MODULE - 5 EQUATIONS AND INEQUALITIES - Maths And Science Lessons

23/09/2017 LINEAR EQUATIONS EQUATIONS AND INEQUALITIES Golden Rule of Equations: “What you do to one side, you do to the other side too” E.g. Solve the following equations: (a) 2s 3 11 2 s 11 3 Linear Equations Quadratic Equations Simultaneous Linear Equations Word Problems Literal Equations Linear Inequalities 2s 8 s 4 (b) 2 3(k 3) 2k 3 2 3k 9 2k 3 7 3k 2k 3 3k 2k 3 7 5k 10 Solving equations with denominators c) 2q 1 3q 2 q 3 6 2 d) 8 2(t 4) t t 8 2(t 4) t t t t 5t 8 2(t 4) 5t 8 2t 8 5 t 2q 1 3q 2 q 6 6 6 3 6 2 2( 2q 1) (3q 2) 3q 5t 2t 8 8 4q 2 3q 2 3q 3t 0 4q 3q 3q 2 2 t 0 2q 4 q 2 Typical Complex Equations 3 EXERCISE 4 2. Solve the following equations: 1. Solve the following equations: (a) b 15 6 – 2b c – 5 2c 3 3 - 2y 6y - l 2a – 7 5a – 12 3(m 2) 2(m - 1) 4(k - 1) 6(k - 1) 7(q- 2) - 2(7 – 3q) 2 0 4(2y - 7) - 8(5 – y) 3(2y 4) - 5(y 7) (2m - 1) (2m l) (2m - 1) Linear Equations Calculator 5 2 Note! Since variables are in the denominator, t 0 otherwise the fraction will be undefined. Need to find LCD 6 and multiply each term by the LCD to get rid of the denominator (a) (b) (c) (d) (e) (f) (g) (h) (i) Solving Equations k 2 1 (b) p p 2 3 2 3p 1 p 1 4 2 (c) 2m 1 m 5 0 5 (d) 3 (e) 2 m m 6 1 4 3 4 (f) 3 4 5 1 2g 4 g 2 3 4 5 t 2 4 2 Special Case Equations Solving Word Problems Step-by-Step 5 6 1

23/09/2017 QUADRATIC EQUATIONS Standard form: E.g. a) ax2 b) bx c 0 x 2x 0 (x 2)(x-2) 0 (x 2) 0 x -2 Note! Quadratic Equations have 2 real solutions! 2 Factorize x(x-2) 0 c) Solve x 0 or 4x(3x - 1) 0 4x 0 x 0 (x-2) 0 x 2 or (x-2) 0 x 2 12 x 2 4 x 0 12x – 4x 0 When 2 factors are timsed together, the answer is 0. Therefore, if A x B 0, then either A 0 or B 0. x2 4 0 write in standard form factorize or (3x – 1) 0 x l x 1 3 7 Note! 8 d) You could have first divided both sides by 4 to simplify the equation. However, you may never divide both ides by the variable you are solving for. The reason for this is that you will lose one of the solutions if you divide by the variable you are solving for. 3 p 2 9 p 6 3 p2 9 p 6 0 3( p 2 3 p 2) 0 3 0 p2 3 p 2 0 ( p 2)( p 1) 0 12 x 2 4 x 0 3x 2 x 0 x(3x – 1) 0 x 0 or 3x – 1 0 x 13 e) p 2 or p 1 6n 9 n 2 0 n 2 6n 9 0 n 2 6n 9 0 ( n 3)( n 3) 0 n 3 or n 3 You may write the solution as n 3 only. NB: This equation has two equal solutions. 9 f) 10 EXERCISE 1 ( 2m 1)( m 2) 25 2m 4m m 2 25 0 2 Solve the following equations: (a) y(y 3) 0 (b) (y - 2)(y 5) 0 (c) (2y - 5)(3y 1) 0 (d) b(b - 3) 0 (e) (t 5)(t - 7) 0 (f) 3p(4p 5) 0 2m 2 3m 27 0 ( 2m 9)( m 3) 0 9 m or m 3 2 Solving Quadratic Equations 11 12 2

