UNIT 5: SIMULTANEOUS EQUATIONS (SYSTEMS OF EQUATIONS)

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UNIT 5: SIMULTANEOUS EQUATIONS (SYSTEMS OF EQUATIONS) Simultaneous linear equations (Systems of linear equations): A system of linear equations is a collection of linear equations with the same

1 UNIT 5: SIMULTANEOUS EQUATIONS (SYSTEMS OF EQUATIONS) Simultaneous linear equations (Systems of linear equations): A system of linear equations is a collection of linear equations with the same variables. ax by=c a' x b' y=c'} This is a system of two linear equations with two variables. Solving a system of equations means finding the values of the variables that make all the equations true at the same time. 2x 3y=5 x 5y=9 } The solution to this system is x=4 and y=1. Number of solutions of a two-variable system of equations: When you are solving systems, you are, graphically, finding intersection of lines. (Remember that the graph of a linear equation, ax by=c, is a straight line, and its points are the solution of the equation. For two-variables system, there are three possible types of solutions: CASE 1: Independent Systems: 2x y=5 x 4y=6} The two lines cross at exactly one point. This point is the only solution of the system. These systems are called independent systems. P(2,1) x+4y=6 2x+y=5 1

CASE 2: Inconsistent Systems: 2x y=4 4x 2y=2} Since parallel line never cross, the system has no solution. These systems are called inconsistent systems.

2 CASE 2: Inconsistent Systems: 2x y=4 4x 2y=2} Since parallel line never cross, the system has no solution. These systems are called inconsistent systems. 2x+y=4 4x+2y=2 CASE 3: Dependent systems: 2x 3y=3 4x 6y=6} The two lines are the same line. Any point of the line is solution to the system. These systems are called dependent system. 4x+6y=6 2x+3y=3 2

3 Your Turn 1. Use the graphical method to solve the following systems of equations: a) x y=2 x y= 4} b) y=3 2x 2y=6} 3

4 c) y=2 x 2y=3} d) 2x y= 1 4x 2y= 2} e) y=1 2x 3y=4} 4

5 f) x y=5 3x y=3} Solving simultaneous equations: 1. The Substitution Method: Find the value of one unknown in either of the given equation and substitute this value in the other equation. x 2y=3 2x 3y=6} 2. The Equating Method: Find the values of one unknown in both equations and both values equal. 2x 3y=3 3x 5y=14} 5

6 3. The Elimination Method: It consists in adding equations together to eliminate variables. Sometimes you have to add equations by a number before you add them. The goal is to end up with one equation that has just one variable. Then, you can use substitution method to find out the value of the other variable. 3x 4y=17 2x 5y=16} Your Turn 1. Use the substitution method to solve the following systems of equations: a) x 5y=7 3x 5y=11} b) 5x y=8 3x y=11} 6

7 c) 3x 10y=6 x 2y=1 } 2. Use the equating method to solve the following systems of equations: a) x 2y=3 2x 3y=6} b) 2x y=4 4x 3y= 7} c) 5x 2y=4 3x 4y=8} 3. Use the elimination method to solve the following systems of equations: a) 2x 3y=11 3x 2y= 3} 7

8 b) 2x 5y=6 8x y=3 } c) 5x 2y=4 3x 4y=8} 4. Solve the following systems of equations: a) x 3 y 2 =4 =2} x 2 y 4 b) x 2 5 3y 1 10 = 3 } 10 2x 3 y =

9 {x 15 3 y 1 y } = c) 7 x 2 1 y 12 =3 5. The admission fee at a small fair is $1,50 for children and $4 for adults. On a certain day, people enter the fair and $5.050 is collected. How many children and how many adults went to the fair? 6. The sum of the digits of a two-digit number is 7. When the digits are reserved, the number is increased by 27. Find the number. 7. A landscaping company placed two orders with a nursery. The first order was 13 bushes and 4 trees, and totalled $487. The second order was for 6 bushes and 2 trees, and totalled $232. The bill does not list the per-item price. What is the cost of one bush and of one tree? 9

