Cross Multiplication Method

In this page cross multiplication method we are going to see how to solve linear equations.

While solving a pair of linear equations in two unknowns x and y using elimination method,we have to think about how to eliminate any one of the variable in two equations to get value of one unknown and we have to plug it in the second to get the value of other variable. There is another method called cross multiplication which simplifies the procedure.

The following arrow diagram may be useful in remembering the above relation. The arrows between the two numbers indicate that they are multiplied,the second product is to be subtracted from the first product.

By using this formula we can easily solve any two equations of two unknowns without worrying about elimination.This formula can be used if the given equations is  in the form a₁ x + b₁ y + c₁ = 0,a₂ x + b₂ y + c₂ = 0.

Examples of cross multiplication method

Solve the equations by cross multiplication method 2 x + 7 y - 5 = 0, -3 x + 8 y = -11

Solution:

First we have to change the given linear equations in the form
a₁ x + b₁ y + c₁ = 0,a₂ x + b₂ y + c₂ = 0. That is we have to bring the constants with x,y term.

2 x + 7 y - 5 = 0

-3 x + 8 y + 11 = 0 x/[(7(11) - (8) (-5)] = y/[(-5(-3) - (2) (11)] = 1/[(2(8) - (-3) (7)]

x/[77 + 40] = y/[15 - 22] = 1/[16 + 21]

x/ = y/[-7] = 1/

x/ =  1/                  y/[-7]  = 1/

x = 117/37                         y = -7/37

Therefore,the solution is (117/37,-7/37)

Example 2:

3 x + 4 y = 24, 20 x - 11 y = 47

Solution:

First we have to make the given equations in the form of a₁ x + b₁ y + c₁ = 0,a₂ x + b₂ y + c₂ = 0.

3 x + 4 y - 24 = 0    ----- (1)

20 x - 11 y - 47 = 0 ----- (2) x/(-188-264)  = y/(-480 -(-141))   = 1/(-33-80)

x/(-452)  = y/(-480+141))   = 1/(-33-80)

x/(-452)  = y/(-339)   = 1/(-113)

x/(-452)  = 1/(-113)                   y/(-339)   = 1/(-113)

x = (-452)/(-113)                             y = (-339)/(-113)

x = 4                                       y = 3

Therefore solution is (4,3).

Verification:

Now let us apply the answer that we got in the first or second equation to check whether we got correct answer or not.

3 x + 4 y = 24

3(4) + 4(3) = 24

12 + 12 = 24

24 = 24

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