UNIT 7: EQUATIONS. The degree of an equation is the greatest degree of all the monomials involved.

Transcription

1 UNIT 7: EQUATIONS. Equations: In Algebra, letters are used to represent numbers. In a statement like x 3=8, you must find a value of x that makes this true. This is called solving an equation and the value of x is the solution of the equation. Before learning some techniques to solve equations, you can use trial and improvement to find the solutions of some equations. For example: a. 5x=20 (A number multiplied by 5 is 20. It cannot be 3, because. 5 3=15 Then x should be 4). b. x 4=14 (a number added to 4 is 14. Then x should be 10) An equation is formed by two sides (the first side on the left of the equals sign, and the second side on the right), and each sides has many terms. The letters used are the unknowns. The degree of an equation is the greatest degree of all the monomials involved. Two equations are equivalent if they have the same solution(s). 1. Use trial and improvement to solve the following equations: a) 5x 7=47 b) 3x 6=0 c) d) e) x 4 =3 20 x =4 20 x 1 =4 1

2 2. Group the equivalent equations: a) 5x=15 b) 2x 3 =4 c) 2x=6 d) 2x 1=5 e) x=3 f) 2x=12 Simple equations: A simple equation uses just one unknown letter. In this section we will consider only linear equations (not letters with powers). To solve linear equations, treat both sides of the equation in the same way (what you do to one side of the equation you must do to the other). The transformations are: 1. Add or subtract the same expressions to both sides of the equation. This means that the expressions on one side of the equation move to the other side changing their sign. 2. Multiply or divide both sides of the equation by the same number different from zero. This means that all that is multiplying on one side moves to the other side dividing, and vice versa. The aim is to leave the letter (the subject) alone on one side of the equation: Solve the following equations: x 5=6 5=x 2 6 x=6 4x=8 15=3x 12=4x 2

3 x 5 =1 x 6 = 1 8 x 4 = x 3=7 3x 2=4 3x 1=1 6x 4=7 x 3 5x 3 2x=2 x 7 x 3 x= x 3 Equations with parenthesis: SPECIAL CASES 0x=b, when b 0 has no solutions. 0x=0, has infinite solutions. Sometimes you will have to remove brackets following these steps: PROCEDURE 2 2x 3 = x 1. Remove brackets and simplify both sides. 4x 6=7 3 6x 2. Group x-terms on one side and numbers on the other side. 3. Isolate the unknown and calculate the solution. 4x 6x= x=10 x= x=1 4. Check the solution = = = 2 Solve the following equations involving brackets: 3

4 2 x 6 =5x 6 x 3 =12x 3 2x 4 =7x x 1 =2 x 8 3 2x 3 =2 3x 8 2 x 1 4=3 x x 1 =3 x x =3 x x 1 5x=3 2 2x 4 Equations with denominators: Sometimes you will find fractions. In that case, the steps are as follows: PROCEDURE 1. Remove brackets, and simplify every fraction you get, if possible. 2. Remove the denominators by multiplying both sides by its lowest common denominator. 3. Group x-terms on one side and numbers on the other side. 4. Isolate the unknown and calculate the solution. 5. Check the solution. 2 x 4 1 =2 x 5 2x 4 2=2 x 5 x 2 2=2 x 5 5x = 4 2x 10 5x 20=4 2x 5x 2x=20 4 3x= = =2 x= 24 =8 x= =

5 1. Solve the following equations with fractions: x = x x 3x = 4x 5 x 5 x x 6 = x x 1= x 3x 15 x= 3x x 20 5x 2 = 10 x x 10 x 3 x 9 4x 27 = x 9 x 2 x 2 = x x x 12 2x =x 8 x 2 x 3 8 2x 2 16 = x 2 2 5

6 Quadratic Equations: Quadratic equations can be expressed (once reduced) as ax 2 bx c=0, where a 0. They can be complete or incomplete depending on the missing terms. INCOMPLETE EQUATIONS: CASE 1 ax 2 c=0 a 0 You have to solve x 2 as if it were a linear equation. And then calculate the square root of the result (if possible): Examples: Solve the following equations: a) 2x 2 32=0 2x 2 32=0 2x 2 =32 x 2 = 32 2 x2 =16 x=± 16=±4 So there are two solutions, x 1 =4 and x 2 = 4. b) 2x 2 8=0 x 2 = 8 2 = 4 x=± 4. This equation has no solutions, since you cannot calculate the square root of a negative number. This equation is incompatible. CASE 2 ax 2 bx=0 a 0 This type of equations can be solved by factorising them, extracting common factors: Example: x 2 4x=0 x x 4 =0 { x=0 x 4=0 x= 4} So there are two solutions: x 1 =0 and x 2 = 4. These equations are always compatible, and x=0 is always one of the solutions. 6

7 Solve the following incomplete equations: x 2 4=0 4x 2 1=0 x 2 7x=0 x 2 3x=0 x 2 100=0 x 2 100=0 9x 2 4=0 x 2 5x=0 3x 2 9x=0 2x 2 10x=0 2x 2 50=0 2x 2 50x=0 7

