School of Business. Blank Page

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "School of Business. Blank Page"

Transcription

1 Equations 5 The aim of this unit is to equip the learners with the concept of equations. The principal foci of this unit are degree of an equation, inequalities, quadratic equations, simultaneous linear equations, graphical equation and their applications in solving usiness prolems followed y eamples.

2 School of Business Blank Page Unit-5 Page-90

3 Bangladesh Open University Lesson 1: Equation and Identity After studying this lesson, you should e ale to: Eplain the nature and characteristics of equations; Eplain the nature and characteristics of identities; Solve the equations; Solve the inequalities. Introduction Many applications of mathematics involve solving equation. In this lesson we will discuss the equation, identities and uses of equations. Equation An equation is a statement which says that two quantities are equal to each other. An equation consists of two epressions with a sign etween them. In other words, if two sides of an equality are equal only for particular value of the unknown quantity or quantities involved, then the equality is called an equation. For eample, 4 8 is true only for. Hence, it is an equation. An equation which does not contain any variale is either a true statement, such as + 3 5, or a false statement, such as If an equation contains a variale, the solution set of the equation is the set of those values for the variale which gives a true statement when sustituted into the equation. For eample, the solution set of y 4 is (, ), ecause ( ) 4, and 4, ut y 4 if y is any numer other than or. Identities The equations signify relation etween two algeraic epressions symolized y the sign of equality. If two sides of an equality are equal for all values of the unknown quantity or quantities involved, then the equality is called an identity. For eample, y ( + y) ( y) is an identity. We can prove that identities hold true for whatever are values of the variales sustituted in these. It we use and y 3 in the aove identity, we have () (3) ( + 3) ( 3) or, 4 9 (5) ( 1) or, 5 5 Again, y sustituting the values of 4 and y 6, we have ( 4) ( 6) ( 4 6) ( 4 + 6) or, ( 10) () or, 0 0 Hence, identities hold true for whatever value is put for variales. An equation is a statement which says that two quantities are equal to each other. If two sides of an equality are equal for all values of the unknown quantity or quantities involved, then the equality is called an identity. Business Mathematics Page-91

4 School of Business Derived identities are the identities derived y transposing the values in the asic identities and are very useful in tackling some prolems in mathematics. Derived Identities Derived identities are the identities derived y transposing the values in the asic identities and are very useful in tackling some prolems in mathematics. For eample, (1) Identity ( + y) + y + y... (i) Derived Identities + y ( + y) y and y ( + y) ( + y ) () Identity ( y) y y... (ii) Derived Identities + y ( y) + y and y + y ( y) By adding (i) and (ii) ( + y) + ( y) ( + y )... (iii) By sustituting (ii) from (i), we get ( + y) ( y) 4y... (iv) By dividing oth (i) and (ii) y 4 and then sutracting (ii) from (i), we have [( + y) / 4] [( y) ] The following section of this lesson contains some model applications of equations. Eample 1: Solve, By cross multiplying, we have or, or, Squaring oth sides, we have, 4(1 + ) 16 (1 ) or, Transposing the term 16 and or, 0 1 Unit-5 Page-9

5 Bangladesh Open University or, 5 3 Therefore, 5 3 is the solution of the given equation. Eample : The sum of two numers is 45 and their ratio is 7:8. Find the numers. Let, one of the numers e The other numer is (45 ) Using the given information, we get, By cross multiplication, we get 8 7 (45 ) or, Transposing the term 7, we have or, or, 1 15 Hence, the one numer is 1 and the other numer is (45 1) 4. Eample 3: The ages of a mother and a daughter are 31 and 7 years respectively. In 3 how many years will the mother s age e times that of the daughter? Let the required numer of years e. Mother s age after years 31 + Daughter s age after years 7 + Using the given information, we get (7 + ) or, 31 + By cross multiplication, we have (31 + ) or, Business Mathematics Page-93

6 School of Business Transposing the terms 3 and or, 41 or, 41 Hence, in 41 years, the mother s age will e daughter s. 3 times age of the Eample 4: The distance etween two stations is 340 km. Two trains start at the same time from these two stations on parallel tracks to cross each other. The speed of one train is greater than that of other y 5 km / hr. If the distance etween the two trains after hours of their start is 30 km, find the speed of each trains? Let the speed of the first train e km / hr. Then the speed of the second train e ( + 5) km / hr. Distance covered y the first train in hrs km. Distance covered y the second train in hrs ( + 5) km ( + 10) km. Since, oth the trains are in opposite directions. Total distance etween the two stations 340 km. Distance etween the two trains after hours 30 km. Therefore, distance covered y two trains in two hours from opposite directions (340 30) 310 km + ( + 10) 310 or, Hence, speed of the first train 75 km / hr and the speed of the second train 80 km / hr. Unit-5 Page-94

7 Bangladesh Open University Questions For Review: These questions are designed to help you assess how far you have understood and can apply the learning you have accomplished y answering (in written form) the following questions: 1. What is an equation and inequalities? Mention the characteristics of equation.. What is the difference etween identity and equation? Give eamples. 3. Solve the following equations: (i) ( + 1) + 7 / (+1) 18 (ii) (iii) Wasifa s mother is four times as old as Wasifa. After five years, her mother will e three times as old as she will e then, what are their present ages? 5. A steamer goes downstream and covers the distance etween two parties in 4 hours while it covers the same distance upstream in 5 hours. If the speed of the stream is 1 cm/per hour, find the speed of the steamer in still water. 6. Three prizes are to e distriuted in a quiz contest. The value of the nd prize is five-siths the value of the first prize and the value of the third prize is four fifths that of the second prize. If the total value of the three prizes is Tk.15,000, find the value of each prize. 7. The price of two cows and five horses is Tk.68,000. If the price of horse eceeds that of a cow y Tk.800, find the price of each. Multiple choice questions ( the appropriate answers) 1. The difference etween two numers is 9 and the difference etween the squres is 07. The numers are: a) 17, 8 ) 16, 7 c) 3, 14. The ratio of two numers is 3 : 5. If each is increased y 10 the ratio ecomes 5 : 7, the numers are: a) 6, 10 ) 15, 5 c) 9, The sum of three numers is 10. If the ratio etween the first and the second e : 3 and that of etween the second and the third e 5 : 3, the second numer is a) 30 ) 45 c) 7 4. Taka 49 were divided among 150 children. Each girl got 50 paisa and each oy got 5 paisa. How much oys were there? a) 104 ) 105 c) If and, then the value of y is: a) 1/3 ) 1 c) 3 Business Mathematics Page-95

8 School of Business Lesson-: Inequality After studying this lesson, you should e ale to: Descrie the nature of inequalities; Eplain the properties of inequality; Solve the inequalities. Nature of Inequality Relationship of two epressions with an inequality sign etween them is called inequality. Relationship of two epressions with an inequality sign ( or, < or >) etween them is called inequality. For eample, > y is greater than y < y is smaller than y > y is not greater than y < y is not smaller than y y is smaller than or equal to y y is greater than or equal to y Properties of Inequalities The fundamental properties of inequalities are as follows: (a) Order Aioms: If and are only elements, then (i) One and only one of the following is true:, < y and > y (ii) If < y and y < z, then y < c (iii) If < y and < z, then z < yz Since, > y and y < are the same statements, the aove aioms can e replaced in terms of > y. As shown earlier sometimes equality signs are comined with inequality signs < y means y or < y. Again < y means is not less than y and that means either > y. So, < y means y. We also say that is positive when 0 and is negative, when < 0. All equals may e add or sutracted from oth sides of inequalities and the inequality is preserved. () Operation Aioms: (i) All equals may e add or sutracted from oth sides of inequalities and the inequality is preserved. For eample, if 5 9 < 1 We may add 5 to oth sides and we get < or, 5 4 < 17 Unit-5 Page-96

