Solving Systems of Equations by Graphing Section 4.1 Solving Systems of Equations by Graphing

Objectives Determine whether a given ordered pair is a solution of a system Solve systems of linear equations by graphing Use graphing to identify inconsistent systems and dependent equations Identify the number of solutions of a linear system without graphing Use a graphing calculator to solve a linear system (optional)

Objective 1: Determine Whether a Given Ordered Pair Is a Solution of a System We have previously discussed equations in two variables, such as x + y = 3. Because there are infinitely many pairs of numbers whose sum is 3, there are infinitely many pairs (x, y) that satisfy this equation. Some of these pairs are listed in table (a). Now consider the equation x − y = 1. Because there are infinitely many pairs of numbers whose difference is 1, there are infinitely many pairs (x, y) that satisfy x − y = 1. Some of these pairs are listed in table (b). From the two tables, we see that (2, 1) satisfies both equations.

Objective 1: Determine Whether a Given Ordered Pair Is a Solution of a System When two equations with the same variables are considered simultaneously (at the same time), we say that they form a system of equations. Using a left brace { , we can write the equations from the previous example as a system: Because the ordered pair (2, 1) satisfies both of these equations, it is called a solution of the system. In general, a system of linear equations can have exactly one solution, no solution, or infinitely many solutions.

EXAMPLE 1 Determine whether (−2, 5) is a solution of each system of equations.

Objective 2: Solve Systems of Linear Equations by Graphing To solve a system of equations means to find all of the solutions of the system. One way to solve a system of linear equations is to graph the equations on the same set rectangular coordinate system. The Graphing Method: 1. Carefully graph each equation on the same rectangular coordinate system. 2. If the lines intersect, determine the coordinates of the point of intersection of the graphs. That ordered pair is the solution of the system. 3. Check the proposed solution in each equation of the original system.

EXAMPLE 2 Solve the system of equations by graphing:

Objective 3: Use Graphing to Identify Inconsistent Systems and Dependent Equations A system of equations that has at least one solution, like that in Example 2, is called a consistent system. A system with no solution is called an inconsistent system.

Objective 3: Use Graphing to Identify Inconsistent Systems and Dependent Equations In the next example, the graphs of the equations of the system are different lines. We call equations with different graphs independent equations. Some equations have the same graph and are equivalent. Because they are different forms of the same equation, they are called dependent equations.

Objective 3: Use Graphing to Identify Inconsistent Systems and Dependent Equations There are three possible outcomes when we solve a system of two linear equations using the graphing method:

EXAMPLE 3 Solve the system of equations by graphing:

Objective 4: Identify the Number of Solutions of a Linear System Without Graphing We can determine the number of solutions that a system of two linear equations has by writing each equation in slope–intercept form. If the lines have different slopes, they intersect, and the system has one solution. (See Example 2.) If the lines have the same slope and different y-intercepts, they are parallel, and the system has no solution. (See Example 3.) If the lines have the same slope and same y-intercept, they are the same line, and the system has infinitely many solutions. (See Example 4.)

EXAMPLE 5 Without graphing, determine the number of solutions of: