Special Cases of Systems of Equations The following is an explanation of two special cases of systems of equations. When solving by algebraic methods, either substitution or elimination, you find that you have “knocked out” both the x and the y variable and your resulting equation which contains only constant numbers is false like 18 = 7, then the answer to the system is “no solution” or the empty set. The lines are parallel and therefore inconsistent. On the other hand, if you are solving by algebraic methods and you find that you have “knocked out” both the x and the y variable and your resulting equation its true like 6 = 6 or 0 = 0, then the answer to the system is infinite solutions. The lines are the same and therefore dependent. Solve the system, 2x + 10y = 3 and x = 1 – 5y, by substitution. Since the second equation is solved for x, I use that value and substitute it into the first equation: 2(1 – 5y) + 10y = 3 2 – 10y + 10y = 3
2 = 3 … which is Solve the system, 6x + 9y = 6 and 2x + 3y = 2, by elimination. I choose to multiply the second line by – 3 in order to eliminate the x variable. (– 3) 2x + 3y(– 3) = (– 3) 2 … the second line now becomes – 6x – 9y = – 6 Now place the equations in column form add the two equations: 6x + 9y = 6 true and therefore the system has
infinite solutions.The views and
opinions expressed in this page are strictly those of Mary Lou Baker. This page was edited on 19-Sep-2007 |