9 Bangladesh Open University (ii) Any term in an inequality can e moved from one side to the other provided that its sign is changed. For eample, if 5 4 < 17. or, 5 < And, again if z > y or, > y + z Both sides of an inequality may e multiplied or divided y a positive numer and the inequality is preserved. For eample, if 1 < 36. After Multiplying the oth sides of inequality y 5, we get 1 5 < 36 5 or, 60 < 180 Again, after dividing oth sides of inequality y 3, we get 1 3 < 36 3 or, 4 < 1 (iii) Both sides on an inequality are multiplied or divided y a negative numer and the direction of the inequality is reversed. For eample, if 7 < 40, (where 4). (iv) (v) By multiplying oth sides of the inequality y 5, we have 7 ( 5) > 40 ( 5) < to >] or, 35 > 00 [Note that inequality sign has een changed from This is ecause when 4, the inequality is greater than 00. An inequality can y converted into an equation: If > y then y + p Where p is the positive real numer (i.e. p > 0) If z > m, then we write z m + q, where q > 0 Hence,, z (y + p) (m + q) ym + yq + pm + pq Now p and q are positive. If in addition y and m are positive, then every term on the right-hand side is also positive so that. z > y. m If y z m >, then < z m y Both sides of an inequality may e multiplied or divided y a positive numer and the inequality is preserved. Both sides on an inequality are multiplied or divided y a negative numer and the direction of the inequality is reversed. (vi) If < y, then > y (vii) Now if 1 > y 1, > y, 3 > y n > y n, then n > y 1 + y + y y n and n > y 1. y. y y n Business Mathematics Page-97

10 School of Business (viii) If > y and n > 0 then n > y n and 1 n < The following section of this lesson contains some applications of inequality. Eample-1: Solve: 3[4 5( 3)] 7 [ + 3 (4 )] 3[4 5( 3)] 7 [ + 3 (4 )] or, 3[ )] 7 [ + 1 3)] or, Transposing oth sides we have or, or, 1 6 Multiplying oth sides y 1, we get or, 6 31 or, 11 1 y n Eample-: Solve: 5 (3 4) > 4[ 3 (1 3)] 5 (3 4) > 4[ 3 (1 3)] or, 5 6 8) > or, > 1 8 or, 45 > 0, Multiplying oth side we get 1, or, 45 < 0 0 or, < 45 i.e. < 9 4 Unit-5 Page-98

11 Questions for review Bangladesh Open University These questions are designed to help you assess how far you have understood and can apply the learning you have accomplished y answering (in written form) the following questions: 1. Solve the following inequalities: (i) (ii) 3 < (4 ) < 7 4 (1 ) (iii) (5 3) 4 19 (iv) 3 [ (3 + )] 11 3 [ 5 (3 )] (v) [(5 7) / ( 3)] (3 4) < 0. Solve each inequality with a sign graph (i) ( + 3) ( + 4) < 0 (ii) ( 1) ( ) ( 3) < 0 Multiple choice questions ( the appropriate answers) 1. If + y 3, > 0 and y > 0, then one of the solution is a) 1, y ) 1, y 1 c), y 1. The system of linear in equalities + y 0, 0 and y 0, has a) eactly 1 solution ) 3 solutions c) no solution 3. The value of from linear inequality 4 7 > a) < 6 ) < 8 c) < 4 Business Mathematics Page-99

12 School of Business Lesson-3: Degree of an Equation After studying this lesson, you should e ale to: Eplain the nature of degree of an equation; Solve the simultaneous linear equation. Nature of Degree of an Equation An ordinary equation involving only the first power of the unknown quantity is called simple or linear equation or equation of the first degree. An equation involving only one unknown quantity is called ordinary equation. An ordinary equation involving only the first power of the unknown quantity is called simple or linear equation or equation of the first degree. When the highest power of the unknown quantity is, it is called quadratic or the second degree equation; when the highest power of the unknown quantity is 3, the equation is termed as cuic or the third degree equation. When the highest power of is 4, the equation is called iquadratic or the fourth degree equation. For eample, + 18 y Linear equation Quadratic equation Cuic equation Biquadratic equation If an equation in is unaltered y changing to 1, it is known as a reciprocal equation. An equation in which the variale occurs as indices or eponents is called an eponential equation. For eample, 3 1, etc. are called eponential equations. If more than one unknown quantity are involved, the numer in independent equations required for solution is equal to the numer of the unknown quantities. Such set of linear equations is called simultaneous linear equations. The following section of this lesson contains some model applications. Eample-1: Solve Rearranging the terms, we have Dividing oth sides y we have, Unit-5 Page-100

13 Bangladesh Open University 1 1 or, or, Putting y for + 1, we have 4 (y ) 16 (y) or, 4y 8 16y or, 4y 16y or, 4y 10y 6y or, y (y 5) 3 (y 5) 0 or, (y 5) (y 3) 0 either, y 5 0 or, y 3 0 or, y 5 or, y 3 or, y When, y, then + or, or, + 5 or, or, ( ) 1 ( ) 0 or, ( 1) ( ) 0 or, y 3 either, 1 0 or, 0 or, When, y, then or, + 3 or, Business Mathematics Page-101

14 School of Business We know that ± 4ac (Here, a, 3, c ) a ( Here, 1) 3 ± ± 7 3 ± 7i i Hence, Eample-: 4 Solve + 1 y 1, or, 3 ± 7i 4 4 y (i) y 4 y (ii) 4 From equation (i) + 1 y or, y + 4 y 1 4 From equation (ii) y + 5 or, y + 4 y... (iii) y + 4 or, 5 or, y (iv) Sutracting the equation (iii) from equation (iv) 0 5 y or, y 5 Putting the value of y in equation (iii), we get (5) or, or, or, or, 5 (5 4) 1 (5 4) 0 or, (5 4) (5 1) 0 either, or, or, 5 4 or, 5 1 Unit-5 Page-10

15 Bangladesh Open University so, 5 4 so, 5 1 Now 5 4, then y , then y Thus, 5 4, 5 1 y 0, y 5 Eample-3: Find the solution of the system of equations. 4 3y + z 1... (1) y + z 6... () 3 + 4y 4z 1... (3) Let us first eliminate z: We rewrite: (1): 8 6y + z (): y + z 6 Sutraction: 6 5y 4... (4) Now we rewrite: (3) : 3 + 4y 4z 1 (): 4 y + 4z 1 Addition: 7 + y (5) Let us now eliminate y from (4) and (5); i.e. 5 (5) : 35 10y 55 (4) : 1 10y 8 Addition: So, 1. Now, sustituting the value of in (5), we get the value of y, i.e. 7 (1) + y 11 or, y 11 7 or, y 4 so, y Now, sustituting the value of and y in (1), we have: 4 (1) 3 () + z 1 or, z or, z 3 Therefore the solution is: 1, y and z 3. Business Mathematics Page-103

16 School of Business Questions for Review: These questions are designed to help you assess how far you have understood and apply the learning you have accomplished y answering (in written form) the following questions: 1. Eplain the nature of degree of an equation.. Solve the following equations: (i) (ii) y + y 3 y 3 (iii) z 0 3. Solve 7 3y 9z 0 + y + 3z 3 (i) 3 + y (ii) + y 18 7 a y 81 y 43.3 Multiple choice questions ( the appropriate answers) 1. The solution of the simultaneous linear equation and y y 3 6 a), y 3 ), y 3 c), y 3. The solution of the system of simultaneous linear equation 4 3y 7 and 7 + 5y is: a) 1, y 1 ) 1, y 1 c) 1, y 1 3. If 7 + 9y 85 and 4 + 5y 48, then a) 7, y 4 ) 7 40, y 5 c) 37, y The solution of the pair of equations + 1 and y is: y a) 1, y 3 ), y 3 c) 1 1, y 3 Unit-5 Page-104

17 Lesson-4: Graphical Equation After studying this lesson, you should e ale to: State the nature of graph; Solve the equation with the help of graph. Graphical Equation Bangladesh Open University In this lesson, we will discuss aout graphical equations with the help of following practical eamples. Eample 1: Solve the equation graphically, +5y 1 and y 1 +5y 1...(i) y 1...(ii) From equation (i), 5y 1. 1 So, y 5 So for 0, 1, and 3; y.4,, 1.6 and 1. respectively. Plotting the points (0,, 4), (1, ), (, 1.6), (, 1.6), (3, 1.) and jointing them, we otain graph of + 5y 1, which is AB in the following figure Again, from equation (ii), y 1 So, y 1+ So, for 0, 1, and 3; y 1,, 3 and 4. Plotting the points (0, 1), (1,), (3, 4) and jointing them, we otain graph of y 1, which is CD in the following figure-5.1 Y A D X X O M C Y Figure-5.1 B Business Mathematics Page-105

18 School of Business From the aove graph it is clear that AB is the line of equation (i) and CD is the line of equation (ii) they are intersecting each other at the point M. The co-ordinates of M are (1, ). Hence the required equation is 1 and y. This gives the solution of 1, y for the pair of simultaneous equations + 5y 1 and y 1. Eample : Draw the graph and solve the following equations; 4y y + 11 and 4 6y + 0 The given equations are 4 y or, y 11 4 or, y (11 + 4) so, y (i) and 4 6y + 0 or, 6y 4 or, 6y (4+) or, y (ii) We put the values 0, 1, and 3 to, and find corresponding values of y then putting down the values in the following tale: y y (4 + )/ The system of equations has no solution, ecause the two lines of the equations are parallel and distinct. The system of equations has no solution, ecause the two lines of the equations are parallel and distinct. Hence the given equations are inconsistent. Y X X O Y Figure-5. Unit-5 Page-106

19 Bangladesh Open University The system of equations has no solution, ecause the two lines of the equations are parallel and distinct. Hence the given equations are inconsistent. Eample 3: Solve the following quadratic equation graphically; Let y 4 +3 We give values 0, 1,, 3, 4... to and find corresponding values of y put down in the following tale: y Plotting the points (0, 3), (1, 0), (, 1), (3, 0), (4, 3) and jointing them, we get the graph y on shown in the following figure 5.3. Y P N X O X O M Y Figure-5.3 The curve MNOP is the graph of The graph of the function y 4 +3 is the curve MNOP which crosses ais at the points N and O where 1 and 3. Hence 1 and 3 are the solutions of the quadratic equation Business Mathematics Page-107

20 School of Business Question for Review: These questions are designed to help you assess how far you have understood and can apply the learning you have accomplished y answering (in written form) the following questions: 1. Draw the graphs and solve the following questions: (a) 1 7y 6 () 5 + y 6 5 7y 7 y Draw the graph of the solution set for the system of liner inequalities 0 y 0 + y 45 + y Draw the graph of the solution set for the system of liner inequalities + y 30 y y Draw the graphs and solve the equations (i) (ii) 3 y y Unit-5 Page-108

21 Lesson 5: Quadratic Equation After studying this lesson, you should e ale to: State the nature of quadratic equation; Bangladesh Open University Eplain the relationship etween roots and coefficient of quadratic equation; Eplain the formation of quadratic equation; and Solve the quadratic equation. Nature of Quadratic Equation Generally an equation contains the square of unknown variale is called a quadratic equation. The general method of solving a quadratic equation of the form a + + c 0 is given. Since a + + c 0; y transposition a + c. Hence dividing oth sides y a, the co-efficient of we have, Generally an equation contains the square of unknown variale is called a quadratic equation. + a c a Adding to oth sides + a a + a c a a + or, or, a a 4ac + a 4a 4a c a or, + a ± a 4ac or, or, a ± ± 4ac a 4ac a Thus the required roots of a + + c 0 are + 4ac a and 4ac a Business Mathematics Page-109

22 School of Business Relationship etween Roots and the Co-efficient of Quadratic Equation A quadratic equation has eactly two roots. A quadratic equation has thus eactly two roots. The relationship etween the roots and the co-efficient of the quadratic equations as as follows: Let the quadratic equation is a + + c 0, then if α and β denote the roots of this quadratic equation, we have + α 4ac a β 4ac a Therefore y addition, α + β + 4ac a + 4ac a + 4ac a 4ac a a And y multiplication, αβ + 4ac a 4ac a ( ) 4ac 4a + 4ac 4ac 4a 4a c a Thus, we have shown that Sum of the roots (α+β) a Coefficient of Coefficient of Product of the roots (αβ) c a Coefficient of Formation of Quadratic Equation Constant term The formation of the quadratic equation whose roots are given can e eplained as follows: Unit-5 Page-110

23 Bangladesh Open University Let the general form of quadratic equation is a + + c 0 and α and β denote the two given roots, where a, and c are constants whose values we have to find out. The sum of the roots α + β, a So, a (α+β) And product of the roots, αβ a c so, c a αβ The aove relations imply that a + + c 0 or, a +[ a(α+β)] + aαβ 0 or, a aα aβ + aαβ 0 or, (α+β) + αβ 0 [Dividing oth sides y a ] So, the required quadratic equation will e (sum of the roots) + (product of the roots) 0 The following section of this lesson contains some model applications of quadratic equation. Eample 1: Solve We know that ± 4ac a Here a 1, 4, c 13 Sustituting the given values we have ( 4) ± ( 4) 4. ( 1)( 13).1 4 ± 16 5 Business Mathematics Page-111

24 School of Business 4 ± 36 4 ± 6i [ Since, i 1] Therefore, + 3i or, 3i Eample : Solve We have or, (3 ) ( 3 + )( 3 ) 5 or, ( 3) ( ) 5 or, or, We know that ± 4ac a Here a 9; 4; c 5 Sustituting the given values we have ( 4) ± ( 4) 4( 9)( 5) 4 ± ± ± So, or, Unit-5 Page-11

25 Bangladesh Open University Eample 3: If α and β are the roots of the equation p + q 0, find the values of (i) α β (ii) α + β (iii) α 3 β 3. Since α and β are the roots of the equation p + q 0 So, sum of the roots, α + β Coefficient of Coefficient of P P 1 Product of the roots, αβ Constant term Coefficient of q q 1 (i) (α β) α + β αβ α + β + αβ 4αβ (α+β) 4αβ p 4q so, α β p 4q (ii) α + β (α + β) αβ p q (iii) α 3 β 3 (α + β) (α αβ + β ) (α + β) [(α β) + αβ] p (p 4q + q) p(p 3q). Eample 4: The roots of are α and β; from a quadratic equation whose roots are α 4 + β 4 and α + β 4 Since α and β are the roots of the equation Coefficient of 1 Sum of the roots, α + β 1 Coefficient of 1 Constant term 1 And product of the roots, αβ 1 Coefficient of 1 Now, the sum of the roots of the required equation, α + β + α + β 4 (α + β ) + [(α + β ) α β ] Business Mathematics Page-113

26 School of Business [(α+β) αβ] + [(α+β) αβ] (αβ) (1.1) + [(1).1] (1) And the product of the roots of the required equation. (α 4 + β ) (α + β 4 ) α 6 + α 4 β 4 + α β + β 6 (α 6 +β 6 ) (αβ) 4 + (αβ) (α +β ) (α 4 α β + β 4 ) + (αβ) 4 + (αβ) [(α+β) αβ] [(α +β ) 3α β ] + (αβ) + (αβ) [(α+β) αβ] [{(α+β) αβ} 3(αβ) ] + (αβ) 4 + (αβ) [(1) (1)] [{(1) (1)} 3(1) ] + (1) + (1) ( 1) (1 3) Hence the required quadratic equation is, (sum of the roots) + (product of the roots) 0 or, ( ) so, Unit-5 Page-114

27 Bangladesh Open University Questions for Review: These questions are designed to help you assess how far you have understood and can apply the learning you have accomplished y answering (in written form) the following questions: 1. What is a quadratic equation? Give an eample. State the characteristics of a quadratic equation. 3. If α and β are the roots of p + q 0, form a quadratic equation whose roots are (αβ α β). 4. If α and β are the roots of , find the values of (1) α + β (ii) α/β (iii) β/α (iv) α 3 + β 3 5. If the equation a + + c 0 and + c + a 0 have a common root, then prove that a + + c 0 and a c. 6. Solve: (i) (ii) y Multiple Choice Questions ( the appropriate answers) 1. The roots of a quadratic equation are 5 and. The equation is: (a) () (c) The quadratic equation whose one root is 3 + 3, is (a) () (c) If α and β are the roots of the question 8 + P 0 and α +β 40 then P is equal to: (a) 8 () 1 (c) 1 4. If α and β are the roots of the equation the value of the α 3 + β 3 is: (a) 1 () 5 (c) 5. If α and β are the roots of the equation , then the equation whose roots are α/β and β/α is: (a) () (c) If one root of the equation is 1/3, then the other root is: (a) 3 () 1/3 (c) 3 Business Mathematics Page-115

