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CHAPTER OUTLINE 4 Linear Equations and Inequalities in Two Variables 2 4.1Solving Systems by Graphing 4.2Solving Systems by the Substitution Method 4.3Solving Systems by the Elimination Method 4.4Applications of Systems of Two Equations 4.5Linear Inequalities in Two Variables

Objectives 4.1Solving Systems by Graphing 3 1Determine Whether an Ordered Pair Is a Solution of a System 2Solve a Linear System by Graphing 3Solve a Linear System by Graphing: Special Cases 4Determine the Number of Solutions of a System Without Graphing

4.1Solving Systems by Graphing 1Determine Whether an Ordered Pair Is a Solution of a System 4 A system of linear equations consists of two or more linear equations with the same variables.

4.1Solving Systems by Graphing 1Determine Whether an Ordered Pair Is a Solution of a System 5 Definition A solution of a system of two equations in two variables is an ordered pair that is a solution of each equation in the system.

EXAMPLE 1 6 Determine whether (2, 3) is a solution of each system of equations.

EXAMPLE 1 7 Determine whether (2, 3) is a solution of each system of equations. Solution a) Since (2, 3) is a solution of each equation, it is a solution of the system.

EXAMPLE 1 8 Determine whether (2, 3) is a solution of each system of equations. Solution b)

4.1Solving Systems by Graphing 2Solve a Linear System by Graphing 9 To solve a system of equations in two variables means to find the ordered pair (or pairs) that satisfies each equation in the system. If two lines intersect at one point, that point of intersection is a solution of each equation.

4.1Solving Systems by Graphing 2Solve a Linear System by Graphing 10 Definition When solving a system of equations by graphing, the point of intersection is the solution of the system. If a system has at least one solution, we say that the system is consistent. The equations are independent if the system has one solution.

EXAMPLE 2 11

EXAMPLE 2 12 Solution

4.1Solving Systems by Graphing 3Solve a Linear System by Graphing: Special Cases 13 Do two lines always intersect? No! Then if we are trying to solve a system of two linear equations by graphing and the graphs do not intersect, what does this tell us about the solution to the system?

EXAMPLE 3 14 Solution The lines are parallel; they will never intersect. Therefore, there is no solution to the system. We write the solution set as .

4.1Solving Systems by Graphing 3Solve a Linear System by Graphing: Special Cases 15 Definition When solving a system of equations by graphing, if the lines are parallel, then the system has no solution. We write this as . Furthermore, a system that has no solution is inconsistent, and the equations are independent.

EXAMPLE 4 16 What if the graphs of the equations in a system are the same line? Solution The system has an infinite number of solutions of the form

EXAMPLE 4 17 What if the graphs of the equations in a system are the same line? Solution

4.1Solving Systems by Graphing 3Solve a Linear System by Graphing: Special Cases 18 Definition When solving a system of equations by graphing, if the graph of each equation is the same line, then the system has an infinite number of solutions. The system is consistent, and the equations are dependent.

4.1Solving Systems by Graphing 3Solve a Linear System by Graphing: Special Cases 19 Procedure Solving a System by Graphing To solve a system by graphing, graph each line on the same axes. 1)If the lines intersect at a single point, then the point of intersection is the solution of the system. The system is consistent, and the equations are independent.

4.1Solving Systems by Graphing 3Solve a Linear System by Graphing: Special Cases 20

4.1Solving Systems by Graphing 3Solve a Linear System by Graphing: Special Cases 21 Procedure Solving a System by Graphing 2)If the lines are parallel, then the system has no solution. We write the solution set as . The system is inconsistent. The equations are independent.

4.1Solving Systems by Graphing 3Solve a Linear System by Graphing: Special Cases 22

4.1Solving Systems by Graphing 3Solve a Linear System by Graphing: Special Cases 23 Procedure Solving a System by Graphing 3)If the graphs are the same line, then the system has an infinite number of solutions. We say that the system is consistent, and the equations are dependent.

4.1Solving Systems by Graphing 3Solve a Linear System by Graphing: Special Cases 24

4.1Solving Systems by Graphing 4Determine the Number of Solutions of a System Without Graphing 25 Example 4 : if a system has lines with the same slope and the same y-intercept (they are the same line), then the system has an infinite number of solutions. Example 3: if a system contains lines with the same slope and different y-intercepts, then the lines are parallel and the system has no solution.

4.1Solving Systems by Graphing 4Determine the Number of Solutions of a System Without Graphing 26 Example 2: if the lines in a system have different slopes, then they will intersect and the system has one solution.

EXAMPLE 5 27 Without graphing, determine whether each system has no solution, one solution, or an infinite number of solutions.

EXAMPLE 5 28 Solution a)

EXAMPLE 5 29 Solution b) The equations are the same: they have the same slope and y- intercept. Therefore, this system has an infinite number of solutions.

EXAMPLE 5 30 Solution c) The equations have the same slope but different y-intercepts. If we graphed them, the lines would be parallel. Therefore, this system has no solution.