1. SYSTEMS OF LINEAR EQUATIONS 1 UNIT 7 SYSTEMS OF LINEAR EQUATIONS

2. 2 UNIT 7 KEYWORDS Equation linear equation simultaneous equations (system of equations) independent system = SCD dependent system = SCI inconsistent system = SI graphical method numerical methods substitution method equating method elimination method non-linear systems of equations systems of inequalities (inequations) to check a solution to verify a solution to isolate a variable to plug in a number for a letter = substituir una letra por un número to solve simultaneous equations to identify to graph/ to plot to set up = establecer Interval set solution = conjunto solución a<b a is less than b a>b a is greater than b a≤b a is less or equal than b a≥b a is greater or equal than b

3. SYSTEMS OF LINEAR EQUATIONS 3 UNIT 7 INDEX 1.Linear equations with two unknown 2.Systems of linear equations 1.Equivalent systems 2.Number of solutions 3.Solving systems 1.The substitution method 2.The equating method 3.The elimination method 4.Systems problems 1.Linear equations with two unknown 2.Systems of linear equations 1.Equivalent systems 2.Number of solutions 3.Solving systems 1.The substitution method 2.The equating method 3.The elimination method 4.Systems problems

4. SYSTEMS OF LINEAR EQUATIONS 4 UNIT 7 1. Linear equations with two unkonwn. Nos vamos a ocupar ahora de las ecuaciones de primer grado con dos incógnitas. Una ecuación de este tipo es, por ejemplo: 2x + y = 50 Observa que el par de valores x = 20, y = 10, hace cierta la igualdad: 2·20 + 10 = 50 Decimos entonces que ese par de valores es una solución de la ecuación. Sin embargo, la solución no es única. Observa que hay otros pares que también son soluciones de la misma ecuación: x = 5, y = 40 2·5 + 40 = 50 En realidad, la ecuación tiene infinitas soluciones. x = 10, y = 30 2·10 + 30 = 50 x = 15, y = 20 2·15 + 20 = 50

5. SYSTEMS OF LINEAR EQUATIONS 5 UNIT 7 ► An equation with two unknown and degree one is called linear equation. ► The solution of a linear equation is pair of values that make the expression true, (x 0, y 0 ). ► A two variables-equation has infinite solutions. Standard Form Every linear equation can be written as follow: ax + by = c where a, b and c are konwn values. 1. Linear equations with two variables.

6. SYSTEMS OF LINEAR EQUATIONS 6 UNIT 7 Independent term unkonwn A linear equation with two unknown x, y is written as follow: coefficient a x + b y = c Standard Form

7. SYSTEMS OF LINEAR EQUATIONS 7 UNIT 7 1.Check wether either of the pairs x=3, y=4 or x=2, y=1 is a solution to the equation 2x - y=3 : ACTIVITIES 2. Fill in the missing value in each of the following solutions to the equation 3x + y=7 a) x=1, y= b ) x=2, y= c) x=0, y=

8. SYSTEMS OF LINEAR EQUATIONS 8 UNIT 7 Graph 2x + y = 5 y = 5 – 2x Isolate y Value table x y = 5 – 2x 0 1 2 3 –1 y = 5 – 2·0 = 5 y = 5 – 2·1 = 3 y = 5 – 2·2 = 1 y = 5 – 2·3 = –1 y = 5 – 2·(–1) = 7 EXAMPLE With the box, we have found five solutions: (0, 5), (1, 3), (2, 1), (3, –1), y (–1, 7). Plot the points. Realize that are alignments in a line. GRAPH OF A LINEAR EQUATION To obtain diferents pair of values that are solution of a linear equation, 1.The unknown, y, must be isolate. 2.Plug in numbers for the other, x. 3.The values are recolected,in order, in a VALUE TALBLE

