To solve simultaneous equations, the first requirement is to make sure the number of unknown variables are equal or less than the number of unique equations relating the unknowns.
By the use of elimination method, we apply the concept of removing one unknown (randomly selected) at a time. This will reduce the number of unknowns and will definitely make the solution simpler. By elimination, we try to first make any two equations having at least one unknown similar (step 1)
Example: Step 1
3x + 2y = 5 --- (A)
x + y = 2 --- (B)
We will target one unknown to be eliminated (removed) ==> Select "y" to be removed.
To make equations (B) having the "y" similar to equation (A), we need to multiply (B) by 2. Therefore new equation (B) becomes ==> 2x + 2y = 4 -----(C)
New set of equations:
3x + 2y = 5 ---(A)
2x + 2y = 4 --- (C)
After making the "y" term look the same, we move on to eliminating them (step 2).
Now, we see that the "2y" are the same. *** To remove the"y" part, we can now subtract equation (A) and (C). This gives us the result,
(3x - 2x) + (2y - 2y) = 5 - 4 ===> x + 0 = 1 ===> x = 1 (Answer).
Note: This removal of y through subtraction is known as ELIMINATION.
Step 3 and the final part is to replace the found answer for x back into any one of the equations (A), (B) or (C). Let's select equation (B) as it is simpler.
x + y = 2 ==> 1 + y = 2 ===> y = 2 - 1 = 1 . Therefore y = 1 (Answer).
By eliminating one unknown from the equations given, we can see that the solution becomes simpler. The first found answer is later replace back into any one of the equations to extract out the other unknowns.
To conclude NOTE:
1) Do learn the concept and method of elimination as it can be applied to any number of equations not limited to the example above which has only two equations.
2) This elimination method applies only for same order type of equations. (Order means the power of the unknown).
x^2 + y^2 = 18 ---- (A)
3x^2 + 4y^2 = 63 ----(B) ===>;Equation (A) & (B) has same order of 2.
For more information regarding solving simultaneous equations, you are invited to this post.
Happy learning maths. Think simple, and maths will be simple!