28 School of Business Lesson-6: Application of Equation in Business Prolems After studying this lesson, you should e ale to: Apply the principles of equations to solve the usiness prolems Introduction The application of equations in the solution of usiness prolems is of great use. The application of equations in the solution of usiness prolems is of great use. It is usual to take the following steps in the solution of usiness prolems. (i) Read the prolem very carefully and try to understand the relation etween the quantities stated therein. (ii) Denote the unknown quantity y the letter, y, z, etc. (iii) Translate the statements of the prolem step y step into mathematical statements; to the etent it is possile. (iv) Solve the equation for the unknown (v) check whether the otained solution satisfies the condition given in the prolem. The following section of this lesson contains some model applications of equations to solve usiness prolems. Eample-1: A man says to his son, Seven years ago I was seven times as old as you were, and three years hence, I shall e three times as old as you will e. Find their present ages. Let the present age of the son e years. His age seven years ago 7 The father s age 7 years ago 7 ( 7) Father s present age 7 ( 7) + 7. Son s age 3 years hence + 3 Father s age 3 years hence 7 ( 7) ( 7) + 10 Using the given information we can write 7 ( 7) ( + 3) or, or, or, 4 48 so, 1 So, son s present age 1 years. Father s present age 7 (1 7) years. Unit-5 Page-116

29 Bangladesh Open University Eample-: A man s annual income has increased for Tk.7,500 ut there is no change in the income ta payale for him since the rate of income ta has een reduced from 10% to 7%. Find his present income. Let previous annual income Present income His income ta of previous year 10% of 0.10 His income ta for present year 70% of ( ) According to the question we can write or, or, or, ( ) so, 17,500 So, the present income ( ) (17, ) Tk.5,000. Eample-3: Tk.0,000 is invested in two shares. The first yield is Tk.1% p.a. and the second yield is 14% p.a. If the total yield at the end of one year is 13% p.a.; how much was invested at each rate? Let in Tk. e invested at 1%. Then (Tk.0000 ) is invested at 14% Interest at 1% 1% of 0.1 Interest at 14% 14% of (0,000 ) Total Interest on 0,000 at 13% 13% of 0,000,600 According to the given information we can write or, or, (00 0.0) 10,000 Hence Tk.10,000 is invested at 1% and (0,000 10,000) Tk.10,000 is invested at 14%. Eample-4: Demand and supply equations are p + q 11 and p + q 7 respectively. Find the equilirium price and quantity. (Where p stands for price and for quantity). Business Mathematics Page-117

30 School of Business The demand function is : p + q (1) And supply function is : p + q 7 So, p 7 q... () Putting the value of p in equation (1) we have, (7 q) + q 11 or, (49 8q + 4q ) + q 11 or, 98 56q + 8q + q 11 or, 9q 56q or, 9q 56q or, 9q 7q 9q or, 9q (q 3) 9 (q 3) 0 or, (q 3) (9q 9) 0 either, q 3 0 or, 9q 9 9 so, q 3 or, q 9 When q 3; the price is, p 7 q When q, the price is, p Therefore the equilirium price and quantity are: 5 9 (p, q) (1, 3) or, 9 9 Unit-5 Page-118

31 Questions for Review: Bangladesh Open University These questions are designed to help you assess how far you have understood and apply the learning you have accomplished y answering (in written form) the following questions: 1. Mr. Saji invested Tk.16,000 in two types of deentures of Tk.100 each. 13% deentures were purchased at Tk.150 each and 10% deentures were purchased at Tk.10 each. If he got interest of Tk.970 after 1 year, find the sum invested in each type of deenture.. The ages (in years) of Ramesh and Rahim are in the ratio 5 : 7. If Ramesh were 9 years older and Rahim were 9 years younger, the age of Ramesh would have een were twice the age of Rahim. Find their ages. 3. If the total manufacturing cost y of making units of a toy is: y (i) What is the variale cost per unit? (ii) What is the fied cost? (iii) What is the average cost of manufacturing 5,000 units? (iv) What is the marginal cost of producing 3,000 units? (v) What will e the effect on average cost per unit if volume changes? Multiple choice questions ( the appropriate answers) 1. The total cost of 6 ooks and 4 pencils is Tk.34 and that of 5 ooks and 5 pencils is Tk.30. The costs of each ook and each pencil respectively are: a) Tk.5 and Tk.1 ) Tk.1 and Tk.5 c) Tk.6 and Tk.1. If 3 chairs and tues cost Tk.100 and 5 chairs and 3 tues cost Tk.1900, then the cost of chairs and tues is: a) Tk.1000 ) Tk.900 c) Tk The monthly income of A & B are in the ratio 4 : 3, Each of them saves Tk.600. If the ratio of the ependitures is 3 :, then the monthly income of A is: a) Tk.1800 ) Tk.400 c) Tk If the laws of demand and supply are respectively given y the q equation 4q + 9p 48 and p 9 +, then the value of the equilirium price and quantity respectively are: a) 8 / 3 and 6 ) 3 / 8 and 5 c) 8 and 6 Business Mathematics Page-119

Polynomial Degree and Finite Differences

Polynomial Degree and Finite Differences CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson, you Learn the terminology associated with polynomials Use the finite differences method to determine the degree of a polynomial

More information

OBJECTIVE 4 EXPONENTIAL FORM SHAPE OF 5/19/2016. An exponential function is a function of the form. where b > 0 and b 1. Exponential & Log Functions

OBJECTIVE 4 EXPONENTIAL FORM SHAPE OF 5/19/2016. An exponential function is a function of the form. where b > 0 and b 1. Exponential & Log Functions OBJECTIVE 4 Eponential & Log Functions EXPONENTIAL FORM An eponential function is a function of the form where > 0 and. f ( ) SHAPE OF > increasing 0 < < decreasing PROPERTIES OF THE BASIC EXPONENTIAL

More information

Algebra I Notes Unit Nine: Exponential Expressions

Algebra I Notes Unit Nine: Exponential Expressions Algera I Notes Unit Nine: Eponential Epressions Syllaus Ojectives: 7. The student will determine the value of eponential epressions using a variety of methods. 7. The student will simplify algeraic epressions

More information

Module 9: Further Numbers and Equations. Numbers and Indices. The aim of this lesson is to enable you to: work with rational and irrational numbers

Module 9: Further Numbers and Equations. Numbers and Indices. The aim of this lesson is to enable you to: work with rational and irrational numbers Module 9: Further Numers and Equations Lesson Aims The aim of this lesson is to enale you to: wor with rational and irrational numers wor with surds to rationalise the denominator when calculating interest,

More information

Solving Systems of Linear Equations Symbolically

Solving Systems of Linear Equations Symbolically " Solving Systems of Linear Equations Symolically Every day of the year, thousands of airline flights crisscross the United States to connect large and small cities. Each flight follows a plan filed with

More information

Number Plane Graphs and Coordinate Geometry

Number Plane Graphs and Coordinate Geometry Numer Plane Graphs and Coordinate Geometr Now this is m kind of paraola! Chapter Contents :0 The paraola PS, PS, PS Investigation: The graphs of paraolas :0 Paraolas of the form = a + + c PS Fun Spot:

More information

Completing the Square

Completing the Square 3.5 Completing the Square Essential Question How can you complete the square for a quadratic epression? Using Algera Tiles to Complete the Square Work with a partner. Use algera tiles to complete the square

More information

Egyptian Fractions: Part II

Egyptian Fractions: Part II Egyptian Fractions: Part II Prepared y: Eli Jaffe Octoer 15, 2017 1 Will It End 1. Consider the following algorithm for eating some initial amount of cake. Step 1: Eat 1 of the remaining amount of cake.