9. SYSTEMS OF LINEAR EQUATIONS 9 UNIT 7 ► Each linear equation has a straigh line associated in the plane. ► Each point of this line represent one of the infinitives solutions of the equation. Graph 3x – 2y = 4. Isolate, y, for the process value table: 3x – 2y = 4 3x – 4 = 2y Represent the pair of values in the plane. EXERCISE x y 0 1 2 1 3 2,5 –1 –3,5 –2 –5 The line that pass through the point is the graph of the equation. y =  = –2 3·0 – 4 2 y =  = –0,5 3·1 – 4 2 –2 –0,5 y = (3x- 4):2

10. SYSTEMS OF LINEAR EQUATIONS 10 UNIT 7 A solution is a pair of number that make both expressions true. Solve a system is find out the solutions of the system. coefficient unkonwn x,y a x + b y = c a' x + b' y = c' Two linear equations form a System of linear equations (simultaneous Equation) when we want to find a solution that is common to both. 2. Systems of linear equations

11. SYSTEMS OF LINEAR EQUATIONS 11 UNIT 7 Which of the following systems is a system of linear equation/simultaneous equation: x + y = 5 x – y = 3 y – x = 3 2x + y = 0 x + y = 1 y = 2x + 1 2x + 3y = 12 3x – y = 7 EXAMPLE non-linear systems of equations

12. SYSTEMS OF LINEAR EQUATIONS 12 UNIT 7 ►If the system equations are not in general form, first expressed in the general form and then ….. 3(2x + 1) – 4y = 4(1 – 2y) – 7 5x – 3y = 4x + 10 5x – 4x – 3y = 10 6x – 4y + 8y = 4 – 7 – 3 EXAMPLE 6x + 3 – 4y = 4 – 8y – 7 Transforms the equations until they are in standard form. Now they are in standard sorm. Now apply some method of resolution x – 3y = 10 6x + 4y = –6 STANDARD FORM

13. SYSTEMS OF LINEAR EQUATIONS 13 UNIT 7 Write down in the standard form the following system: 2(x – 1) = 3(y + 1) – 3 x – y = 0 4(2x – 7) – 5y = 0 3(3y – 4) – 4x = 0 3x – 1 = 4(y + 5) + 2 3(x – 2) = y + 7 EXAMPLE

14. SYSTEMS OF LINEAR EQUATIONS 14 UNIT 7 EXAMPLE 2 x - 3y = 3 3x + y = 10 2 · 3 – 3·1 = 3 3·3 + 1 = 10 Its solution is x=3, y=1 because => (3,1) is solution of the system Check that x=3, y=0 is not a solution 2 · 3 – 3·0 3 3·3 + 0 10 Is it (3, 1) or (3,0) a solution of the system?

15. SYSTEMS OF LINEAR EQUATIONS 15 UNIT 7 3. For the following equations, check wether either of the pair x=1, y=4 :or x=1, y=4 is a solution to the system. a) 3x + y = 7 2x - y = 3 ACTIVITIES b) 2x + y = 6 5x - y = 1 c) x + y = 5 x - y = 1 4. Complete the following systems to make the solution the pair of values x=3, y=2. a) 2x + y = x - y = b) 5x - 7y = 3x + 2y = c) 4 x - 5y = x - 2y =

16. SYSTEMS OF LINEAR EQUATIONS 16 UNIT 7 2.1. Equivalents systems Equivalents systems: are those which have the same solutions. Solution: x = 2, y = 1 2x + 3y = 7 x = 4y – 2 –2x + 3y = –1 x – 4y = – 2 They are equivalents Plot the systems and realize that both have the same solution, so they are equivalents To solve a system, there is to obtain another simplified that must be equivalent. EXERCISE

17. SYSTEMS OF LINEAR EQUATIONS 17 UNIT 7 SUM RULE: Si a los dos miembros de una ecuación de un sistema se le suma o resta un mismo número o una misma expresión algebraica, resulta otro sistema equivalente al dado. 3x = –6 + 4y x + 2y = 8 3x – 4y = –6 x + 2y = 8 Restamos 4y a los dos miembros de la primera ecuación

