# SOLUTION: Hi -- I'm a beginning student in Linear Algebra, and am not quite sure how to answer the following question, which is taken from my textbook: "Suppose the system below is con

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 Question 563136: Hi -- I'm a beginning student in Linear Algebra, and am not quite sure how to answer the following question, which is taken from my textbook: "Suppose the system below is consistent for all possible values of f and g. What can you say about the coefficients c and d. Justify your answer. 2x + 4y = f cx + dy = g." Not quite sure what is meant by the question "what can you say." My only thought so far is that If the system is consistent, that means x and y have the same value for both equations: x(2 + c) + y(4 + d) = f + g, which, if c = f+g, fits into the form ax + by = c. But that's about it. Thanks in advance! Cheers, Dave YruetaAnswer by Edwin McCravy(8999)   (Show Source): You can put this solution on YOUR website!```No, they are consistent if they have a solution. So we find what their solution must be by elimination: 2x + 4y = f cx + dy = g Multiply the first equation through by -d and the second equation through by 4 -2dx - 4dy = -df 4cx + 4dy = 4g (4c-2d)x = 4g-df x = Start over: 2x + 4y = f cx + dy = g Multiply the first equation through by c and the second equation through by -2 2cx + 4cy = cf -2cx - 2cy = -2g (4c-2d)y = cf-26 y = Notice that for both x and y, the denominators are the same, 4c-2d In order for those expresions for x and y to be defined solution, their denominators must not be zero, because division by 0 is not defined. But division by any other number is defined. Therefore the only requirement is that their denominator must not be equal to 0, so 4c-2d ≠ 0 divide through by 2 2c-d ≠ 0 2c ≠ d d ≠ 2c As long as d is not equal to 2c, there will be a solution for any values of f and g. The answer to: "What can you say about the coefficients c and d?" is d is not equal to 2c. [If you have studied Cramer's rule, you can just say the determinant of coefficients D must not equal 0. The above was done assuming you have not studied or were not allowed to use Cramer's rule.] Edwin```