More information

CONTENTS. iii v viii FOREWORD PREFACE STUDENTS EVALUATION IN MATHEMATICS AT SECONDARY STAGE

CONTENTS. iii v viii FOREWORD PREFACE STUDENTS EVALUATION IN MATHEMATICS AT SECONDARY STAGE CONTENTS FOREWORD PREFACE STUDENTS EVALUATION IN MATHEMATICS AT SECONDARY STAGE iii v viii CHAPTER Real Numbers CHAPTER Polynomials 8 CHAPTER 3 Pair of Linear Equations in Two Variables 6 CHAPTER 4 Quadratic

More information

Mathematics Background

Mathematics Background UNIT OVERVIEW GOALS AND STANDARDS MATHEMATICS BACKGROUND UNIT INTRODUCTION Patterns of Change and Relationships The introduction to this Unit points out to students that throughout their study of Connected

More information

Algebra I Notes Concept 00b: Review Properties of Integer Exponents

Algebra I Notes Concept 00b: Review Properties of Integer Exponents Algera I Notes Concept 00: Review Properties of Integer Eponents In Algera I, a review of properties of integer eponents may e required. Students egin their eploration of power under the Common Core in

More information

Chapter 3. Q1. Show that x = 2, if = 1 is a solution of the system of simultaneous linear equations.

Chapter 3. Q1. Show that x = 2, if = 1 is a solution of the system of simultaneous linear equations. Chapter 3 Q1. Show that x = 2, if = 1 is a solution of the system of simultaneous linear equations. Q2. Show that x = 2, Y = 1 is not a solution of the system of simultaneous linear equations. Q3. Show

More information

f 0 ab a b: base f

f 0 ab a b: base f Precalculus Notes: Unit Eponential and Logarithmic Functions Sllaus Ojective: 9. The student will sketch the graph of a eponential, logistic, or logarithmic function. 9. The student will evaluate eponential

More information

Graphs and polynomials

Graphs and polynomials 5_6_56_MQVMM - _t Page Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 Graphs and polnomials VCEcoverage Areas of stud Units & Functions and graphs Algera In this chapter A The inomial

More information

1Number ONLINE PAGE PROOFS. systems: real and complex. 1.1 Kick off with CAS

1Number ONLINE PAGE PROOFS. systems: real and complex. 1.1 Kick off with CAS 1Numer systems: real and complex 1.1 Kick off with CAS 1. Review of set notation 1.3 Properties of surds 1. The set of complex numers 1.5 Multiplication and division of complex numers 1.6 Representing

More information

QUADRATIC EQUATIONS EXPECTED BACKGROUND KNOWLEDGE

QUADRATIC EQUATIONS EXPECTED BACKGROUND KNOWLEDGE 6 QUADRATIC EQUATIONS In this lesson, you will study aout quadratic equations. You will learn to identify quadratic equations from a collection of given equations and write them in standard form. You will

More information

Linear Equations in Two Variables

Linear Equations in Two Variables Linear Equations in Two Variables LINEAR EQUATIONS IN TWO VARIABLES An equation of the form ax + by + c = 0, where a, b, c are real numbers (a 0, b 0), is called a linear equation in two variables x and

More information

Section 2.1: Reduce Rational Expressions

Section 2.1: Reduce Rational Expressions CHAPTER Section.: Reduce Rational Expressions Section.: Reduce Rational Expressions Ojective: Reduce rational expressions y dividing out common factors. A rational expression is a quotient of polynomials.

More information

ab is shifted horizontally by h units. ab is shifted vertically by k units.

ab is shifted horizontally by h units. ab is shifted vertically by k units. Algera II Notes Unit Eight: Eponential and Logarithmic Functions Sllaus Ojective: 8. The student will graph logarithmic and eponential functions including ase e. Eponential Function: a, 0, Graph of an

More information

Graphs and polynomials

Graphs and polynomials 1 1A The inomial theorem 1B Polnomials 1C Division of polnomials 1D Linear graphs 1E Quadratic graphs 1F Cuic graphs 1G Quartic graphs Graphs and polnomials AreAS of STud Graphs of polnomial functions

More information

Essential Maths 1. Macquarie University MAFC_Essential_Maths Page 1 of These notes were prepared by Anne Cooper and Catriona March.

Essential Maths 1. Macquarie University MAFC_Essential_Maths Page 1 of These notes were prepared by Anne Cooper and Catriona March. Essential Maths 1 The information in this document is the minimum assumed knowledge for students undertaking the Macquarie University Masters of Applied Finance, Graduate Diploma of Applied Finance, and

More information

UNIT 5 QUADRATIC FUNCTIONS Lesson 2: Creating and Solving Quadratic Equations in One Variable Instruction

UNIT 5 QUADRATIC FUNCTIONS Lesson 2: Creating and Solving Quadratic Equations in One Variable Instruction Lesson : Creating and Solving Quadratic Equations in One Variale Prerequisite Skills This lesson requires the use of the following skills: understanding real numers and complex numers understanding rational

More information

Higher. Polynomials and Quadratics. Polynomials and Quadratics 1

Higher. Polynomials and Quadratics. Polynomials and Quadratics 1 Higher Mathematics Polnomials and Quadratics Contents Polnomials and Quadratics 1 1 Quadratics EF 1 The Discriminant EF Completing the Square EF Sketching Paraolas EF 7 5 Determining the Equation of a

More information

Finding Complex Solutions of Quadratic Equations

Finding Complex Solutions of Quadratic Equations y - y - - - x x Locker LESSON.3 Finding Complex Solutions of Quadratic Equations Texas Math Standards The student is expected to: A..F Solve quadratic and square root equations. Mathematical Processes

More information

MATH 225: Foundations of Higher Matheamatics. Dr. Morton. 3.4: Proof by Cases

MATH 225: Foundations of Higher Matheamatics. Dr. Morton. 3.4: Proof by Cases MATH 225: Foundations of Higher Matheamatics Dr. Morton 3.4: Proof y Cases Chapter 3 handout page 12 prolem 21: Prove that for all real values of y, the following inequality holds: 7 2y + 2 2y 5 7. You

More information

5Higher-degree ONLINE PAGE PROOFS. polynomials

5Higher-degree ONLINE PAGE PROOFS. polynomials 5Higher-degree polnomials 5. Kick off with CAS 5.2 Quartic polnomials 5.3 Families of polnomials 5.4 Numerical approimations to roots of polnomial equations 5.5 Review 5. Kick off with CAS Quartic transformations

More information

ERASMUS UNIVERSITY ROTTERDAM Information concerning the Entrance examination Mathematics level 2 for International Business Administration (IBA)

ERASMUS UNIVERSITY ROTTERDAM Information concerning the Entrance examination Mathematics level 2 for International Business Administration (IBA) ERASMUS UNIVERSITY ROTTERDAM Information concerning the Entrance examination Mathematics level 2 for International Business Administration (IBA) General information Availale time: 2.5 hours (150 minutes).

More information

Student s Name : Class Roll No. ADDRESS: R-1, Opp. Raiway Track, New Corner Glass Building, Zone-2, M.P. NAGAR, Bhopal

Student s Name : Class Roll No. ADDRESS: R-1, Opp. Raiway Track, New Corner Glass Building, Zone-2, M.P. NAGAR, Bhopal fo/u fopkjr Hkh# tu] ugha vkjehks dke] foifr ns[k NksM+s rqjar e/;e eu dj ';kea iq#"k flag ladyi dj] lgrs foifr vusd] ^cuk^ u NksM+s /;s; dks] j?kqcj jk[ks VsdAA jfpr% ekuo /kez izksrk ln~q# Jh jknksm+nklth

More information

Goal: To graph points in the Cartesian plane, identify functions by graphs and equations, use function notation

Goal: To graph points in the Cartesian plane, identify functions by graphs and equations, use function notation Section -1 Functions Goal: To graph points in the Cartesian plane, identify functions by graphs and equations, use function notation Definition: A rule that produces eactly one output for one input is

More information

Conceptual Explanations: Logarithms

Conceptual Explanations: Logarithms Conceptual Eplanations: Logarithms Suppose you are a iologist investigating a population that doules every year. So if you start with 1 specimen, the population can e epressed as an eponential function:

More information

The Final Exam is comprehensive but identifying the topics covered by it should be simple.