8 COMPLETE EQUATIONS: ax 2 bx c=0 a 0, b 0, c 0 You can use this formula to solve this type of equation: Example: Solve the equation 3x 2 7x 2=0 1. Identify the coefficients a, b and c: x= b± b2 4ac 2a In this equation: a=3 b= 7 c=2 2. Substitute them in the expression above: x= 7 ± In this case, we get two solutions. = 7± = 6 7± 25 6 ={7 5 6 = =1 3 Solve the following quadratic equations using the formula: a) x 2 5x 6=0 b) x 2 3x 2=0 8

9 c) 2x 2 3x 1=0 d) 15x 2 x 6=0 e) x 2 6x 8=0 f) 2x 2 3x 5=0 g) 3x 2 5x 2=0 h) x 2 3x 4=0 i) x 2 4x 3=0 9

10 Problems with equations: 1. A number and the number before it add up to 77. What numbers are they? 2. When we add 13 units to the double of a number, the result is 99. What is the number? 3. A pride of lions is made up of 13 lions, and there are 3 females more than males. How many lions and how many lionesses are there in the pride? 4. There are 31 people in a bar. How many men and women are there if we know that there are 5 more men than women? 5. In a cow farm, the number of horns and legs add up to 30. How many cows are there in the farm? 10

11 6. An ice-cream costs 80 cents more than a cake. Kate and George had an ice-cream and two cakes for tea, and it all cost 4,40. How much does a cake cost? And, how much is an icecream? 7. Calculate the dimensions of a rectangular field, knowing that its length is 20 metres long more than its width and there is a fence which is 240 metres long. 8. All sides of an equilateral triangle are x-2 metres long. a) Find an expression for the perimeter of the triangle. b) If the perimeter is 12 metres long, calculate the value of x. 9. The base of a rectangle is x-6 metres long, and its height is 4 metres. Find the value of x when the area of the rectangle is 8m A blouse has x buttons. A shirt has x+2 buttons. a) Write an expression for the number of buttons on four blouses. b) Write an expression for the number of buttons on three shirts. c) Three shirts have the same number of buttons in total as four blouses. Write an equation and solve it to find the value of x. d) How many buttons are there on a shirt? 11

12 11. The sum of the square of a number and six times the number is 91. Find the two possible values of the number. 12. The length of a rectangular lawn is 5 metres longer than its width. If the area of the lawn is 176m 2, find the dimensions of the lawn. 13. Two positives numbers have a difference of 3 and their squares have a sum of 317. Let one of the numbers be x. Write down an equation in x based on this information and solve it to find the two numbers. 14. The legs of a right-angled triangle are (2x+1) and (2x+3)cm long. If the area of this triangle is 13cm 2, calculate the value of x. 15. Arthur has a bag containing 28 candies, some of them are mints and some others are lemon. If the number of mints is three times the number of lemon ones, how many candies of each kind can Arthur eat? 16. John kept 4/5 of the oranges that were in the box. Mary took half of the oranges that were left in the box, leaving only 5 oranges. How many oranges did the box contain originally? 12

13 17. Rachel and Sarah have 45 biscuits between them. Sarah tells Rachel: if you give me 5 biscuits I will have twice as many biscuits as you. How many biscuits does each of them have? 19. Two cyclists leave the same point, at the same time in the same direction. The first one pedals at a constant speed of 35 km/h and the second one at 25 km/h. How long will it take them to be 15 km apart? 20. Two cars leave the same point at the same time, but in opposite directions. The first one is driven at a constant speed of 85 km/h, and the second one at a constant speed of 90 km/h. a) How many kilometres apart will be in two hours? b) How long will it take them to be 300 km apart? 21. Find the dimensions of a rectangular lawn knowing that the perimeter is 180 m long, and one side is 20 m longer than wider. 22. In a lake an electrical tower has been installed. One seventh of its weight is under the ground at the bottom of the lake, while 2/9 of the remaining is immersed in the water. If the tower rises 14 m above the lake surface, what is its total height? 13

14 23. Find two numbers knowing that the difference between them is 8, and the difference between three times the lowest and the highest is A father is 25 years older than his son. Find their ages taking into account that in 15 years the father's age will be twice his son's age. 25. Two integers can be added together to make 9, and the difference between twice the first one and three times the second one is 48. What are these numbers? 26. Find two numbers which can be added together to make 37 and knowing that the difference between them is The difference between the two acute angles of a right-angled triangle is 15º. What is the measure of the angles? 14

15 28. Find the ages of a mother and her daughter taking into account that three years ago the former was three times older than the latter, and in 8 years the mother's age will be twice the daughter's age. 29. The perimeter of a rectangle is 90 cm long, but if its base doubles and its height triples, it will be 206 m long. Calculate the dimensions of the initial rectangle. Keywords: Equation= Ecuación solution= Solución sides= miembros (de la ecuación) terms= términos unknown= incógnita variable= variable degree= grado equivalent equation= ecuación equivalente simple equation= ecuación de primer grado to isolate= despejar, aislar (la incógnita) to check= comprobar, revisar quadratic equation= ecuación cuadrática o de segundo grado complete equation= ecuación completa incomplete equation= ecuación incompleta compatible equation= ecuación compatible (con solución) incompatible equation= ecuación incompatible (sin solución) 15