Question

Find the solutions of the form $x = a,y = 0$ and $x = 0,y = b$ for the following equations:
$2x + 5y = 10$ and $2x + 3y = 6$.

Answer 1

Hint: Substitute y = 0 in both the equations to check if x is constant. If it does, then (x = constant value, y = 0) is the required solution. Similarly, put x = 0 in both the equations to check if y is constant. Then the (x = 0, constant value of y) will be the required answer.

Complete step by step answer:
We have been given a pair of linear equations in 2 variables x and y.
$2x + 5y = 10$
$2x + 3y = 6$
The solution for these equations would be a pair of values, one for x and one for y, such that the left hand side equals the right hand side.
We will work on these equations by making appropriate substitutions to obtain the required solutions.
We know that geometrically, a linear equation represents a line.
Therefore, it can be understood that the no. of solutions of a linear equation = the no. of points satisfying the equation = the no. of points lying on the line.
This implies that a linear equation has infinitely many solutions because there are an infinite number of points lying on a line.
Let us number these equations $2x + 5y = 10$ and $2x + 3y = 6$ as (1) and (2) respectively.
In this question, we are asked to find only a particular type of solution.
For equation (1), the expected set of solutions must be of the form (i) \[x = a,y = 0\] and (ii) \[x = 0,y = b\]
(i)\[x = a,y = 0\]
Substitute y = 0 in equations (1) and (2)
From equation (1), we get 2x = 10 which implies that x = 5.
From equation (2), we get 2x = 6 which implies that x = 3.
As we can see the value of x is variable for equations (1) and (2), when the value of y is 0.
Therefore, we can conclude that the solution of the form \[x = a,y = 0\] does not exist for the given equations.

(ii)\[x = 0,y = b\]
Substitute x = 0 in equations (1) and (2)
From equation (1), we get 5y = 10 which implies that y = 2.
From equation (2), we get 3y = 6 which implies that y = 2.
Thus, the solution of the form \[x = 0,y = b\] for equations (1) and (2) is \[x = 0,y = 2\].

Note: We must refrain from solving the given equations simultaneously as it will not lead us anywhere close to the expected solutions. This would be a wrong method for solving such questions.