Systems of Linear Equations in Two Variables

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1 Name Period Date: Topic: 3-5 Systems of Linear Equations in Two Variables Essential Question: Economists often talk about supply and demand curves. Given two linear equations for supply and demand where price is the domain, how could you determine the optimum amount of a product to produce? Standard: A-REI.6 Objective: Linear System: Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. To solve systems of linear equations in two variables. A set of linear equations in the same two variables is called a system of linear equations, or a linear system. Any ordered pair of numbers that is a solution of each equation in the system is called a solution of the system, or a simultaneous solution of the equations. Sometimes, a system of equations is called simultaneous equations. When you graph a linear system with two equations in the same coordinate plane, there are three possibilities: 1. The two lines will intersect. 2. The two lines will be parallel. 3. The two equations will be equivalent, and there will be only one line. In this case, the two equations are said to coincide. Summary

2 The three types of graphs are illustrated below: Intersecting lines: Parallel lines: Coinciding lines: 2

3 Algebraically: These geometric relationships mean: 1. When the graphs intersect in only one point, the system has one solution. 2. When the graphs are parallel, the system has no solution. 3. When the graphs coincide. The solution set for the system contains infinitely many ordered pairs. Equivalent systems: Transformations that Produce Equivalent Systems: We can extend the definition of equivalent equations to systems. Equivalent systems are systems that have the same solution set. To solve a system of equations, you transform the system into an equivalent system whose solution is easily seen. Transformations that produce equivalent systems are listed below. 1. Replacing an equation by an equivalent equation. (Recall that multiplying each side of an equation by the same nonzero number produces an equivalent equation.) 2. Substituting for one variable in any equation an equivalent expression for that variable obtained from another equation in the system. (This expression may be the value of the variable, if known.) 3. Replacing any equation by the sum of that equation and another equation in the system. (Recall that to add two equations, you add their left sides, add their right sides, and equate the results.) We will study two methods that use these transformations to solve linear systems of equations. 3

4 Substitution Method: Sometimes one of the equations in a linear system can easily be solved for one of the variables. For example, the equation can be solved easily for y: Now, you can substitute this expression ( equation and solve the resulting equation for x. ) for y into the other This is called the Substitution Method, and the next example illustrates its use. Example 1: Solve this system: Step 1: Since the coefficient of x in the first equation is 1, it is convenient to express x in terms of y in the first equation. Step 2: Now, substitute, the expression, Solve for y., for x in the second equation. ( ) Step 3: Substitute 1 for y in either of the original equations to find x. ( ) Solution: The solution of the system is (3, 1). 4

5 Exercise 1: Use the substitution method to solve these systems: 5

6 Linear Combinations: When you add two equations, the result is a linear combination of the equations. To solve a system of equations, a useful goal is to obtain a linear combination that has fewer variables than the given system. To solve a linear system of two equations using the method of linear combinations, we usually multiply one or both equations by a constant. Then we add the resulting, equivalent equations. We choose the constants in order to eliminate one of the variables in the resulting equation. The next example illustrates this process. Example 2: Find a linear combination that eliminates x from the following system: ( ) 6

7 Exercise 2: Find a linear combination to eliminate one of the variables in these systems: 7

8 Example 3: Use the linear combinations method to solve this system: In Example 2 we used the linear combinations method to find an equivalent equation in which the variable x had been eliminated from this system: Solve for y, Substitute this value for y into the first equation and solve for x. Check: Check that ( 1, 3) is a solution of this system: ( ) ( ) ( ) ( ) 8

9 Exercise 3: Use the linear combinations method to solve this system: 9

10 No Solution: When the graphs of the equations of a system are parallel, the system has no solution. If you try to solve such a system algebraically, both variables are eliminated and the result is a false statement. This is illustrated in the next example. Example 4: Solve this system: Substitute the expression for y from the second equation into the first equation: ( ) Solve for x: FALSE this system has no solution. The graphs of the two equations are parallel. 10

11 Exercise 4: Solve the following systems. If there is no solution, say so. 11

12 Infinite Solutions: When the graphs of the equations of a system coincide, the system has infinitely many solutions. If you try to solve such a system algebraically, you obtain an identity and all solutions of one equation are solutions of the other. This is illustrated in the next example Example 5: Solve this system: Multiply the first equation by 2, and then add the equations: IDENTITY Since is an identity, the solution set of the system is the same as the solution set of either of the equations, namely, the infinite set {( ) }. Three of the solutions are, ( ) ( ) ( ) 12

13 Exercise 5: Solve the following systems. If the system has an infinite solution set, specify it and give three solutions. If there is no solution, say so. ( ) ( ) 13

14 Solution Types: 1. If a system of equations has at least one solution, the equations in the system are called consistent. 2. If the system has no solution, the equations are called inconsistent, and their graphs are parallel lines. 3. If a consistent system has infinitely many solutions, then the equations are called dependent, and their graphs are coinciding lines. Example 6: Write three linear systems: one consistent, one inconsistent, and one dependent. Consistent The ratios of the two x-coefficients and the two y-coefficients must be different. Note that Inconsistent The ratios of the two x-coefficients and the two y-coefficients must be equal, but the ratio of the two constant coefficients must be different. Note that 14

15 Dependent The ratios of the two x-coefficients, the two y-coefficients, and the two constant coefficients must be equal. Note that Exercise 6: Write three linear systems: one consistent, one inconsistent, and one dependent. Class work: p 128 Oral Exercises: 1-17 Homework: p 129 Written Exercises: 1-31 odd p 130 Mixed Review: 1-12 p 130 Written Exercises: