Question

The solution of the equation $2x - 3y = 7$ and $4x - 6y = 20$ is

Answer 1

Hint: These are two linear equations in two variables. These equations can be solved by various methods such as substitution method, elimination method, graphical method.

Complete step-by-step answer:
According to question, we have
$2x - 3y = 7$ …(1)
$4x - 6y = 20$ …(2)
As we can see the coefficients of equation (1) are $a_1 = 2$, $b_1 = - 3$ and $c_1 = -7$ whereas the coefficients of equation (2) are $a_2 = 4$, $b_2 = - 6$ and $c_2 = -20$.
Now taking the ratios of coefficients, then
$\dfrac{{a_1}}{{a_2}} = \dfrac{2}{4} = \dfrac{1}{2}$ … (3);
  $\dfrac{{b_1}}{{b_2}} = \dfrac{{ - 3}}{{ - 6}} = \dfrac{1}{2}$ …(4)
  and $\dfrac{{c_1}}{{c_2}} = \dfrac{-7}{{-20}} = \dfrac{7}{{20}}$ …(5)
From equation (3), (4) and (5) we see that
$\dfrac{{a_1}}{{a_2}} = \dfrac{{b_1}}{{b_2}} \ne \dfrac{{c_1}}{{c_2}}$
Hence it can be said the given lines are parallel to each other and they do not intersect at any point, hence no solution exists.

Note: It can be solved by other methods as well. Here we have used a graphical method by comparing the coefficients. It should be noted that,
If $\dfrac{{a_1}}{{a_2}} \ne \dfrac{{b_1}}{{b_2}} \ne \dfrac{{c_1}}{{c_2}}$, then lines are intersecting in nature. Hence, one solution exists.
If $\dfrac{{a_1}}{{a_2}} = \dfrac{{b_1}}{{b_2}} \ne \dfrac{{c_1}}{{c_2}}$, then lines are parallel to each other. Hence, no solution exists.
If $\dfrac{{a_1}}{{a_2}} = \dfrac{{b_1}}{{b_2}} = \dfrac{{c_1}}{{c_2}}$, then lines are coincident in nature. Hence, infinite solutions exist.