Unit 6. System of equations.

Transcription

1 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 compatible indeterminado Inconsistent system: Sistema incompatible Graphical method: Método gráfico Substitution method: Método de sustitución Matching method: Método de igualación Elimination method: Método de reducción Straight line: Recta 6.1. Linear Equations with two variables. Solutions. Definition We call a linear equation with two variables (from now on, a linear equation) to an equation that can be written in the form ax + by = c where a, b and c are real numbers. The form ax + by = c is called standard form of a linear equation. We call a solution of a linear equation to a two values that when they are substituted into their corresponding variables, make a true statement. We usually write the solutions to linear equations in ordered pairs. De termine which ordered pair of values is a solution of the linear equation 3x - 2y = 4: (0, -2), (2, 1), (1, 1) The pair (0, -2) (- 2) = 4, so (0, -2) is a solution of the equation. The pair (2, 1) = 4, so (2, 1) is a solution of the equation. The pair (1, 1) = 1 4, so (1, 1) is not a solution of the equation. In fact, we could prove with a lot of pairs of numbers and a lot of them would be a solution of the equation. In conclusion, every linear equation has infinite solutions. As the solutions are given in pairs, they can be represented in a coordinate axes, so when all the solutions are drawn together we get a straight line : Let's see all the solutions of the previous example.

2 All the points in this straight line satisfy the equation, so all of them are solutions of it. In conclusion, we can have infinite solutions with an equation, but what if we want an unique solution? We must, therefore, add another linear equation Systems of equations. Definition We call a system of equations to a set of equations with several varaibles. Solving a system of equations is finding the pair of values that satisfies every linear equation. Let's see what happen when we join a new linear equation to the previous one:

3 So the solution of the system will be the intersection of both straight lines. This method of resolution of systems of equations is called graphical method. It has a main problem: In the most of the cases is impossible to calculate the exact solution but we only can draw it Types of systems of equations Every system of equations can be of one of the following types: The system has an only solution as in the previous example. These systems are called independent systems. The system has no solution. In this case, they will be called inconsistent systems. In all these cases, the straight lines will be parallel. Te system has infinite solutions. These systems are called dependent systems. In this case, they both the liner equations are the same Methods to solve systems of equations are: There are three methods to solve systems of equations apart from the graphical method. They

4 Substitution method: To solve a system using this method: - we choose a variable of one of the equations and clear it up. - Then we substitute the resulting expression in the other equation, so we get a linear equation with just one variable. - We solve this equation, so we get the value of one variable. - Finally, we substitute this value in the cleared up expression, so we get the value of the other variable. We clear the 'y' up from the second equation: (1) We substitute this expression in the other equation: We substitute this value in the expression (1): So, the solution of this system is Matching method: - We choose the same variable in both the equations and clear them up. - Then, we make the both expressions equals and solve the resultant linear equation, so we get the value of one variable. - Finally, we substitute this value in one f the equations, so we can get the value of the other variable. We clear the 'x' up from the both of the equations: (1) (2) Now, we make (1) = (2), so we get a linear equation: We substitute the value of 'y' in whatever equation we want, for example, the first one:

5 So, the solution to this system of equations is Elimination method: This method consists in searching equivalent equations to the first one, so we can eliminate one of the variables by adding both of the manipulated equations. You can apply this method twice to get the value of both of the variables. Manipulating an equation means, to multiply it by a number, so the same variable of both the equations have opposite coefficients. We are going to eliminate the 'x' variable first, so we indicate the number that we are multiplying by: Now, we add both the equations: Then, we repeat the same process with the other variable: So, the solution of this system is Exercises: 1. Are the ordered pairs (6, 4), (6, 12) and (0, 5) solutions of the linear equation 4x - 3y = 12? 2. Solve 2x - y = 6 and x + y = graphically. What is the common solution to both equations? 3. Look at the equations of the following systems and determine if they are independent, inconsistent or dependent systems: a. c. b. d. 4. Use the graphical method to solve the following systems of equations: a. b. 5. Use the substitution method to solve the following systems of equations: a. c. b. 6. Use the elimination method to solve the following systems of equations: a. c. b.

6 7. Solve the following systems of equations: a. b. 8. In a sweet shop, I spend $3.20 on three cans of soft drink and four bars of chocolate. The next day, I buy a can of soft drink and four bars of chocolate for $2. How much does each item cost? 9. Twice the larger or two numbers is three more than five times the smaller, and the sum of four times the larger and three times the smaller is 71. What are the numbers? students and 29 staff are going on a school trip. They travel by large and small coaches. The large coaches seat 55 and the small ones seat 39. If there are no spare seats and five coaches are to make the journey, how many of each coach are used? 11. In an isosceles trapezium, the smallest base is 6 cm and the length of the largest one is the same as two equal sides together. If the perimeter of the trapezium is 38 cm, find its dimensions.