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Unformatted text preview: 50 PROBLEMS FOR CHAPTER 1 Obviously, it depends on whether we treat 2:1: = 3 as an equation in one variable or as an equation in two variables with one of the coefficients 0. You should have come across this problem before. The statement :3 = 3 might be the equation of a line in the number plane. Now consider a linear equation in one variable — as: = b with {1,5 E IR. There are three essentially different cases. 1. If a 75 0 then there is a unique solution a: = E. The solution set is {2 } 2. If a = 0 (Le. the coefficient of :1: is 0) and b = 0 then every value of a: is a solution and the solution set is R. 3. If a = 0 and b 75 0 then there is no solution. The solution set is the empty set (ll. How about a linear equation in two unknowns, 2:1 and 3:2,. that is an equation of the form elm] + [1.2.3172 = b, where (11,0.2, and b are scalars? There are four exhaustive but not exclusive cases. _fl2)‘+ i 1. If {11 75 0 then the solution set is {( ”1 A '11) :A E R}. 2. If {12 7E 0 then the solution set is {( (11;; b) :A E R}. ...
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