877 9.1 Systems of Linear Equations NOTE In the algebraic solution for Example 4, we wrote the solution set with the variable x arbitrary. We could write the solution set with y arbitrary. e ay - 2 4 , yb f Solve -4x + y = 2 for x. By selecting values for y and solving for x in this ordered pair, we can find individual solutions. Verify again that 10, 22 is a solution by letting y = 2 and solving for x to obtain 2 - 2 4 = 0. EXAMPLE 4 Solving a System with Infinitely Many Solutions Solve the system. 8x - 2y = -4 (1) -4x + y = 2 (2) ALGEBRAIC SOLUTION Divide each side of equation (1) by 2, and add the result to equation (2). 4x - y = -2 Divide (1) by 2. -4x + y = 2 (2) 0 = 0 True The result, 0 = 0, is a true statement. In the original system, equation (1) can be obtained by multiplying equation (2) by -2. Thus the equations are equivalent, and any ordered pair 1x, y2 that satisfies either equation will satisfy the system. To express the solution set, we solve equation (2) for y. -4x + y = 2 (2) y = 4x + 2 The solutions of the system can be written in the form of a set of ordered pairs 1x, 4x + 22, for any real number x. Some ordered pairs in the solution set are 10, 4 # 0 + 22, or 10, 22, and 11, 4 # 1 + 22, or 11, 62, as well as 13, 142, and 1-2, -62. As shown in Figure 4, the equations of the original system are dependent and lead to the same straight-line graph. The solution set can be written 51x, 4x + 226. GRAPHING CALCULATOR SOLUTION Solving the equations for y gives y1 = 4x + 2 (1) and y2 = 4x + 2. (2) When written in this form, we can immediately determine that the equations are identical. Each has slope 4 and y-intercept 10, 22. As expected, the graphs coincide. See the top screen in Figure 5. The table indicates that y1 = y2 for selected values of x, providing another way to show that the two equations lead to the same graph. 1 –2 –1 2 x 0 –4x + y = 2 8x – 2y = –4 y Consistent system with infinitely many solutions Figure 4 Refer to the algebraic solution to see how the solution set can be written using an arbitrary variable. S Now Try Exercise 33. 8x − 2y = −4 −4x + y = 2 −10 −10 10 10 Figure 5
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