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3 Special Kinds of Theorems > Use of Counterexamples - Pg. 40

40 CHAPTER 3 Special Kinds of Theorems Prove this statement by proving "If i, then ii," "If ii, then iii," and "If iii, then i." 7. Let x and y be two distinct negative real numbers. The following statements are equivalent: i. x < y, ii. |x | > |y |, and iii. x 2 > y 2 . 8. Consider the two systems of linear equations ( a 1 x + b 1 y = c 1 S 1 : a 2 x + b 2 y = c 2 and S 2 : ( a 1 x + b 1 y = c 1 ða 1 + ba 2 Þx + ðb 1 + bb 2 Þy = c 1 + bc 2 with a 1 , a 2 , b 1 , b 2 , c 1 , c 2 real numbers and b 0. The pair of values (x 0 , y 0 ) is a solution of S 1 if and only if it is a solution of S 2 . USE OF COUNTEREXAMPLES An example can be very useful when trying to make a point or explain the result obtained in a proof, but it cannot be used as a proof of the fact that a statement is true. Let us see what might happen if we used and