Question

What is the purpose of using the elimination method?

Answer 1

Hint: Here we will understand the elimination method to solve a system of linear equations. Now, we will further understand the conditions in which we prefer to use the elimination method and finally we will take an example of two linear equations and check how they are solved using this method.

Complete step-by-step solution:
Here we have been asked to describe the purpose of using the elimination method. First we need to understand the meaning of this method.
Now, in mathematics the elimination method is one of the three methods to solve a system of linear equations. In this method, if we are provided with two linear equations in two variables, we try to eliminate one of the variables by either subtracting or adding the two equations after making their coefficients equal. This elimination results in the formation of a linear equation in one variable which is then solved to find its value. After finding the value of this variable we substitute it in any of the two equations to determine the value of the eliminated variable. Let us try an example to understand it clearly.
Let us assume two equations as: -
\[\Rightarrow 3x+4y=20\] - (1)
\[\Rightarrow 2x+3y=12\] - (2)
Here, let us eliminate the variable y. We can see that the coefficients of the variable y in both the equations are not the same so let us make them equal. Multiplying equation (1) with 3 and equation (2) with 4 and then adding we get,
\[\begin{align}
  & \Rightarrow \left( 9x+12y \right)-\left( 8x+12y \right)=60-48 \\
 & \Rightarrow x=12 \\
\end{align}\]
So we have obtained the value of x therefore substituting this value in equation (1) we get,
\[\begin{align}
  & \Rightarrow 3\times 12+4y=20 \\
 & \Rightarrow 36+4y=20 \\
 & \Rightarrow 4y=20-36 \\
 & \Rightarrow y=-4 \\
\end{align}\]
Hence, the solution of the given system of equations is given as \[\left( x,y \right)=\left( 12,-4 \right)\].

Note: Note that there are two other methods also to solve a system of linear equations namely: the substitution and the cross multiplication method. The use of these methods depends on the situation. Sometimes it is better to use the substitution or the cross multiplication method instead of elimination of variables. You cannot add or subtract the two equations before making the coefficient of one of the variables equal in the two equations.