................. ........................................................................... !!"
!!" ##
!#! !
(c) (d) "
"#
!
.................................................................. ........................................................................... .................................................................. ........................................................................... .................................................................. ........................................................................... When dividing algebraic fractions: 1. Take the reciprocal (flip) of the second expression 2. Replace the division for a multiplication 3. Multiply as before Example 6 ! !
!"! $"
÷
(a) ÷ (b) ! "
#
!%
.................................................................. ........................................................................... .................................................................. ........................................................................... .................................................................. ........................................................................... .................................................................. ........................................................................... 4 General Mathematics (Preliminary Course) | Basic Algebraic Skills (AM1) Gener
(c) !! #! !
÷
" " ! $" "
.................................................................. .................................................................. .................................................................. .................................................................. "!
(e) !!" ÷
#"
(d) ! !"
÷ $"
"" #
(HINT: ! ! =
!!
) "
........................................................................... ........................................................................... ........................................................................... ........................................................................... (f) !! #
÷ !" "" $
!
!"
(HINT: !" =
) "
"
.................................................................. ........................................................................... .................................................................. ........................................................................... .................................................................. ........................................................................... .................................................................. ........................................................................... Example 7 Daniel has made two errors in his working out in the following expression. Explain what they are and how he should have proceeded. #
!!" #"
.......................................................................................................... ÷
"! $%!
#
.......................................................................................................... !!" #"
=
!
"! $%!
.......................................................................................................... $%!" "
=
"%! #
.......................................................................................................... = "!" "
.......................................................................................................... !
!
Activity Ex 2.02 ALL 5 General Mathematics (Preliminary Course) | Basic Algebraic Skills (AM1) Gener
• Expansion and Simplification of Expressions When expanding expressions with grouping symbols: 1. Follow order of operations 2. Multiply all terms inside the brackets with the term directly outside the bracket 3. Simplify Example 8 Expand and simplify the following expressions. (a) 3(x + 4) (b) !!" ! +!# .................................................................. ........................................................................... (c) !!" ! ! #$ (d) !! ! +"# .................................................................. ........................................................................... (e) !" ! + #$+%" ! !&$ (f) ! ! "# ! ! $% .................................................................. ........................................................................... .................................................................. ........................................................................... (g) !" ! !#$! %" ! !!$ (h) ! ! " ! ! ! #$!" ! ! #$ .................................................................. ........................................................................... .................................................................. ........................................................................... Example 9 Two trees in a forest had their heights recorded as (x – 5) metres and (2x + 3) metres. What is the difference in height between the two trees given that the tree with height (2x + 3) metres is taller? .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... Activity Ex 2.03 Q 1, 2, 3 6 General Mathematics (Preliminary Course) | Basic Algebraic Skills (AM1) Gener
Substitute Numerical Values into Algebraic Expressions Substitute Given Values for the other Pronumerals in a Mathematical Formula from a Vocational or Other Context to Find the Value of the Subject of the Formula Tips for solving substitution questions: 1. Always show your substitution in your working out 2. Be careful with negative numbers Example 10 ! +"
Evaluate if: !"
(a) x = 3 and y = 2 (b) x = –2 and y = 8 .................................................................. ........................................................................... .................................................................. ........................................................................... .................................................................. ........................................................................... Example 11 Evaluate the following expressions if a = 2, b = –4 and c = 3. (a) ! ! " (b) ! ! ! " .................................................................. ........................................................................... .................................................................. ........................................................................... .................................................................. ........................................................................... •
•
(c) ! +"
!# (correct to 3 sig figs) .................................................................. .................................................................. .................................................................. (d) 5(2 – c) ........................................................................... ........................................................................... ........................................................................... 7 General Mathematics (Preliminary Course) | Basic Algebraic Skills (AM1) Gener
Example 12 The library has an equation for charging its borrowers overdue fines. The equation is !
! = #" +"$ , where F is the fine and n is the number of days the book is overdue. "
What is the minimum fine for a book that is overdue? .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... Example 13 The growth of an amount invested in a high interest account can be calculated using the formula A = P (1 + r)n, where A is the final amount, P is the initial amount, r is the interest rate and n is the time period. If $1000 is invested for 6 years at 5% p.a., what is the final amount of the investment? .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... Example 14 #
! !"" $
R is related to V by the formula R = #
& "! %
When V decreases from 50 to 40, the value of R: (A) Increases by 9 (B) Decreases by 9 (C) Increases by 81 (D) Deceases by 3600 .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... Activity Ex 2.04 Q 1, 3, 5, 8, 9, 10, 12, 13, 14, 17, 18, 19 8 General Mathematics (Preliminary Course) | Basic Algebraic Skills (AM1) Gener
Solve Linear Equations involving Two Steps An equation is a statement where you must find a value for the pronumeral in the equation. The method of solving equations can vary significantly. The below examples illustrate the range of questions that may come up in the General Mathematics course. Example 15 Solve the following equations. (a) a + 4 = 7 (b) 2a + 3 = 9 .................................................................. ........................................................................... .................................................................. ........................................................................... .................................................................. ........................................................................... (c) 12 = 5x – 3 (d) 3x – 2 = 4x .................................................................. ........................................................................... .................................................................. ........................................................................... .................................................................. ........................................................................... !
(e) 4x = 48 (f) = "# !
.................................................................. ........................................................................... .................................................................. ........................................................................... ! !!
= # (g) 3(f – 2) = 24 (h) "
.................................................................. ........................................................................... .................................................................. ........................................................................... .................................................................. ........................................................................... .................................................................. ........................................................................... •
9 General Mathematics (Preliminary Course) | Basic Algebraic Skills (AM1) Gener
(i) !
! !!
= # "
.................................................................. .................................................................. .................................................................. .................................................................. (k) –10 = –2c – 4 .................................................................. .................................................................. .................................................................. .................................................................. (m) 10 = – 4g .................................................................. .................................................................. .................................................................. .................................................................. Activity Ex 2.05 ALL !
(j) ! " = # !
........................................................................... ........................................................................... ........................................................................... ........................................................................... (l) !!
= $# "#
........................................................................... ........................................................................... ........................................................................... ........................................................................... (n) !!
+ # = ! "
........................................................................... ........................................................................... ........................................................................... ........................................................................... 10 General Mathematics (Preliminary Course) | Basic Algebraic Skills (AM1) Example 16 .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... 11 General Mathematics (Preliminary Course) | Basic Algebraic Skills (AM1)