Question

The equations $ 3x - 5y + 2 = 0 $ and $ 6x + 4 = 10y $ have
A.No solutions
B.A single solution
C.Two solutions
D.An infinite number of solutions

Answer 1

Hint: Linear equations are defined for the lines of the coordinate system and represent straight lines. These equations are of the first order. By comparing the ratio of coefficients of variables and constants with each other, we can make many observations regarding the two equations. Using these ratios we can find out the correct answer.

Complete step-by-step answer:
If the ratio of the x-coefficients is equal to the ratio of y-coefficients and also to that of the constants of the two equations $ (\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}}) $ then it means that the two lines represented by that equations are coincident and thus have infinite solutions and the two linear equations are said to be dependent consistent.
The first equation is $ 3x - 5y + 2 = 0 $ that is $ {a_1} = 3,\,{b_1} = - 5\,and\,{c_1} = 2 $
The second equation is $ 6x - 10y + 4 = 0 $ that is $ {a_2} = 6,\,{b_2} = - 10\,and\,{c_2} = 4 $
So we get –
 $
\Rightarrow \dfrac{{{a_1}}}{{{a_2}}} = \dfrac{3}{6} = \dfrac{1}{2} \\
\Rightarrow \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{ - 5}}{{ - 10}} = \dfrac{1}{2} \\
\Rightarrow \dfrac{{{c_1}}}{{{c_2}}} = \dfrac{2}{4} = \dfrac{1}{2} \\
  $
We see that $ \dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}} $ which means these two equations have infinitely many solutions.
So, the correct answer is “Option D”.

Note: If the ratio of the x-coefficients of the two equations is not equal to the ratio of y-coefficients of the two equations $ (\dfrac{{{a_1}}}{{{a_2}}} \ne \dfrac{{{b_1}}}{{{b_2}}}) $ then it means the two lines represented by that equations intersect each other at one point and thus have exactly one solution or unique solution and the two linear equations are said to be consistent.