Question

Find the solution of the equation $x + 2y = 6$ from the options given below-
$A. (0, 3), (6, 0), (2, 2) and (4, 1)$
$B. (0, 2), (6, 0), (2, 2) and (4, 1)$
$C. (0, 3), (6, 0), (2, 2) and (3, 1)$
$D. (0, 3), (6, 0), (1, 2) and (4, 1)$

Answer 1

Hint: The solution(s) of an equation are those values which satisfy the equation when we substitute them into the equation. For example- $(0, 0, 0)$ is a solution of $x + y + z = 0$. Also, an equation can have zero to upto infinite roots, so we should consider all possible cases while solving.

Complete step-by-step answer:
We have been given the equation $x + 2y = 6$. We will substitute all the solutions given in the options and check which option has all the solutions which satisfy the equation.
Considering option $A. (0, 3), (6, 0), (2, 2) and (4, 1)$
At $(0, 3)$,
$x + 2y = 0 + 2(3) = 6$
At $(6, 0)$,
$x + 2y = 6 + 2(0) = 6$
At $(2, 2)$,
$x + 2y = 2 + 2(2) = 6$
At $(4, 1)$,
$x + 2y = 4 + 2(1) = 6$

Considering option $B. (0, 2), (6, 0), (2, 2) and (4, 1)$
At $(0, 2)$,
$x + 2y = 0 + 2(2) = 4$
At $(6, 0)$,
$x + 2y = 6 + 2(0) = 6$
At $(2, 2)$,
$x + 2y = 2 + 2(2) = 6$
At $(4, 1)$,
$x + 2y = 4 + 2(1) = 6$

Considering option $C. (0, 3), (6, 0), (2, 2) and (3, 1)$
At $(0, 3)$,
$x + 2y = 0 + 2(3) = 6$
At $(6, 0)$,
$x + 2y = 6 + 2(0) = 6$
At $(2, 2)$,
$x + 2y = 2 + 2(2) = 6$
At $(3, 1)$,
$x + 2y = 3 + 2(1) = 5$

Considering option $D. (0, 3), (6, 0), (1, 2) and (4, 1)$
At $(0, 3)$,
$x + 2y = 0 + 2(3) = 6$
At $(6, 0)$,
$x + 2y = 6 + 2(0) = 6$
At $(1, 2)$,
$x + 2y = 1 + 2(2) = 5$
At $(4, 1)$,
$x + 2y = 4 + 2(1) = 6$

We can see that only option A has all the four solutions satisfying the equation. In options B, C and D only three of the four are satisfying. Hence, the correct answer is option A.

Note: In such types of questions, we should always remember that one equation may have more than one or even zero solutions, so we should be open to all the possibilities. In this case, $x + 2y = 6$ can have an infinite number of real solutions. For example,
$(0, 3), (1, 2.5), (2, 2), (3, 1.5)$ and so on. As a general notation, the solutions are in the form $(x, y)$, which means that the first number is the value of x, the second is the value of y.