23/09/2017 EXERCISE 2 EXERCISE 3 2.Solve the following equations: (a) p 1 (b) k 25 (c) g 2 9 g (d) 4 x 2 16 (e) m 2 14m 16 0 (f) 2m 2 m 10 2 3. Solve the following equations: (a) (x - 7)(x - 5) 0 (b) (k - 7)(k 3) 0 (c) (r - 3)(r - 2) 12 (d) (x - 2)² 0 (e) (y – 7)(y 5) 0 (f) (w - 7)(w 3) 24 (g) 6(1 – f )² 5f (h) (q – 2)² 16 2 (g) j 2 10 j 25 Quadratic Equations Calculator 13 SIMULTANEOUS LINEAR EQUATIONS 14 Substitution Method E.g. Solve: 3x – y l0 and x y 6 Determining the values of x and y that will solve the equation simultaneously E.g. Solve simultaneously: 3x - y 10 and x y 6 x 4 and y 2 satisfy the equations simultaneously. This can be checked by substitution 3x – y l0 and x y 6 3(4) - (2) 10 (4 ) (2) 6 True statement True statement Label each equation 3x – y l0 . (1) x y 6 . (2) Now pick either one of the equations and solve for one of the variables (choose the easier equation) 3x – y 10 (1) y 3x – 10 (3) Now replace the variable y in equation (2) with (3) x y 6 x (3x - 10) 6 4x - l0 6 4x 16 x 4 15 16 Elimination method Given: 3x – y l0 . (1) x y 6 . . (2) E.g. a) Solve: 3x – y l0 and x y 6 Now substitute x 4 into either equation (1) or (2) to solve for y x y 6 . (2) (4) y 6 y 6–4 y 2 17 Label each equation 3x – y l0 . (1) x y 6 (2) Add the 2 equations vertically, in order to eliminate one of the variables: 4x 0y 16 4x 16 x 4 Now substitute x 4 into either equation (1) or (2) to solve for y x y 6 . (2) (4) y 6 y 2 18 3

23/09/2017 Given: 3x – 2y l0 2x 4y 12 E.g. b) Solve: 3x – 2y l0 and 2x 4y 6 Label each equation 3x – 2y l0 . (1) 2x 4y 12 . (2) Multiply one of the equations by a factor, in order to eliminate a variable. So, multiply (1) by 2, so that the y’s can be eliminated, to solve for x: 3x – 2y l0 . (1) x 2 6x – 4y 20 . (3) Add the two equations (3) (2): 6x – 4y 20 . (3) 2x 4y 12 . (2) 8x 0y 32 x 4 . (1) . (2) Now substitute x 4 into either equation (1) or (2) to solve for y 3x – 2y 10 . (1) 3(4) – 2y 10 12 – 2y 10 -2y 10 – 12 -2y -2 y 1 Solving Simultaneous Equations by means of Elimination 19 EXERCISE 1. Solve for x and y by using the method of substitution: (a) x – y 2 and 2x y 10 (b) y - 3x - 2 and 7x - 2y 8 (c) 3x 5y 8 and x - 2y - l (d) 7x - 3y 41 and 3x – y 17 2. Solve for x and y by using the method of elimination: (a) x – y 2 and 2x y l0 (b) x y - 5 and 3x y - 9 (c) x 2y 5 and x – y - l (d) 3x 5y 8 and x - 2y -l (e) 2x - 3y l0 and 4x 5y 42 Points of Intersection Calculator 20 3. Solve for x and y using any method of your choice: (a) x y 1 and x - 2y 1 (b) 3x 2y 2 and 5x - 2y - l8 (c) x 4y 14 and 3x 2y 12 (d) 2y - 3x 7 and 4y - 5x 21 (e) 3x 2y 6 and 5x 3y l1 (f) y 2 x 1 1 and y 4 x 3 1 3 4 21 22 WORD PROBLEMS Meters Linear Word Problems Example 1: 5 2 Consecutive Numbers Word Problem Example 2: A curtain manufacturing company sold a client 12 meters of material. The cost of the good material was R8 per meter. Some of the material was inferior and was sold for R7 per meter. The total cost of the material was R100. How many meters of material was inferior? (Let the good material x.) 23 Inferior material 12-x Price per meter 7 Good material x 8 Cost 7(12-x) 8x 7(12-x) 8x 100 96 – 7x 8x 100 x 100 – 96 x 4 The length of the inferior material is 4 meters . 24 4

23/09/2017 Simultaneous Word Problems Example The perimeter of a rectangular floor is 6 meters. The length of the floor must be twice the length of its breadth. Equation for the perimeter: 2x 2y 6 Equation relating the length to the breadth: x 2y Solve simultaneously: 2x 2y 6 .(1) x - 2y 0. (2) 3x 6 x 2 Determine the dimensions of the floor. 2 – 2y 0 -2y -2 y l Therefore: Length 2m and Breadth 1m. 25 26 Quadratic Word Problems Example A rectangle has an area of 8 meters. Its breadth is 2 meters less than its length. Find the dimensions of the rectangle. Equation relating to area x(x - 2) 8 x²- 2x 8 x²- 2x – 8 0 (x - 4)(x 2) 0 x 4 or x - 2 Since x -2 is not a solution, as the length cannot be negative. So, Length 4m and the Breadth 2m. 27 28 Exercise Complex Word Problems 1. There were 600 spectators at a tennis match. Of these 144 were children and there were twice as many men as women. Determine how many women were there. 2. Tickets to the DJ Kwa Kwa concert cost R200 and R300. A total of 250 tickets were sold. The total amount taken for the show was R55 000. Determine how many of each ticket were sold. 3. The length of a rectangular field is 3 meters more than its breadth. The area of the field is 70 m . Calculate the length of the field. Age word problem Train word problem Algebraic word problem 29 30 5