10 8. A boy has a number of coins in both hands. If he passes two coins from the right hand to the left hand, there will be the same amount of coins in both hands. If he passes three coins from the left hand to the right hand, he will have double the amount of coins in his right hand than in his left hand. At the beginning, how many coins did the boy have in each hand? 9. Find the ages of two people knowing that ten years ago, the age of the first person was four times the age of the second person, and in twenty years, the age of the first person will be the double of the age of the second person. 10. A group of students paid 144 for 3 tickets in the shade and 6 tickets in the sun to a bull fight. Another group paid 66 for 2 tickets in the shade and 2 tickets in the sun. Calculate the price of each ticket. 11. There are 420 students in a high school. Forty-two percent of ESO and 52% of Bachillerato are girls, which means 196 in total. Calculate the amount of students in ESO and in Bachillerato. 12. A total of 925 tickets are sold for $5925. If adult tickets cost $7,50 and children's tickets $3,00, How many tickets of each kind were sold? 10

11 13. Mr. Brian has $ to invest. He invests part at 6%, the rest at 7% and he earns $1 280 interest. How much did he invest at each rate? 14. There are 13 animals in the barn. Some are chickens and some are pigs. There are 40 legs in all. How many of each animals are there? 15. Jerry was 1/3 as young as his grandfather 15 years ago. If the sum of their ages is 110, how old is Jerry's grandfather? 16. In three more years, Miguel's grandfather will be six time as old as Miguel was last year. When Miguel's present age is added to his grandfather's present age, the total is 68. How old is each one now? 17. A man is 6 times as old as his son. In 4 years he will be four times as his son will be. How old are they now? 11

12 Non-linear systems of equations: In this kind of systems at least one of the equation is not linear, we usually solve these systems using the substitution method. x y=1 x 2 y =5} 2 Your Turn 1. Solve the following non-linear systems of equations: a) x y=10 x y=7} b) x 2 y 2 =58 x 2 y =40} 2 12

13 c) 1 x 1 y = 1 } 20 x 2y=3 d) 2x y=2 xy x =0} 2 x 1 y=2x 1 e) { 2x y=11 } 2. The perimeter of a rectangle is 26 cm and its surface is 40 cm 2. Calculate its dimensions. 13

14 3. The sum of two real numbers is 10 and their product is 24. Calculate both numbers. 4. Find two consecutive odd numbers whose product is 49 times the square of the smaller. 5. Calculate the surface of a square knowing that the diagonal length exceeds 5 cm its side length. 6. Calculate the perimeter and surface of a rectangle if the diagonal exceeds in 8 cm the base and the diagonal exceeds in 16 cm the height. 14

15 Systems of inequalities: A system of inequalities is a collection of inequalities with the same group of variables. Solving a system of inequalities means finding the values of the variables that make all the inequalities true at the same time. x 3 7 x 2 0} Your Turn 1. Solve the following systems of inequalities: a) 2x 4 0 } 2x 7 x

16 2x 5 3 b) { 3 x 1 x 3 0} c) 7x x 2x 3 3x 1 } d) x 8 3x 2 x 2 2 x 2 } e) x 4 5 2x 3x 2 2x 1} 16

17 Keywords: equation = ecuación linear equation = ecuación lineal simultaneous equations (system of equations) = sistema de ecuaciones independent system = sistema compatible determinado (única solución) dependent system = sistema compatible indeterminado (infinitas soluciones) inconsistent system = sistema incompatible (sin solución) graphical method = método gráfico numerical methods = métodos numéricos substitution method = método de substitución equating method = emétodo de igualación elimination method = método de reducción non-linear systems of equations = sistemas de ecuaciones no lineales systems of inequalities (inequations)= sistemas de inecuaciones to check a solution = comprobar una solución to verify a solution = verificar, comprobar una solución to isolate a variable = despejar una incógnita o variable to plug in a number for a letter = substituir una letra por un número to solve simultaneous equations = resolver un sistema de ecuaciones to identify = identificar to graph = reprentar gráficamente to set up = establecer interval = intervalo set solution = conjunto solución a<b a is less than b = a es menor que b a>b a is greater than b = a es mayor que b a b a is less or equal than b = a es menor o igual que b a b a is greater or equal than b = a es mayor o igual que b 17

Unit 6. System of equations.

Unit 6. System of equations. Keywords: Linear equation: Ecuación de primer grado System of equations: Sistema de ecuaciones Independent system: Sistema compatible determinado Dependent system: Sistema