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Algebraic Methods of Solving a Pair of Linear Equations

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Pair of linear equations

A pair of the linear equation is in the following form-

linearequation7

The most commonly used algebraic methods of solving a pair of linear equations in two variables are –

Cross Multiplication Method

linearequation1Let  a1x + b1y + c1=0

a2x + b2y + c2=0


be a system of simultaneous linear equations in two variables x and y such that a1/a2 ≠ b1/b2

        

i.e. a1b2 – a2b1 ≠ 0. Then the system has a unique solution given by

linearequation8

Here are the steps which we follow while solving a pair of linear equations by cross multiplication method:

Step I – Obtain the two equations.

Step II – Shift all terms on LHS in the two equations to introduce zeros on RHS i.e., write the two equations in the following form:

a1x + b1y + c1=0

a2x + b2y + c2=0

Step III – In the above system of equations there are three columns viz.

linearequation9

To obtain the solution, write x, -y and 1 separated by equality signs as shown below:

  linearequation2

In the denominator of x leave column containing x and write remaining two columns in the same order, in the denominator of –y leave column containing y and write the remaining two columns. Similarly, in the denominator of one write columns containing x and y.


Step IV – To obtain the denominators of x, -y and 1, cross multiply the numbers and subtract the product. Applying this, we get

linearequation3
Step V
– Obtain the value of x by equating first and third expression in step IV. The value of y is obtained by equating second and third expression in step IV.

To get a more clear idea, let’s explain with an example:

Example:Solve the following system of equations by using the method of cross multiplication:

x + y = 7                                                                                      

5x + 12y = 7

The given system of equations is

x + y – 7 = 0

5x +12y – 7 = 0

By cross – multiplication, we get

  linearequation5

linearequation6
 
x = 11 and y = -4

Try these questions now:

1. Solve the following system of equations by using the method of cross multiplication:

2x + 3y = 17

3x - 2y = 6

(Answer: x = 4 and y = 3)

2. Solve the following system of equations by using the method of cross multiplication:

2x - y = 3

4x + y = 3

(Answer: x = 1 and y = -1)

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Reference Links :

http://en.wikipedia.org/wiki/Linear_equation

http://en.wikipedia.org/wiki/Cross-multiplication

http://en.wikipedia.org/wiki/Simultaneous_equations

    

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