SOLUTION: Tell how to determine whether the system has a unique solution, and solve it with the Cramer method: ax + by = c dx + ey = f The answer is: the system has a unique solution

Algebra ->  -> SOLUTION: Tell how to determine whether the system has a unique solution, and solve it with the Cramer method: ax + by = c dx + ey = f The answer is: the system has a unique solution       Log On

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 Click here to see ALL problems on Matrices-and-determiminant Question 549768: Tell how to determine whether the system has a unique solution, and solve it with the Cramer method: ax + by = c dx + ey = f The answer is: the system has a unique solution if: delta = ae - bd not equal to 0 and the solution is: x = (ce - bf) / (ae - bd) y = (af - cd) / (ae - bd) How do we explain that? Answer by Edwin McCravy(9716)   (Show Source): You can put this solution on YOUR website!``` ax + by = c dx + ey = f By Cramer's rule: Delta = = ae - bd Dx = = ce - bf Dy = = af - cd x = = = y = = = Denominators cannot be 0, so Delta = = ae - bd ≠ 0. and both and have unique values when that denominator is not zero. ------------------------------------------------------ In case you need to explain why Cramer's rule works: To eliminate y, multiply the first equation by e and the second equation by -b, and add the equations vertically term by term: aex + bey = ce -bdx - bey = -bf ----------------------- aex-bdx = ce-bf (ae-bd)x = ce-bf x = To eliminate x, multiply the first equation by -d and the second equation by a, and add the equations vertically term by term: -adx - bdy = -cd adx + aey = af ----------------------- aey-bdy = af-cd (ae-bd)y = af-cd y = a and y are the same using the elimination method as they are using Cramer's rule. Edwin```