The Final Exam is comprehensive but identifying the topics covered by it should be simple. Math 10 Final Eam Study Guide The Final Eam is comprehensive ut identifying the topics covered y it should e simple. Use the previous eams as your primary reviewing tool! This document is to help provide

More information

( ) ( ) or ( ) ( ) Review Exercise 1. 3 a 80 Use. 1 a. bc = b c 8 = 2 = 4. b 8. Use = 16 = First find 8 = 1+ = 21 8 = =

( ) ( ) or ( ) ( ) Review Exercise 1. 3 a 80 Use. 1 a. bc = b c 8 = 2 = 4. b 8. Use = 16 = First find 8 = 1+ = 21 8 = = Review Eercise a Use m m a a, so a a a Use c c 6 5 ( a ) 5 a First find Use a 5 m n m n m a m ( a ) or ( a) 5 5 65 m n m a n m a m a a n m or m n (Use a a a ) cancelling y 6 ecause n n ( 5) ( 5)( 5) (

More information

Section 0.4 Inverse functions and logarithms

Section 0.4 Inverse functions and logarithms Section 0.4 Inverse functions and logarithms (5/3/07) Overview: Some applications require not onl a function that converts a numer into a numer, ut also its inverse, which converts ack into. In this section

More information

NCERT. not to be republished SIMPLE EQUATIONS UNIT 4. (A) Main Concepts and Results

NCERT. not to be republished SIMPLE EQUATIONS UNIT 4. (A) Main Concepts and Results SIMPLE EQUATIONS (A) Main Concepts and Results The word variable means something that can vary i.e., change and constant means that does not vary. The value of a variable is not fixed. Variables are denoted

More information

ERASMUS UNIVERSITY ROTTERDAM

ERASMUS UNIVERSITY ROTTERDAM Information concerning Colloquium doctum Mathematics level 2 for International Business Administration (IBA) and International Bachelor Economics & Business Economics (IBEB) General information ERASMUS

More information

a b a b ab b b b Math 154B Elementary Algebra Spring 2012

a b a b ab b b b Math 154B Elementary Algebra Spring 2012 Math 154B Elementar Algera Spring 01 Stud Guide for Eam 4 Eam 4 is scheduled for Thursda, Ma rd. You ma use a " 5" note card (oth sides) and a scientific calculator. You are epected to know (or have written

More information

Solve each system by graphing. Check your solution. y =-3x x + y = 5 y =-7

Solve each system by graphing. Check your solution. y =-3x x + y = 5 y =-7 Practice Solving Sstems b Graphing Solve each sstem b graphing. Check our solution. 1. =- + 3 = - (1, ). = 1 - (, 1) =-3 + 5 3. = 3 + + = 1 (, 3). =-5 = - 7. = 3-5 3 - = 0 (1, 5) 5. -3 + = 5 =-7 (, 7).

More information

We are working with degree two or

We are working with degree two or page 4 4 We are working with degree two or quadratic epressions (a + b + c) and equations (a + b + c = 0). We see techniques such as multiplying and factoring epressions and solving equations using factoring

More information

Higher Tier - Algebra revision

Higher Tier - Algebra revision Higher Tier - Algebra revision Contents: Indices Epanding single brackets Epanding double brackets Substitution Solving equations Solving equations from angle probs Finding nth term of a sequence Simultaneous

More information

At first numbers were used only for counting, and 1, 2, 3,... were all that was needed. These are called positive integers.

At first numbers were used only for counting, and 1, 2, 3,... were all that was needed. These are called positive integers. 1 Numers One thread in the history of mathematics has een the extension of what is meant y a numer. This has led to the invention of new symols and techniques of calculation. When you have completed this

More information

Factor and Solve Polynomial Equations. In Chapter 4, you learned how to factor the following types of quadratic expressions.

Factor and Solve Polynomial Equations. In Chapter 4, you learned how to factor the following types of quadratic expressions. TEKS 5.4 2A.1.A, 2A.2.A; P..A, P..B Factor and Solve Polynomial Equations Before You factored and solved quadratic equations. Now You will factor and solve other polynomial equations. Why? So you can find

More information

When two letters name a vector, the first indicates the and the second indicates the of the vector.

When two letters name a vector, the first indicates the and the second indicates the of the vector. 8-8 Chapter 8 Applications of Trigonometry 8.3 Vectors, Operations, and the Dot Product Basic Terminology Algeraic Interpretation of Vectors Operations with Vectors Dot Product and the Angle etween Vectors

More information

STRAND F: ALGEBRA. UNIT F4 Solving Quadratic Equations: Text * * Contents. Section. F4.1 Factorisation. F4.2 Using the Formula

STRAND F: ALGEBRA. UNIT F4 Solving Quadratic Equations: Text * * Contents. Section. F4.1 Factorisation. F4.2 Using the Formula UNIT F4 Solving Quadratic Equations: Tet STRAND F: ALGEBRA Unit F4 Solving Quadratic Equations Tet Contents * * Section F4. Factorisation F4. Using the Formula F4. Completing the Square UNIT F4 Solving

More information

PAIR OF LINEAR EQUATIONS

PAIR OF LINEAR EQUATIONS 38 MATHEMATICS PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 3 3.1 Introduction You must have come across situations like the one given below : Akhila went to a fair in her village. She wanted to enjoy rides

More information

Objectives To solve equations by completing the square To rewrite functions by completing the square

Objectives To solve equations by completing the square To rewrite functions by completing the square 4-6 Completing the Square Content Standard Reviews A.REI.4. Solve quadratic equations y... completing the square... Ojectives To solve equations y completing the square To rewrite functions y completing

More information

STRAND: ALGEBRA Unit 2 Solving Quadratic Equations

STRAND: ALGEBRA Unit 2 Solving Quadratic Equations CMM Suject Support Strand: ALGEBRA Unit Solving Quadratic Equations: Tet STRAND: ALGEBRA Unit Solving Quadratic Equations TEXT Contents Section. Factorisation. Using the Formula. Completing the Square

More information

Math 216 Second Midterm 28 March, 2013

Math 216 Second Midterm 28 March, 2013 Math 26 Second Midterm 28 March, 23 This sample exam is provided to serve as one component of your studying for this exam in this course. Please note that it is not guaranteed to cover the material that

More information

Egyptian Fractions: Part II

Egyptian Fractions: Part II Egyptian Fractions: Part II Prepared y: Eli Jaffe Octoer 15, 2017 1 Will It End 1. Consider the following algorithm for eating some initial amount of cake. Step 1: Eat 1 of the remaining amount of cake.

More information

Lesson 6: Solving Exponential Equations Day 2 Unit 4 Exponential Functions

Lesson 6: Solving Exponential Equations Day 2 Unit 4 Exponential Functions (A) Lesson Contet BIG PICTURE of this UNIT: CONTEXT of this LESSON: How can I analyze growth or decay patterns in data sets & contetual problems? How can I algebraically & graphically summarize growth

More information

ALGEBRAIC EXPRESSIONS AND POLYNOMIALS

ALGEBRAIC EXPRESSIONS AND POLYNOMIALS MODULE - ic Epressions and Polynomials ALGEBRAIC EXPRESSIONS AND POLYNOMIALS So far, you had been using arithmetical numbers, which included natural numbers, whole numbers, fractional numbers, etc. and

More information

Chapter 7. 1 a The length is a function of time, so we are looking for the value of the function when t = 2:

Chapter 7. 1 a The length is a function of time, so we are looking for the value of the function when t = 2: Practice questions Solution Paper type a The length is a function of time, so we are looking for the value of the function when t = : L( ) = 0 + cos ( ) = 0 + cos ( ) = 0 + = cm We are looking for the

More information

4. Multiply: (5x 2)(3x + 4) 6 y z 16 = 11. Simplify. All exponents must be positive. 12. Solve: 7z 3z = 8z Solve: -4(3 + 2m) m = -3m

4. Multiply: (5x 2)(3x + 4) 6 y z 16 = 11. Simplify. All exponents must be positive. 12. Solve: 7z 3z = 8z Solve: -4(3 + 2m) m = -3m - - REVIEW PROBLEMS FOR INTERMEDIATE ALGEBRA ASSESSMENT TEST-Rev. Simplify and combine like terms: -8 ( 8) 9 ( ). Multiply: ( )( ). Multiply: (a )(a a 9). Multiply: ( )( ). y. 7 y 0 ( ) y 7. ( y ) 7a (

