How to solve Linear Equations consisting of 2 variables by Eliminating Variable?
Let’s suppose I have equations:
(1)
(2)
We want to solve them by variable elimination method. The method is to multiply one or both of the equations by such a number that coefficients of
or
for both the equations become equal.
Let’s suppose, I multiply equation (2) by 2, we get
(3)
We can see that equations (3) and (1) now have equal coefficients for variable
.
Let’s write equation (1) and (3):
(1)
(3)
Now, I can eliminate variable
by just subtracting equation (3) from (1).
Now, we have found value of variable
. We can put value of
in (1), (2) or (3) to find value of
.
Putting value of y in (2), we get
Therefore,
and
(This is the solution)
Now let’s solve 2nd example using the same method which is called variable elimination method. This time we will eliminate variable y.
I have following equations to solve:
(1)
(2)
Multiplying equation (1) by 2 and equation (2) by 3, we get
and
(3) and
(4)
At this point, the coefficients of variable y are same for both the equations, so we can eliminate variable y by just subtracting equation (4) from equation (3).
So, subtracting equation (4) from equation (3), we get
Putting value of
in equation (1), we get
Therefore,
and
(This is the solution)
Sometimes students get confused with what number to use to make coefficients equal for both of the equations. We can easily deal with this problem. Read the following carefully.
Let’s suppose two equations are:
(1)
(2)
Let’s suppose b and d are prime numbers and you want to eliminate y. Then, we just need to do this.
Doing so will make coefficients of variable
equal to bd for both of the equations. Now, we can subtract one equation from the other to eliminate variable y.
If, you want to eliminate
and you notice that a and c are prime numbers. We can do the following:
Doing so will make coefficients of variable
equal to ac for both the equations. Now, we can subtract one equation from the other to eliminate variable
.
Note that the above strategy can be used anytime even if (a,c) and (b,d) are not prime. I recommend using smallest possible numbers to multiply with equations but if you are stuck and cannot think of a number which will make coefficients equal then simply use above mentioned technique which is usually used in case of prime coefficients.
Note: We do not necessarily need to subtract equations in order to eliminate variable. We can also add equations in order to eliminate one variable.
Example, I have equations:
(1)
(2)
We can see that coefficients of variable
are 1 and -1 for equations (1) and (2) respectively.
If, we multiply equation (1) by -1 then it will make the coefficients equal. Then, we can subtract equation (2) from (1) to eliminate y.
But, this will take more time. Why not to simply add equations (1) and (2) to eliminate variable y.
Adding equations (1) and (2), we get
Putting value of
in equation (1), we get
Therefore,
and
(This is the solution)
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