The elimination method is one of prominent methods in solution to the system of linear equations with two variables. The elimination method for solving equations may be apparent in certain cases where, the coefficients of a particular variable or variables are same.
In such a case you automatically use this technique. These cases of equations are called solving elimination equations. For, example, if x + y = 5 and x – y = 1, you eliminate ‘y’ by just adding both the equations to get the solution as x = 3. Similarly you eliminate ‘x’ by subtracting the second equation from first and figure out y = 2. But all systems are not as simple as this. Hence let us take a broader look on solving by elimination method.
Suppose, you have a system of equations in which all the coefficients of the variables are different. Select a variable for elimination. Your skill and experience will tell you which will be the ideal variable to eliminate. Take the LCM of the coefficients of that variable and multiply each equation with the ‘missing’ factor of the LCM in each case.
Now the given set of equations is transformed in such a way that the new set has the same coefficient for the selected variable in each equation. Now do the subtraction or addition operations on the transformed equations, so that the selected variable is eliminated and you get the solution of the remaining variable. Subsequently, plug in that in any of the given equations to figure out the solution of the other variable.
Let us elaborate with an example. Let the system be 17x + 2y = 49 and 19x + 3y = 54. It is prudent to decide to eliminate ‘y’. Eliminating ‘x’ not an incorrect step. But that will lead to cumbersome working. The LCM of 2 and 3 is 6.
Multiplying the first by 3 and the second by 2 (the missing factors of 6), the equations are transformed to 51x + 6y = 147 and 38x + 6y = 108.. A subtraction operation gives you the result as 13x = 39, means x = 3. Now plugging in x = 3 back, you can figure out y = -1.
The method of solving equations by elimination is the basis for the concept of use of determinants for solution of system of equations.
let us make a simple illustration. Suppose a1x + b1y = c1 and a2x + b2y = c2 . By elimination method you can establish, x = (c1b2 – c2b1 )/(a1b2 – a2b1 ), which is the concept behind Cramer’s rule of solution by using determinants.