More information

Module 2, Section 2 Solving Equations

Module 2, Section 2 Solving Equations Principles of Mathematics Section, Introduction 03 Introduction Module, Section Solving Equations In this section, you will learn to solve quadratic equations graphically, by factoring, and by applying

More information

MTH 65 WS 3 ( ) Radical Expressions

MTH 65 WS 3 ( ) Radical Expressions MTH 65 WS 3 (9.1-9.4) Radical Expressions Name: The next thing we need to develop is some new ways of talking aout the expression 3 2 = 9 or, more generally, 2 = a. We understand that 9 is 3 squared and

More information

SOLUTION OF QUADRATIC EQUATIONS

SOLUTION OF QUADRATIC EQUATIONS SOLUTION OF QUADRATIC EQUATIONS * The standard form of a quadratic equation is a + b + c = 0 * The solutions to an equation are called the roots of the equation. * There are methods used to solve these

More information

ACCUPLACER MATH 0310

ACCUPLACER MATH 0310 The University of Teas at El Paso Tutoring and Learning Center ACCUPLACER MATH 00 http://www.academics.utep.edu/tlc MATH 00 Page Linear Equations Linear Equations Eercises 5 Linear Equations Answer to

More information

3.5 Solving Quadratic Equations by the

3.5 Solving Quadratic Equations by the www.ck1.org Chapter 3. Quadratic Equations and Quadratic Functions 3.5 Solving Quadratic Equations y the Quadratic Formula Learning ojectives Solve quadratic equations using the quadratic formula. Identify

More information

Math 080 Final Exam Review

Math 080 Final Exam Review Math 080 Final Ea Review Note: If you have difficulty with any of these proles, get help, then go ack to the appropriate sections in the tet and work ore proles! For proles through 0, solve for, writing

More information

Lesson 6.1 Recursive Routines

Lesson 6.1 Recursive Routines Lesson 6. Recursive Routines. Give the starting value and constant multiplier for each sequence. Then find the fifth term. a. 4800, 200, 300,... b. 2, 44., 92.6,... c. 00, 90, 8,... d. 00, 0, 02.0,...

More information

Worked solutions. 1 Algebra and functions 1: Manipulating algebraic expressions. Prior knowledge 1 page 2. Exercise 1.2A page 8. Exercise 1.

Worked solutions. 1 Algebra and functions 1: Manipulating algebraic expressions. Prior knowledge 1 page 2. Exercise 1.2A page 8. Exercise 1. WORKED SOLUTIONS Worked solutions Algera and functions : Manipulating algeraic epressions Prior knowledge page a (a + ) a + c( d) c cd c e ( f g + eh) e f e g + e h d i( + j) + 7(k j) i + ij + 7k j e l(l

More information

THOMAS WHITHAM SIXTH FORM

THOMAS WHITHAM SIXTH FORM THOMAS WHITHAM SIXTH FORM Algebra Foundation & Higher Tier Units & thomaswhitham.pbworks.com Algebra () Collection of like terms. Simplif each of the following epressions a) a a a b) m m m c) d) d d 6d

More information

Pair of Linear Equations in Two Variables

Pair of Linear Equations in Two Variables Exercise. Question. Aftab tells his daughter, Seven ears ago, I was seven times as old as ou were then. Also, three ears from now, I shall be three times as old as ou will be. (Isn t this interesting?)

More information

3. Pair of linear equations in two variables (Key Points) An equation of the form ax + by + c = 0, where a, b, c are real nos (a 0, b 0) is called a linear equation in two variables x and y. Ex : (i) x

More information

4.6 (Part A) Exponential and Logarithmic Equations

4.6 (Part A) Exponential and Logarithmic Equations 4.6 (Part A) Eponential and Logarithmic Equations In this section you will learn to: solve eponential equations using like ases solve eponential equations using logarithms solve logarithmic equations using

More information

6th Grade. Translating to Equations. Slide 1 / 65 Slide 2 / 65. Slide 4 / 65. Slide 3 / 65. Slide 5 / 65. Slide 6 / 65

6th Grade. Translating to Equations. Slide 1 / 65 Slide 2 / 65. Slide 4 / 65. Slide 3 / 65. Slide 5 / 65. Slide 6 / 65 Slide 1 / 65 Slide 2 / 65 6th Grade Dependent & Independent Variables 15-11-25 www.njctl.org Slide 3 / 65 Slide 4 / 65 Table of Contents Translating to Equations Dependent and Independent Variables Equations

More information

8-4. Negative Exponents. What Is the Value of a Power with a Negative Exponent? Lesson. Negative Exponent Property

8-4. Negative Exponents. What Is the Value of a Power with a Negative Exponent? Lesson. Negative Exponent Property Lesson 8-4 Negative Exponents BIG IDEA The numbers x n and x n are reciprocals. What Is the Value of a Power with a Negative Exponent? You have used base 10 with a negative exponent to represent small

More information

GRADE 7 MATH LEARNING GUIDE. Lesson 26: Solving Linear Equations and Inequalities in One Variable Using

GRADE 7 MATH LEARNING GUIDE. Lesson 26: Solving Linear Equations and Inequalities in One Variable Using GRADE 7 MATH LEARNING GUIDE Lesson 26: Solving Linear Equations and Inequalities in One Variable Using Guess and Check Time: 1 hour Prerequisite Concepts: Evaluation of algebraic expressions given values

More information

University of Colorado at Colorado Springs Math 090 Fundamentals of College Algebra

University of Colorado at Colorado Springs Math 090 Fundamentals of College Algebra University of Colorado at Colorado Springs Math 090 Fundamentals of College Algebra Table of Contents Chapter The Algebra of Polynomials Chapter Factoring 7 Chapter 3 Fractions Chapter 4 Eponents and Radicals

More information

The following document was developed by Learning Materials Production, OTEN, DET.

The following document was developed by Learning Materials Production, OTEN, DET. NOTE CAREFULLY The following document was developed y Learning Materials Production, OTEN, DET. This material does not contain any 3 rd party copyright items. Consequently, you may use this material in

More information

Paula s Peaches (Learning Task)

Paula s Peaches (Learning Task) Paula s Peaches (Learning Task) Name Date Mathematical Goals Factorization Solving quadratic equations Essential Questions How do we use quadratic functions to represent contetual situations? How do we

More information

Math 117, Spring 2003, Math for Business and Economics Final Examination

Math 117, Spring 2003, Math for Business and Economics Final Examination Math 117, Spring 2003, Math for Business and Economics Final Eamination Instructions: Try all of the problems and show all of your work. Answers given with little or no indication of how they were obtained

More information

A population of 50 flies is expected to double every week, leading to a function of the x

A population of 50 flies is expected to double every week, leading to a function of the x 4 Setion 4.3 Logarithmi Funtions population of 50 flies is epeted to doule every week, leading to a funtion of the form f ( ) 50(), where represents the numer of weeks that have passed. When will this

More information

we make slices perpendicular to the x-axis. If the slices are thin enough, they resemble x cylinders or discs. The formula for the x

we make slices perpendicular to the x-axis. If the slices are thin enough, they resemble x cylinders or discs. The formula for the x Math Learning Centre Solids of Revolution When we rotate a curve around a defined ais, the -D shape created is called a solid of revolution. In the same wa that we can find the area under a curve calculating

More information

Essential Question: How can you solve equations involving variable exponents? Explore 1 Solving Exponential Equations Graphically

Essential Question: How can you solve equations involving variable exponents? Explore 1 Solving Exponential Equations Graphically 6 7 6 y 7 8 0 y 7 8 0 Locker LESSON 1 1 Using Graphs and Properties to Solve Equations with Eponents Common Core Math Standards The student is epected to: A-CED1 Create equations and inequalities in one

More information

Name. Unit 1 Worksheets Math 150 College Algebra and Trig

Name. Unit 1 Worksheets Math 150 College Algebra and Trig Name Unit 1 Worksheets Math 10 College Algebra and Trig Revised: Fall 009 Worksheet 1: Integral Eponents Simplify each epression. Write all answers in eponential form. 1. (8 ). ( y). (a b ). y 6. (7 8

More information

Inequalities and Accuracy

Inequalities and Accuracy Lesson Fourteen Inequalities and Accuracy Aims The aim of this lesson is to enable you to: solve problems involving inequalities without resorting to graphical methods Contet There are many areas of work

More information

Intersections and the graphing calculator.

