Solving Linear Simultaneous Equations of Two Variables: How to Choose the Right Method for Your Problem

Linear simultaneous equations involve two or more equations that have two or more unknown variables. The solutions to these equations are the values of the unknown variables that satisfy all the equations simultaneously. There are several methods to solve linear simultaneous equations, including the comparison method and the cross-multiplication method. Here, we will explain each of these methods in detail and provide examples to help students understand the concepts easily.

1. Comparison Method:

The comparison method involves comparing the coefficients of one variable in two different equations and then eliminating that variable to find the value of the other variable.

Let's take an example of two simultaneous linear equations:

x + y = 6
x - y = 4

To solve these equations using the comparison method, we can follow these steps:

From first equation we get

x = 6 - y

From second equation we get

\(x = 4 + y\)

On comparing we get

\(6 - y\) = \(4 + y\)
- 2y = - 2
 y = 1

Now put the value of y in any one of the original equation we get

x +1 = 6

x = 5

Therefore, the solution of the simultaneous equations is x = 5 and y = 1

2. Cross Multiplication Method:

The cross-multiplication method involves multiplying the coefficients of one variable in each equation by the coefficient of the other variable in the other equation. This method eliminates one variable and simplifies the other variable.

Let's take the same example to solve the equation through cross multiplication:

x + y = 6
x - y = 4

We can write the above equation in the following format

x + y - 6 = 0
x - y - 4 = 0

Now applying the general formula of cross multiplication 

\(\frac{x}{b_1c_2-b_2c_1}=\frac{y}{c_1a_2-c_2a_1}=\frac{1}{a_1b_2-a_2b_1}\)

\(\frac{x}{1.(-4) -(-1)(-6)}=\frac{y}{(-6).1 - 1(-4)}=\frac{1}{1.(-1) - 1.1}\)

\(\frac{x}{-10}=\frac{y}{-2}=\frac{1}{-2}\)

x = 5 and y = 1

Therefore, the solution of the simultaneous equations is x = 5 and y = 1

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