Question

Solve the following pair of equations by elimination method
$\begin{align}
  & 3x+2y=11 \\
 & 2x+3y=4 \\
\end{align}$

Answer 1

Hint: In this question, there are two variables x and y and two equations, therefore in elimination method we have to equalize the coefficients of one variable by making the two coefficients of a variable as their Lowest Common Multiple (LCM) by multiplying the equations by suitable factors. Then, the chosen variable can be eliminated through subtraction of one equation from the other and the value of the other variable can be calculated. Using this value, we can get the value of the firstly chosen variable.

Complete step-by-step answer:

We are given two equations in two variables.
$\begin{align}
  & 3x+2y=11..................(1.1) \\
 & 2x+3y=4....................(1.2) \\
\end{align}$
Now, in the method of elimination, first we have to equalize the coefficients of one variable so as to eliminate it.
Let us choose x for this. Coefficients of x are 3(in 1.1) and 2(in 1.2). Lowest Common Multiple (LCM) of 2 and 3 is 6. Now,
$eq(1.1)\times 2=2(3x+2y=11)=6x+4y=22..............(1.3)$
$eq(1.2)\times 3=3(2x+3y=4)=6x+9y=12................(1.4)$
Now subtracting eq (1.3) from eq (1.4), we get
$\begin{align}
  & (6x+9y=12)-(6x+4y=22) \\
 & \Rightarrow 6x+9y-6x-4y=12-22 \\
 & \Rightarrow 5y=-10 \\
 & \Rightarrow y=-2..........................(1.5) \\
\end{align}$
Now, we can put the obtained value of y in either of the equations (1.1) or (1.2) and get the value of x.
Putting value of y from eq(1.5) in eq(1.1), we get
$\begin{align}
  & 3x+2y=11 \\
 & \Rightarrow 3x+2(-2)=11 \\
 & \Rightarrow 3x=11+4 \\
 & \Rightarrow x=\dfrac{15}{3}=5.....................(1.6) \\
\end{align}$
Therefore, from this we get the required values of x and y as 5 and (-2) respectively.

Note: We could also have obtained the value of x by putting the value of y equation (1.2) and it would have given the same value of x. However, while subtracting the equations we must be careful to change the sign of the equation to be subtracted and add it to the other equation.