Intersections and the graphing calculator. Chapter 4 Analytical Algebra 2 1 Intersections and the graphing calculator. Find all intersections of the graphs of the following equations: x y = 3 2y = x 2 y = 4x 2 Find all intersections of the graphs

More information

This is Solving Linear Systems, chapter 4 from the book Beginning Algebra (index.html) (v. 1.0).

This is Solving Linear Systems, chapter 4 from the book Beginning Algebra (index.html) (v. 1.0). This is Solving Linear Systems, chapter 4 from the book Beginning Algebra (index.html) (v. 1.0). This book is licensed under a Creative Commons by-nc-sa 3.0 (http://creativecommons.org/licenses/by-nc-sa/

More information

Algebra Review C H A P T E R. To solve an algebraic equation with one variable, find the value of the unknown variable.

Algebra Review C H A P T E R. To solve an algebraic equation with one variable, find the value of the unknown variable. C H A P T E R 6 Algebra Review This chapter reviews key skills and concepts of algebra that you need to know for the SAT. Throughout the chapter are sample questions in the style of SAT questions. Each

More information

( ) ( ) x. The exponential function f(x) with base b is denoted by x

( ) ( ) x. The exponential function f(x) with base b is denoted by x Page of 7 Eponential and Logarithmic Functions Eponential Functions and Their Graphs: Section Objectives: Students will know how to recognize, graph, and evaluate eponential functions. The eponential function

More information

EQUATIONS REDUCIBLE TO QUADRATICS EQUATIONS

EQUATIONS REDUCIBLE TO QUADRATICS EQUATIONS CHAPTER EQUATIONS REDUCIBLE TO QUADRATICS EQUATIONS Numbers The numbers,, are called natural numbers or positive integers. a is called a fraction where a and b are any two positive integers. b The number

More information

ACCUPLACER MATH 0311 OR MATH 0120

ACCUPLACER MATH 0311 OR MATH 0120 The University of Teas at El Paso Tutoring and Learning Center ACCUPLACER MATH 0 OR MATH 00 http://www.academics.utep.edu/tlc MATH 0 OR MATH 00 Page Factoring Factoring Eercises 8 Factoring Answer to Eercises

More information

Recall that when you multiply or divide both sides of an inequality by a negative number, you must

Recall that when you multiply or divide both sides of an inequality by a negative number, you must Unit 3, Lesson 5.3 Creating Rational Inequalities Recall that a rational equation is an equation that includes the ratio of two rational epressions, in which a variable appears in the denominator of at

More information

Exponent Laws Unit 4 Exponential Functions - Lesson 18

Exponent Laws Unit 4 Exponential Functions - Lesson 18 (A) Lesson Contet BIG PICTURE of this UNIT: CONTEXT of this LESSON: What are & how & why do we use eponential and logarithic functions? proficiency with algeraic anipulations/calculations pertinent to

More information

Inequalities. Inequalities. Curriculum Ready.

Inequalities. Inequalities. Curriculum Ready. Curriculum Ready www.mathletics.com Copyright 009 3P Learning. All rights reserved. First edition printed 009 in Australia. A catalogue record for this ook is availale from 3P Learning Ltd. ISBN 978--986-60-4

More information

Math 120. x x 4 x. . In this problem, we are combining fractions. To do this, we must have

Math 120. x x 4 x. . In this problem, we are combining fractions. To do this, we must have Math 10 Final Eam Review 1. 4 5 6 5 4 4 4 7 5 Worked out solutions. In this problem, we are subtracting one polynomial from another. When adding or subtracting polynomials, we combine like terms. Remember

More information

Zeroing the baseball indicator and the chirality of triples

Zeroing the baseball indicator and the chirality of triples 1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 7 (2004), Article 04.1.7 Zeroing the aseall indicator and the chirality of triples Christopher S. Simons and Marcus Wright Department of Mathematics

More information

1.3 Exponential Functions

1.3 Exponential Functions Section. Eponential Functions. Eponential Functions You will be to model eponential growth and decay with functions of the form y = k a and recognize eponential growth and decay in algebraic, numerical,

More information

Math 0210 Common Final Review Questions (2 5 i)(2 5 i )

Math 0210 Common Final Review Questions (2 5 i)(2 5 i ) Math 0 Common Final Review Questions In problems 1 6, perform the indicated operations and simplif if necessar. 1. ( 8)(4) ( )(9) 4 7 4 6( ). 18 6 8. ( i) ( 1 4 i ) 4. (8 i ). ( 9 i)( 7 i) 6. ( i)( i )

More information

SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 2 NOVEMBER 2016

SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 2 NOVEMBER 2016 SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY NOVEMBER 06 Mark Scheme: Each part of Question is worth 4 marks which are awarded solely for the correct answer.

More information

Algebra. Robert Taggart

Algebra. Robert Taggart Algebra Robert Taggart Table of Contents To the Student.............................................. v Unit 1: Algebra Basics Lesson 1: Negative and Positive Numbers....................... Lesson 2: Operations

More information

Patterns, Functions, & Relations Long-Term Memory Review Review 1

Patterns, Functions, & Relations Long-Term Memory Review Review 1 Long-Term Memor Review Review 1 1. What are the net three terms in the pattern? 9, 5, 1, 3,. Fill in the Blank: A function is a relation that has eactl one for ever. 3. True or False: If the statement

More information

4.2 SOLVING A LINEAR INEQUALITY

4.2 SOLVING A LINEAR INEQUALITY Algebra - II UNIT 4 INEQUALITIES Structure 4.0 Introduction 4.1 Objectives 4. Solving a Linear Inequality 4.3 Inequalities and Absolute Value 4.4 Linear Inequalities in two Variables 4.5 Procedure to Graph

More information

Summer Review Packet for Students Entering AP Calculus BC. Complex Fractions

Summer Review Packet for Students Entering AP Calculus BC. Complex Fractions Summer Review Packet for Students Entering AP Calculus BC Comple Fractions When simplifying comple fractions, multiply by a fraction equal to 1 which has a numerator and denominator composed of the common

More information

b) 2( ) a) Write an equation that models this situation. Let S = yearly salary and n = number of yrs since 1990.

b) 2( ) a) Write an equation that models this situation. Let S = yearly salary and n = number of yrs since 1990. Precalculus Worksheet 3.1 and 3.2 1. When we first start writing eponential equations, we write them as y= a. However, that isn t the only way an eponential equation can e written. Graph the following

More information

Non-Linear Regression Samuel L. Baker

Non-Linear Regression Samuel L. Baker NON-LINEAR REGRESSION 1 Non-Linear Regression 2006-2008 Samuel L. Baker The linear least squares method that you have een using fits a straight line or a flat plane to a unch of data points. Sometimes

More information

CHAPTER - 2 EQUATIONS. Copyright -The Institute of Chartered Accountants of India

CHAPTER - 2 EQUATIONS. Copyright -The Institute of Chartered Accountants of India CHAPTER - EQUATIONS EQUATIONS LEARNING OBJECTIVES After studying this chapter, you will be able to: u Understand the concept of equations and its various degrees linear, simultaneous, quadratic and cubic

More information

The Mean Version One way to write the One True Regression Line is: Equation 1 - The One True Line

The Mean Version One way to write the One True Regression Line is: Equation 1 - The One True Line Chapter 27: Inferences for Regression And so, there is one more thing which might vary one more thing aout which we might want to make some inference: the slope of the least squares regression line. The

More information

Question 1. (8 points) The following diagram shows the graphs of eight equations.

Question 1. (8 points) The following diagram shows the graphs of eight equations. MAC 2233/-6 Business Calculus, Spring 2 Final Eam Name: Date: 5/3/2 Time: :am-2:nn Section: Show ALL steps. One hundred points equal % Question. (8 points) The following diagram shows the graphs of eight

More information

8-1 Exploring Exponential Models

8-1 Exploring Exponential Models 8- Eploring Eponential Models Eponential Function A function with the general form, where is a real number, a 0, b > 0 and b. Eample: y = 4() Growth Factor When b >, b is the growth factor Eample: y =

More information