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Elimination Method by Equating Coefficients
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Elimination Method (by Equating Coefficients)
There is another method of eliminating a variable, than often used method i. e --------Suppose you are to solve
23x - 17y + 11=0 ------(1)
and 31x + 13y - 57 = 0 -------(2)
Now expressing x in terms of y would involve division by 23 or 31. Express y in terms of x, it would involve division by 17 or 13. You know that multiplication is more convenient than division, better to convert the division process into a multiplication process.
Multiplying the first equation by 13 viz., coefficient of y in (2), and second by 17 viz., coefficient of y in (1), you will get an equivalent system of equations. The new system has the advantage that y has the same numerical coefficient 17x13 in both the equations. When you add these new equations, the terms containing y would cancel out as these have opposite signs and the same numerical coefficient. Thus, y has been eliminated. Now proceed as before, and solve the system. This method of elimination is also called elimination by equating coefficients for obvious reasons.

Example: Solve the following system of equations using the elimination method by equating coefficients:
11x - 5y + 61 = 0 (1)
3x - 20y - 2 = 0 (2)
Solution: Let us multiply equation (1) by 3 and equation (2) by 11. This gives
33x - 15y + 183 = 0 (3)
and 33x - 220y - 22 = 0 (4)
Subtracting (4) from (3), you will get 205y + 205 = 0
, or y = - 1
Substituting this value of y in equation (2), you will get
3x - 20 * (- 1) - 2 = 0
or 3x = -18
or x = - 6
Thus, the required solution is
x = - 6 and y = -1.
Now you should verify; substitute x = - 6 and y = -1 in the given equations, you will notice both the equations are satisfied. Hence, the solution is correct

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