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Question

Explanation of the elimination method of linear equation in 2 variable.


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Solution

Step 1: Standard form of linear equation in two variables

Linear equation in two variables x and y is of the form

ax+by=c

where a,b,c are constant.

Step 2: Explanation of the elimination method

Suppose we have two simultaneous linear equations in two variables.

a1x+b1y=c1-----(1)a2x+b2y=c2-----(2)

We have to eliminate one variable, either x or y from the two equations.

Suppose we eliminate x. Then from the resulting equation we have to find the value of y.

Then substitute the value of y in any one of the equation (1) or (2) and solve the equation to find the value of x.

Step 3: Elimination of x

Let us eliminate x.

Multiply equation (1) by a2. We get,

a2(a1x+b1y)=a2c1or,a2a1x+a2b1y=a2c1------(3)

Multiply equation (2) by a1. We get,

a1(a2x+b2y)=a1c2or,a1a2x+a1b2y=a1c2-----(4)

Now, subtract equation (3) from equation (4)

a1a2x+a1b2y=a1c2a2a1x+a2b1y=a2c1a1b2y-a2b1y=a1c2-a2c1

a1a2x and a2a1xgets canceled out and we get the equation,

a1b2y-a2b1y=a1c2-a2c1

Step 4: Find out the value of y

a1b2y-a2b1y=a1c2-a2c1or,(a1b2-a2b1)y=a1c2-a2c1or,y=a1c2-a2c1a1b2-a2b1(Dividingbothsidebya1b2-a2b1)

Step 5: Find out the value of x

Thus eliminating x we get the value of y.

Now we will put the value of y in any one of the equation (1)or (2) and solve to get the value of x.

Put the value of y in equation (1)

a1x+b1(a1c2-a2c1a1b2-a2b1)=c1or,a1x=c1-b1(a1c2-a2c1a1b2-a2b1)or,a1x=c1(a1b2-a2b1)-b1(a1c2-a2c1)a1b2-a2b1or,a1x=(c1a1b2-c1a2b1)-(b1a1c2-b1a2c1)a1b2-a2b1or,a1x=c1a1b2-b1a1c2a1b2-a2b1(c1a2b1andb1a2c1getscancelledout)or,a1x=a1(c1b2-b1c2)a1b2-a2b1or,x=c1b2-b1c2a1b2-a2b1(Cancellingouta1frombothside)

Step 6: Conclusion

By elimination method x=c1b2-b1c2a1b2-a2b1

y=a1c2-a2c1a1b2-a2b1

Hence, elimination method of linear equation in 2 variables is explained.


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