Solving Systems of Linear Equations by Graphing 7.3 Solving Systems of Linear Equations by Graphing

Solving Systems of Linear Equations by Graphing 7.3 Solving Systems of Linear Equations by Graphing 1. Decide whether a given ordered pair is a solution of a system. 2. Solve linear systems by graphing. 3. Solve special systems by graphing. 4. Identify special systems without graphing.

Definitions A system of linear equations, or linear system, consists of two or more linear equations with the same variables. In the system on the right, think of y = 3 as an equation in two variables by writing it as 0x + y = 3.

Decide whether a given ordered pair is a solution of a system. Objective 1 Decide whether a given ordered pair is a solution of a system.

Definitions A solution of a system of linear equations is an ordered pair that makes both equations true at the same time. A solution of an equation is said to satisfy the equation.

Determining Whether an Ordered Pair Is a Solution Classroom Example 1 Determining Whether an Ordered Pair Is a Solution Decide whether the ordered pair (4, –1) is a solution of each system. a. b. a. To decide whether (4, –1) is a solution of the system, substitute 4 for x and –1 for y in each equation. (4, –1) is a solution.

Determining Whether an Ordered Pair Is a Solution (cont.) Classroom Example 1 Determining Whether an Ordered Pair Is a Solution (cont.) Decide whether the ordered pair (4, –1) is a solution of each system. a. b. b. To decide whether (4, –1) is a solution of the system, substitute 4 for x and –1 for y in each equation. (4, –1) is NOT a solution.

Solve linear systems by graphing. Objective 2 Solve linear systems by graphing.

Graphing linear systems. The set of all ordered pairs that are solutions of a system is its solution set. One way to find the solution set of a system of two linear equations is to graph both equations on the same axes. Any intersection point would be on both lines and would therefore be a solution of both equations. Thus, the coordinates of any point at which the lines intersect give a solution of the system.

Solving a Linear System by Graphing Step 1 Graph each equation of the system on the same coordinate axes. Step 2 Find the coordinates of the point of intersection of the graphs if possible, and write it as an ordered pair. Step 3 Check that the ordered pair is the solution by substituting it in both of the original equations. If it satisfies both equations, write the solution set.

Solving a System by Graphing Classroom Example 2 Solving a System by Graphing Solve the system by graphing. Graph each line using any method. The lines suggest that the graphs intersect at (3, 2). Check in both equations. (3, 2) is the solution of the system.

Solve special systems by graphing. Objective 3 Solve special systems by graphing.

Solving Special Systems by Graphing Classroom Example 3 Solving Special Systems by Graphing Solve each system by graphing. a. The two lines are parallel and have no points in common. There is no solution. The solution set is

Solving Special Systems by Graphing (cont) Classroom Example 3 Solving Special Systems by Graphing (cont) Solve each system by graphing. b. The graphs of these two equations are the same line. The solution set is {(x, y) | 2x – 5y = 8}.

Three Cases for Solutions of Linear Systems with Two Variables

Three Cases for Solutions of Linear Systems with Two Variables

Identify special systems without graphing. Objective 4 Identify special systems without graphing.

Identify special systems. We can recognize special systems without graphing by comparing their slopes and y-intercepts. We do this by writing each equation in slope- intercept form, solving for y.

Identifying the Three Cases Using Slopes Classroom Example 4 Identifying the Three Cases Using Slopes Describe each system without graphing. State the number of solutions. a. Write each equation in slope-intercept form. The slopes are the same; the lines are parallel. The system has no solution.

Identifying the Three Cases Using Slopes (cont.) Classroom Example 4 Identifying the Three Cases Using Slopes (cont.) Describe each system without graphing. State the number of solutions. b. Write each equation in slope-intercept form. The equations represent the same line. The system has an infinite number of solutions.

Identifying the Three Cases Using Slopes (cont.) Classroom Example 4 Identifying the Three Cases Using Slopes (cont.) Describe each system without graphing. State the number of solutions. c. Write each equation in slope-intercept form. The equations represent lines that are neither parallel nor the same line. The system has exactly one solution.