Among the various methods to solve simultaneous equations, the **Elimination Method** is
one of the most popular and easiest methods to solve the pair of simultaneous
equations in two variables x and y.

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In this method, one variable either x or y is eliminated to find the value of another variable. And, for that, we need to make coefficients of any one variable equal by multiplying or dividing the equations by the least number so that the coefficients of x or y become equal in both the new equations.

Then, by adding
or subtracting according to these equal coefficients have opposite or same sign
respectively, we get an equation which will have only one variable. Then, simplification
will give the value of that variable.

After solving
the value of either x or y obtained from the above steps is substituted in any one of
the given equations to get the value of the other variable.

**Steps for solving simultaneous equations by elimination
method:**

__Step
1__: Convert one variable of both equations with the same
coefficient.

__Step
2__: Eliminate the variable having the same coefficient by
subtracting or adding.

__Step
3__: Solve the equation having one variable.

__Step
4__: Substitute the value of one variable so obtained in any one of the
given equations to calculate the value of another variable.

This process
of **Elimination Method** will be clear
by the following worked out examples.

*Worked Out Examples*

*Worked Out Examples*

*Example 1:**
Solve: x + 3y = 13 and x + y = 7 by elimination method.*

*Solution:** Here,*

*x + 3y = 13
…………. (i)*

*x + y = 7
………….. (ii)*

*Subtracting equation (ii) from (i),
we get,*

*Now, substituting the value of y = 3
in equation (ii), we get,*

* x
+ 3 = 7*

*or, x
= 7 – 3*

*or, x
= 4*

*Hence, x = 4 and y = 3.*

*Example 2:**
Solve: x + 2y = 5 and 5x – 2y = 13 by elimination method.*

*Solution:** Here,*

*x + 2y = 5
…………. (i)*

*5x – 2y = 13
………….. (ii)*

*Adding equations (i) and (ii) we get,*

*Now, substituting the value of x = 3
in equation (i) we get,*

* 3
+2y = 5*

*or, 2y
= 5 – 3*

*or, 2y
= 2*

*or, y
= 1*

*Hence, x = 3 and y = 1.*

*Example 3:**
Solve: 3x + 5y = 7 and 7x + 10y = 13 by elimination method.*

*Solution:** Here,*

* 3x
+ 5y = 7 …………. (i) × 2*

* 7x
+ 10y = 13 ………….. (ii)*

*Multiplying equation (i) by 2 and subtracting
from equation (ii) we get,*

*Now, substituting the value of x = –1
in equation (i) we get,*

* 3
× –1 + 5y = 7*

*or, –3
+ 5y = 7*

*or, 5y
= 7 + 3*

*or, 5y
= 10*

*or, y
= 2*

*Hence, x = **–1** and y = 2.*

*Example 4:**
Solve: 5x + 3y = 5 and 3x – 4y = 32 by elimination method.*

*Solution:** Here,*

* 5x
+ 3y = 5 …………. (i) × 4*

* 3x
– 4y = 32 ………….. (ii) × 3*

*Multiplying equation (i) by 4 and (ii)
by 3 and then adding we get,*

*Now, putting x = 4 in equation (i) we
get,*

* 5
× 4 + 3y = 5*

*or, 20
+ 3y = 5*

*or, 3y
= 5 **–** 20*

*or, 3y
= **–**15*

*or, y
= **–**5*

*Hence, x = **4** and y = –5.*

* *

* *

*Solution:** Here,*

*Multiplying equation (i) by 3 and (ii)
by 2 and subtracting (ii) from (i), we get,*

*Now, substituting the value of x = 6
in equation (i), we get,*

* 3/6
**+ 2/y
= 1*

*or, 1/2
**+ 2/y
= 1*

*or, 2/y
**=
1– 1/2*

*or, 2/y
**= 1/2*

*or, y
= **4*

*Hence, x = **6** and y = 4.*

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