18. SYSTEMS OF LINEAR EQUATIONS 18 UNIT 7 PRODUCT RULE: Si se multiplican o dividen los dos miembros de una ecuación de un sistema por un mismo número distinto de cero, resulta otro sistema equivalente al dado. Multiplicamos los dos miembros de la segunda ecuación por 2. 3x = –6 + 4y x + 2y = 8 3x – 4y = –6 x + 2y = 8 3x – 4y = –6 2x + 4y = 16 ·2

19. SYSTEMS OF LINEAR EQUATIONS 19 UNIT 7 Si a una ecuación de un sistema se le suma o resta otra ecuación del mismo, resulta otro sistema equivalente al dado Sumamos a la segunda ecuación la primera. 3x – 4y = –6 (1) 2x + 4y = 16 (2) 3x – 4y = –6 5x = 10 3x – 4y = –6 x = 2 y = 3 x = 2 (2) + (1)

20. SYSTEMS OF LINEAR EQUATIONS 20 UNIT 7 2.2. Number of solutions 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. CASE 1: Some systems have an unique solution. They are called independent system. For two-variables system, there are three possible types of solutions: CASE 2: Some systems have infinitive solutions. They are called dependent System. CASE 3: Some systems have no solution. They are called inconsistent system. Sistema Compatible Indeterminado (SCI) Sistema Incompatible(SI)Sistema Compatible Determinado (SCD)

21. SYSTEMS OF LINEAR EQUATIONS 21 UNIT 7

22. SYSTEMS OF LINEAR EQUATIONS 22 UNIT 7 (1) x + y = 7  y = 7 – x (2) 3x – y = 9  y = 3x – 9 La solución del sistema es el punto común: x = 4, y = 3 El par de valores x = 4, y = 3 satisface ambas igual- dades, es decir, es solución de las dos ecuaciones: x = 4 4 + 3 = 7 y = 3 3·4 – 3 = 9 CASE 1 x + y = 7 3x – y = 9 Gráficamente, la solución del sistema es el punto de corte de las rectas que representan a las ecuaciones. Nombro las ecuaciones x 2 3 4 5 6 7.... y 5 4 3 2 1 0.... x 1 2 3 4 5 6.... y –6 –3 0 3 6 9.... (1) (2) SYSTEMS WITH A SOLUTION

23. SYSTEMS OF LINEAR EQUATIONS 23 UNIT 7 SYSTEMS WITH INFINITIVE SOLUTIONS También puede ocurrir que las dos ecuaciones sean portadoras de informaciones idénticas. x + y = 3 2x + 2y = 6 En este caso, cualquier par de valores que haga cierta la primera igualdad, también hace cierta la segunda. El sistema tiene infinitas soluciones. ► Systems with infinitive solutions are dependent (indeterminados). ► Graphically, the lines representing equations have all their points in common, ie, match. CASE 2

24. SYSTEMS OF LINEAR EQUATIONS 24 UNIT 7 Ya has visto que la solución de un sistema lineal es el punto de corte de dos rectas. Por tanto, un sistema lineal tendrá generalmente una solución única. Sin embargo, como verás a continuación, hay casos especiales. SYSTEMS WITHOUT SOLUTION Puede ocurrir que las dos ecuaciones del sistema sean portadoras de información contradictoria. x + y = 3 x + y = 6 En este caso es imposible encontrar un par de valores ( x, y ) que haga ciertas ambas igualdades a la vez. El sistema no tiene solución. ► Systems without solution are called inconsistent. ► Graphically, the lines representing the equations have no common point, ie, they are parallel. CASE 3

25. SYSTEMS OF LINEAR EQUATIONS 25 UNIT 7 6. Write down a linear system that is: a) inconsistent b) dependent. Plot them. 7. Plot the system and indicate which is independent, inconsistent and dependent. a) x + y = 3 x – y = 5 ACTIVITIES b) 2x – y = 1 2x – y = 5 c) x – 2y = 3 3x – 6y = 9 a) x + y = 5 x – y = 3 5. Solve graphically : c) y – x = 3 2x + y = 0 b) x + y = 1 y = 2x + 1 d) 2x + 3y = 12 3x – y = 7