Use an acronym to help you remember the three cases. For example CI for one, II for none, CD for infinite. x y c) Infi nite number of solutions Consistent system Dependent equations y x a) One solution—the point of intersection Consistent system Independent equations Figure 5.1 y x b) No solution Inconsistent system Independent equations YOU TRY 3 Solve each system by graphing. a) x y 4 b) 2x 6y 9 x y 1 4x 12y 18 4 Determine the Number of Solutions of a System Without Graphing The graphs of lines can lead us to the solution of a system. We can also determine the number of solutions a system has without graphing. We saw in Example 4 that if a system has lines with the same slope and the same y-intercept (they are the same line), then the system has an infi nite number of solutions. Example 3 shows that if a system contains lines with the same slope and different y-intercepts, then the lines are parallel and the system has no solution. Finally, we learned in Example 2 that if the lines in a system have different slopes, then they will intersect and the system has one solution. EXAMPLE 5 In-Class Example 5 Without graphing, determine whether each system has no solution, one solution, or an infinite number of solutions. a) 10x 8y 12 15x 12y 18 b) 4x 3y 8 x 2y 5 c) 2x 3y 15 8x 12y 3 Answer: a) infinite number of solutions b) one solution c) no solution Without graphing, determine whether each system has no solution, one solution, or an infi nite number of solutions. a) y 3 4 x 7 b) 4x 8y 10 c) 9x 6y 13 5x 8y 8 6x 12y 15 3x 2y 4 Solution a) The fi rst equation is already in slope-intercept form, so write the second equation in slope-intercept form. 5x 8y 8 8y 5x 8 y 5 8 x 1 The slopes, 3 4 and 5 8 , are different; therefore, this system has one solution. www.mhhe.com/messersmith SECTION 4.1 Solving Systems by Graphing 249
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