Ace Your Math Tests: Solving Linear Simultaneous Equations with Confidence

Linear Simultaneous Equations are a system of equations that contain two or more variables. These equations can be solved simultaneously to find the values of the variables that satisfy all equations in the system. A solution is a set of values of the variables that satisfy all the equations in the system.

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:

3x + 2y = 2
x + y = 4

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

From first equation we get

3x = 2 - 2y
\(x = \frac{(2 -2y)}{3}\)

From second equation we get

\(x = 4 - y\)

On comparing we get

\(\frac{(2 -2y)}{3}\) = \(4 - y\)
2 - 2y = 12 - 3y
y = 10

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

x +10 = 4

x = - 6

Therefore, the solution of the simultaneous equations is x = - 6 and y = 10

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:

3x + 2y = 2
x + y = 4

We can write the above equation in the following format

3x + 2y - 2 = 0
x + y - 4 = 0

Now applying the general formula of cross multiplication 

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

\(\frac{x}{2.(-4) -1(-2)}=\frac{y}{(-2).1 - 3(-4)}=\frac{1}{1.3 - 2.1}\)

\(\frac{x}{-6}=\frac{y}{10}=\frac{1}{1}\)

x = - 6 and y = 10

Therefore, the solution of the simultaneous equations is x = - 6 and y = 10

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