there are these things called systems of equations, whereas there are 2 equations, like 2 x + 4 = 3 y and x + y = 15 my question is: is there a very simple formula for a x + b y = c and d x + e y = f? i've tried, like 5 times, all with no reliable results.

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there are these things called systems of equations, whereas there are 2 equations, like   2  x  +  4  =  3  y and   x  +  y  =  15 my question is: is there a very simple formula for   a  x  +  b  y  =  c and   d  x  +  e  y  =  f? i&#039;ve tried, like 5 times, all with no reliable results.

victorollaa5 · Accepted Answer

Yes, there is a formula, for example in terms of matrices. Let  A  =      (                            a                          b                                      d                          e                      )    ,    v  =      (                            x                                      y                      )    ,    w  =      (                            c                                      f                      )    .Then the system of equations is summarized as matrix equation   A  v  =  w. Suppose that   det  (  A  )  =  a  e  −  b  d  ≠  0. Then   v  =      A          −      1        w is the formula for the unique solution   v. For   det  (  A  )  =  0 we can make a case distinction, but the solutions may not be unique, or there may be no solution, e.g., for   a  =  b  =  0 and   c  =  1.

Llubanipo · Accepted Answer

Yes. They&#039;re called Cramer&#039;s formulae: first rewrite the linear system in standard form:      {                            x          −          3          y          =          −          4          ,                                      x          +          y          =          15.                        As the determinant of the system       |                            1                          −          3                                      1                          1                      |    ≠  0, the solutions are:  x  =            |                                    −            4                                −            3                                                15                                1                              |              |                                    1                                −            3                                                1                                1                              |        ,    y  =            |                                    1                                −            4                                                1                                15                              |              |                                    1                                −            3                                                1                                1                              |        .This generalises to systems of   n linear equations with   n unknowns if their determinant is non-zero.

there are these things called systems of equations, whereas there are 2 equations, like 2 x

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