Let a , b , c be real numbers. I must consider the following system of linear equations: { x + 2 b y = a b x + ( 1 − b ) y = c 2 + c For which set of values of a is the following proposition true: "For every b there exists a c such that the system has a solution"?

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Let   a  ,  b  ,  c be real numbers. I must consider the following system of linear equations:      {                            x          +          2          b          y          =          a                                      b          x          +          (          1          −          b          )          y          =                      c            2                    +          c                        For which set of values of a is the following proposition true: &quot;For every   b there exists a   c such that the system has a solution&quot;?

Melina Glover · Accepted Answer

For a   c with a solution to exist, it must be possible satisfy the second equation, which is a quadratic for   c.So we know that there is a solution   c precisely when the associated discriminant is nonnegative. This discriminant is   Δ  =  1  −  4  (  b  x  +  (  1  −  b  )  y  )When we solve for   x using the other equation, i.e.   x  =  a  −  2  b  y, we obtain:                    Δ                            =        1        −        4        (        b        (        a        −        2        b        y        )        +        (        1        −        b        )        y        )                                          =        1        −        4        a        b        +        8                  b          2                y        +        y        −        b        y                                          =        1        −        4        a        b        +        y        (        8                  b          2                −        b        +        1        )            Because we may vary   y to obtain a solution, the only way   Δ is guaranteed to be negative is when   8      b    2    −  b  +  1 is zero. This you can use to provide conditions on   a to ensure that   Δ  ≥  0 even in that case.

Let a , b , c be real numbers. I must consider the following system of linear equa

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