Linear Simultaneous Equations: Tips for Finding the Best Method

Linear simultaneous equations are equations that involve two or more unknown variables and they need to be solved simultaneously. A solution to a system of linear equations is an assignment of values to the unknown variables that satisfies all the equations in the system. There are different methods to solve systems of linear equations, but two common methods are the Comparison Method and Cross Multiplication method.

1. Comparison Method:

The comparison method involves comparing the coefficients of one variable in both equations to eliminate that variable. Once one variable is eliminated, the other variable can be easily solved.

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

2x + y = 12
6x + 5y = 40

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

From first equation we get

2x = 12 - y
\(x = \frac{(12 -y)}{2}\)

From second equation we get

6x = 40 - 5y
\(x = \frac{(40 -5y)}{6}\)

On comparing we get

\(\frac{(12 -y)}{2}\) = \(\frac{(40 -5y)}{6}\)
72 - 6y = 80 - 10y
10y - 6y = 80 - 72
4y = 8
y = 2

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

2x + 2 = 12
2x = 10
x = 5

Therefore, the solution of the simultaneous equations is x = 5 and y = 2

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:

2x + y = 12
6x + 5y = 40

We can write the above equation in the following format

2x + y - 12 = 0
6x + 5y - 40 = 0

Now applying the general formula of cross multiplication 

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

\(\frac{x}{1.(-40) - 5(-12)}=\frac{y}{(-12).6 - 2(-40)}=\frac{1}{2.5 - 1.6}\)

\(\frac{x}{20}=\frac{y}{8}=\frac{1}{4}\)

x = 20/4 = 5 and y = 8/4 = 2

Therefore, the solution of the simultaneous equations is x = 5 and y = 2

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