Question

Solve the following pair of linear equations by the elimination method:
\[\dfrac{x}{a} + \dfrac{y}{b} = a + b:{\text{ }}\dfrac{x}{{{a^2}}} + \dfrac{y}{{{b^2}}} = 2,{\text{ }}a \ne 0,{\text{ }}b \ne 0\]

Answer 1

Hint: We had to multiply any one equation by any constant such that the coefficient of x oy y became equal in both the equations. And after that we can find the value of one variable and then putting that value in another equation we will get the required value of another variable.

Complete step-by-step answer:
As we know that we are given two equations and have to find the value of x and y by elimination method.
So, equations are,
\[ \Rightarrow \dfrac{x}{a} + \dfrac{y}{b} = a + b\] -----(1)
\[ \Rightarrow \dfrac{x}{{{a^2}}} + \dfrac{y}{{{b^2}}} = 2\] --------(2)
Now according to the elimination method we had to first, multiply any one equation or both the given equations by some suitable non-zero constants to make the coefficient of any one of the variables (i.e. x or y) equal in both equations.
So, multiplying equation 1 by \[\dfrac{1}{a}\].
\[ \Rightarrow \dfrac{x}{{{a^2}}} + \dfrac{y}{{ab}} = \dfrac{{a + b}}{a}\] ----------(3)
Now we have to add or subtract one equation from another in such a way that one variable gets eliminated. So, we will get the equation of one variable.
So, now subtracting equation 2 from 3.
 \[ \Rightarrow \dfrac{x}{{{a^2}}} + \dfrac{y}{{ab}} - \dfrac{x}{{{a^2}}} - \dfrac{y}{{{b^2}}} = \dfrac{{a + b}}{a} - 2\]
\[ \Rightarrow \dfrac{y}{b}\left( {\dfrac{1}{a} - \dfrac{1}{b}} \right) = \dfrac{{a + b - 2a}}{a}\]
Now taking LCM on LHS and RHS of the above equation.
\[ \Rightarrow \dfrac{y}{b}\left( {\dfrac{{b - a}}{{ab}}} \right) = \dfrac{{b - a}}{a}\]
Now cross-multiplying the above equation.
\[ \Rightarrow y = {b^2}\]
Now we have to substitute the value of one variable we get to any of the given equations.
So, putting the value of y in equation 2.
\[ \Rightarrow \dfrac{x}{{{a^2}}} + \dfrac{{{b^2}}}{{{b^2}}} = 2\]
\[ \Rightarrow \dfrac{x}{{{a^2}}} = 1\]
Cross-multiplying above equation.
\[ \Rightarrow x = {a^2}\]
Hence, the solution of the equation \[\dfrac{x}{a} + \dfrac{y}{b} = a + b\] and \[\dfrac{x}{{{a^2}}} + \dfrac{y}{{{b^2}}} = 2\] will be \[x = {a^2}\] and \[y = {b^2}\].

Note: - Whenever we come up with this type of problem, if a particular method to solve the linear equations is given then we have to use only that method. Otherwise there are easier methods than elimination methods to solve linear equations. And one of them is a substitution method for that we had to take one of the equations and from that find the value of variable x in terms of y. And then put that value of x in another equation to find the value of y. Now we will get the value of x by putting the value of y in the previous equation of x. This will be the easiest and efficient method to find the solution of the problem if the method to be used is not given.