10/2/2010. Objectives. Solving Systems of Equations and Inequalities. Solving Systems of Equations by Graphing S E C T I O N 4.1


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1 Solving Systems of Equations and Inequalities 4 S E C T I O N 4.1 Solving Systems of Equations by Graphing Objectives 1. Determine whether a given ordered pair is a solution of a system. 2. Solve systems of linear equations by graphing. 3. Use graphing to identify inconsistent systems and dependent equations. 4. Identify the number of solutions of a linear system without graphing. 3 1
2 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. 4 1 Determine whether a given ordered pair is a solution of a system 5 Determine whether a given ordered pair is a solution of a system Liner equations: with infinitely many pairs of numbers whose sum is 3, there are infinitely many pairs (x, y) that satisfy this equation. 6 2
3 Determine whether a given ordered pair is a solution of a system System of equations: 7 Determine whether a given ordered pair is a solution of a system Because the ordered pair (2, 1) satisfies both of these equations, it is called a solution of the system. 8 Example 1 Determine whether ( 2, 5) is a solution of each system of equations. a. b. 9 3
4 Example 1(a) Solution Recall that in an ordered pair, the first number is the xcoordinate and the second number is the ycoordinate. To determine whether ( 2, 5) is a solution, we substitute 2 for x and 5 for y in each equation. Check: 3x + 2y = 4 3( 2) + 2(5) = 4 The first equation. 10 Example 1(a) Solution x y = 7 The second equation = 7 Since ( 2, 5) satisfies both equations, it is a solution of the system. 11 Example 1(b) Solution Again, we substitute 2 for x and 5 for y in each equation. Check: 4y = 18 x 4(5) 18 ( 2) = 20 The first equation. 12 4
5 Example 1(b) Solution y = 2x 5 2( 2) 5 = 4 The second equation. False Although ( 2, 5) satisfies the first equation, it does not satisfy the second. Because it does not satisfy both equations, ( 2, 5) is not a solution of the system Solve systems of linear equations by graphing 14 Solve systems of linear equations by graphing To use the graphing method to solve we graph both equations on one set of coordinate axes using the intercept method, as shown below. 15 5
6 Solve systems of linear equations by graphing Although there are infinitely many pairs (x, y) that satisfy x + y = 3, and infinitely many pairs (x, y) that satisfy 3x y = 1, only the coordinates of the point where their graphs intersect satisfy both equations simultaneously. Thus, the solution of the system is (1, 2). 16 Solve systems of linear equations by graphing To check this result, we substitute 1 for x and 2 for y in each equation and verify that the pair (1, 2) satisfies each equation. Check: First equation Second equation x + y = 3 3x y = (1) = = 1 When the graphs of two equations in a system are different lines, the equations are called independent equations. 17 Solve systems of linear equations by graphing 18 6
7 Example 2 Solve the system of equations by graphing: 19 Example 2 Solution 20 Example 2 Solution From the graph, the solution appears to be (4, 2). 21 7
8 Example 2 Solution To check, we substitute 4 for x and 2 for y in each equation and verify that the pair (4, 2) satisfies each equation. Check: 2x + 3y = 2 2(4) + 3( 2) = 2 This is the first equation. 3x = 2y (4) 2( 2) = 12 This is the second equation. The equations in this system are independent equations, and the system is a consistent system of equations Use graphing to identify inconsistent systems and dependent equations 23 Example 4 Solve the system of equations by graphing: 24 8
9 Example 4 Solution Since y = 2x 6 is written in slope intercept form, we can graph it by plotting the yintercept (0, 6) and then drawing a slope of 2. (The rise is 2, and the run is 1.) y = 2x 6 So m = 2 = and b = 6. We graph 4x + 2y = 8 using the intercept method. 25 Example 4 Solution The system is graphed below. Since the lines in the figure are parallel, they have the same slope. 26 Example 4 Solution We can verify this by writing the second equation in slope intercept form and observing that the coefficients of x in each equation are equal and the yintercepts are different, (0, 6) and (0, 4). y = 2x 6 4x + 2y = 8 2y = 4x + 8 y = 2x + 4 Because parallel lines do not intersect, this system has no solution and is inconsistent. Since the graphs are different lines, the equations of the system are independent. 27 9
10 Use graphing to identify inconsistent systems and dependent equations Sometimes a system of equations has no solution. Such systems are called inconsistent systems. 28 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 Identify the number of solutions of a linear system without graphing 30 10
11 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. If the lines have the same slope and different yintercepts, they are parallel, and the system has no solution. If the lines have the same slope and same yintercept, they are the same line, and the system has infinitely many solutions. 31 Example 6 Without graphing, determine the number of solutions of: Strategy: We will write both equations in slope intercept form. Solution: To write each equation in slope intercept form, we solve for y. 5x + y = 5 The first equation. 3x + 2y = 8 The second equation. 32 Example 6 Solution Different slopes Since the slopes are different, the lines are neither parallel nor identical. Therefore, they will intersect at one point and the system has one solution
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