23/09/2017 4. The area of a room in a house has the following measurements as shown below. LITERAL EQUATIONS A literal equation is one in which letters of the alphabet are used as coefficients and constants. These equations, usually referred to formulae, are used a great deal in Science and Technology. The aim is to solve the equation or formula for a specific letter or to make that letter the subject of the formula. If the total area of the room is 43 m², calculate the value of x. E.g. a) Make 31 a the subject of the formula: v u at 32 c) Make r the subject of the formula: A v u at v u at A r v u a t v u a t b) Make V the subject of the formula: 1 A r2 A P1V1 PV 2 2 T1 T2 r 2 2 r r V 1 A P1V1 PV T1T2 2 2 T1T2 T1 T2 P1V1T2 P2V2T1 P1V1T2 PV T 2 2 1 P1T2 P1T2 V1 P2V2T1 P1T2 33 EXERCISE 34 LINEAR INEQUALITIES 1. Make h the subject of the formula S 2 r h r 2. Make a the subject of the formula v 2 u 2 2as 3. Make x the subject of the formula (a) p kx s (b) 2 px 3kx f (c) px kx 0 2 4. The area of a circle is 47t. Calculate the length of the radius of the circle. 35 Consider the following true statement: -3 1 If we now multiply (or divide) both sides by - 1, the statement will become 3 - 1 which is false! Therefore, when you multiply or divide by a negative number, the inequality sign changes direction e.g. 3 - 1 which is true! Multiplication and Division in Inequalities 36 6

23/09/2017 E.g. Solve the following inequalities and then represent the solution(s) on a number line: a) b) 2k 1 2k 1 3 4(2m 1) 5m 2 Times through by the LCD 3, in order to get rid of the denominator. 8m 4 5m 2 8m 5m 2 4 2k 1 2k 1 3 2k 1 6k 3 3m 6 m 2 Solving Linear Inequalities 2k 6k 3 1 4k 4 k 1 Linear Inequalities and the Number Line 37 38 c) 3 3m 1 5 EXERCISE Add 1 to each part of the inequality. 1. Solve the following inequalities and represent your solution on a number line. a) 3 1 3m 1 1 5 1 2 3m 6 x 5 6 2x Now divide each part by 3. 2 3m 6 b) x 5 2x 3 2 m 2 3 c) 4( x 1) 6( x 1) Solving Complex Inequalities d) 4( x 3) 2( x 1) 0 39 40 3. Solve the following inequalities and represent 2. Solve the following inequalities and represent your solution on a number line. your solution on a number line. (a) m m 1 2 3 (a) 2 y y 1 5 (b) 3m 1 m 1 4 2 (b) 3 a 2 4 (c) m 5 m 1 3 (c) 5 2k 1 5 (d) 3m 2 m 6 0 4 3 (d) 8 3p 2 4 (e) 9 l 5 d (f) 2 2 k 2 (e) (x 3)(x - 4) (x - 3)(x 4) 41 42 7

23/09/2017 Interval Notation E.g. Write the following in interval notation: Is a way of representing real numbers by use of brackets: Round brackets ( ; ) indicates not included (equivalent of or ) Square brackets [ ; ] indicates included (equivalent of or ) Interval Notation 43 EXERCISE 2. Represent the following on a number line: 1. Represent the following sets on a number line: (a) {x : x 6; x Z } (b) {x : x 4; x N } (c) {x : 5 ; x Z } (d) {x : l x; x N } (e) {x : x 7; x R} (f) 1 {x : 3 x; x R} 2 44 45 (a) (-3 ; 6) (b) (-2 ; 11) (c) (-2 ; 5) (d) (-8 ; 8) (e) ( 5; ) (f) ( ;3] 46 8

EQUATIONS AND INEQUALITIES Golden Rule of Equations: "What you do to one side, you do to the other side too" Linear Equations Quadratic Equations Simultaneous Linear Equations Word Problems Literal Equations Linear Inequalities 1 LINEAR EQUATIONS E.g. Solve the following equations: (a) (b) 2s 3 11 4 2 8 2 11 3 s