# Elimination Method

**Elimination by Equating the Coefficients**

In this method, we multiply both the equations by suitable non-zero constants so that the coefficients of the variable to be eliminated becomes equal. We now add or subtract the equations so that the variable with equal coefficients gets eliminated.Â Â

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1. 2x + 3y = 13Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 2. 3x + 2y = 7

Â Â Â 5x + 2y = 16Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 4x - 3y = -2

1. 2x + 3y = 13Â Â .....(i)

Â Â Â 5x + 2y = 16Â .....(ii)

Multiplying (i) by 5 and (ii) by 2, and we subtract (ii) from (i).

Â Â Â 10x + 15yÂ = 65

Â Â __ -10x -Â 4yÂ Â = -32__

Â Â Â Â Â Â Â Â Â Â Â 11yÂ Â Â = 33

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â y = 3

Substituting value of y in (i)

We get 2x + 3(3) = 13

Â 2x = 4

Â Â Â x = 2

Therefore, the solution is

x = 2, y = 3

2.Â 3x + 2y = 7Â .....(i)

Â Â Â Â Â 4x - 3y = -2Â .....(ii)

Multiplying (i) by 4 and (ii) by 3, we subtract (ii) from (i)

Â 12x + 8y = 28

__Â -12x + 9y = 6__

Â Â Â Â Â Â Â Â 17y = 34

Â Â Â Â Â Â Â Â Â Â Â Â y = 2

Putting this value of y in (i),

We get 3x + 2(2) = 7

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 3x = 3Â Â

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â x = 1

Therefore, the solution is x = 1, y = 2Â Â Â

Â

Â Â Â

Let us assumeÂ Â andÂ Â Â ...........(A)

The equations now become

2u + v = 2Â Â Â Â Â Â Â ---------- (i)

Â u - v = 4Â Â Â Â Â Â Â Â Â ---------- (ii)

Adding (i) and (ii), we get

Â 3u = 6Â Â Â Â Â Â

Â u = 2

From(1)Â Â Â Â Â Â 2(2) + v = 2

Â Â Â Â Â Â Â Â Â v = -2Â Â Â

Now x = ua.....from (A)

Hence x = 2a,Â Â Â Â

Now y = v .....from (A).Â

Hence y = -2b

Therefore, the solution is x = 2a, y = -2b