Question

On comparing the ratios, $\dfrac{{{a_1}}}{{{a_2}}}, \dfrac{{{b_1}}}{{{b_2}}}$, $and$ $\dfrac{{{c_1}}}{{{c_2}}}$ find out whether the following pair of linear equations are consistent or inconsistent:
(i). $3x + 2y = 5;2x - 3y = 7$
(ii). $2x - 3y = 8;4x - 6y = 9$
(iii). $\dfrac{3}{2}x + \dfrac{5}{3}y = 7;9x - 10y = 14$
(iv). $5x - 3y = 11; - 10x + 6y = - 22$
(v). \[\dfrac{4}{3}x + 2y = 8;2x + 3y = 12\]

Answer 1

Hint: First we have consider the linear equations will be consistent if $\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}} or \dfrac{{{a_1}}}{{{a_2}}} \ne \dfrac{{{b_1}}}{{{b_2}}}$.
A set of linear equations is said to be consistent, if there exists at least one solution for these equations. A set of linear equations is said to be inconsistent, if there are no solutions for these equations.
Finally we compared the equation and we will get the answer.

Complete step-by-step answer:
$3x + 2y = 5;2x - 3y = 7$
We can write it as in the form $3x + 2y - 5 = 0$, $2x - 3y - 7 = 0$
Linear equations are always in the form of ${a_1}x + {b_1}y + {c_1} = 0,{a_2}x + {b_2}y + {c_2} = 0$,
So we have to whether the linear equation is consistent,
We need to compare the variables of the given equation with the mentioned condition $\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}}or\dfrac{{{a_1}}}{{{a_2}}} \ne \dfrac{{{b_1}}}{{{b_2}}}$
   So, $\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{3}{2},\dfrac{{{b_1}}}{{{b_2}}} = - \dfrac{2}{3},\dfrac{{{c_1}}}{{{c_2}}} = \dfrac{5}{7}$
∴ $\dfrac{{{a_1}}}{{{a_2}}} \ne \dfrac{{{b_1}}}{{{b_2}}}$
Thus, this pair of linear equations is consistent and it has a unique solution and represents a pair of intersecting lines.
$2x - 3y = 8;4x - 6y = 9$
       We can write it as $2x - 3y - 8 = 0,4x - 6y - 9 = 0$
Linear equations are always in the form of ${a_1}x + {b_1}y + {c_1} = 0,{a_2}x + {b_2}y + {c_2} = 0$,
So to check whether the linear equation is consistent
We need to compare the variables of the given equation with the mentioned condition.
So, $\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{2}{4} = \dfrac{1}{2},\dfrac{{{b_1}}}{{{b_2}}} = \dfrac{3}{6} = \dfrac{1}{2}$,$\dfrac{{{c_1}}}{{{c_2}}} = \dfrac{8}{9}$
Since, $\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} \ne \dfrac{{{c_1}}}{{{c_2}}}$so the above linear equation will be inconsistent and do not have any solution and represents a pair of parallel lines.
$\dfrac{3}{2}x + \dfrac{5}{3}y = 7;9x - 10y = 14$
We can write it as, $\dfrac{3}{2}x + \dfrac{5}{3}y - 7 = 0,9x - 10y - 14 = 0$
Linear equations are always in the form of ${a_1}x + {b_1}y + {c_1} = 0,{a_2}x + {b_2}y + {c_2} = 0$,
So to check whether the linear equation is consistent. We need to compare the variables of the given equation with the mentioned condition $\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}}or\dfrac{{{a_1}}}{{{a_2}}} \ne \dfrac{{{b_1}}}{{{b_2}}}$
So, $\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{\dfrac{3}{2}}}{9} = \dfrac{1}{6},\dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{\dfrac{5}{3}}}{{ - 10}} = - \dfrac{1}{6},\dfrac{{{c_1}}}{{{c_2}}} = \dfrac{7}{{14}}$
 ∴$\dfrac{{{a_1}}}{{{a_2}}} \ne \dfrac{{{b_1}}}{{{b_2}}}$
Thus this pair of linear equation is consistent and has unique solution and represents a pair of intersecting lines.
$5x - 3y = 11; - 10x + 6y = - 22$
We can write it as $5x - 3y - 11 = 0, - 10x + 6y + 22 = 0$
Linear equations are always in the form of ${a_1}x + {b_1}y + {c_1} = 0,{a_2}x + {b_2}y + {c_2} = 0$,
So to check whether the linear equation is consistent. We need to compare the variables of the given equation with the mentioned condition $\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}}or\dfrac{{{a_1}}}{{{a_2}}} \ne \dfrac{{{b_1}}}{{{b_2}}}$

So, $\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{5}{{ - {{10}_{}}}} = - \dfrac{1}{2},\dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{ - 3}}{6} = - \dfrac{1}{2},\dfrac{{{c_1}}}{{{c_2}}} = - \dfrac{{11}}{{22}} = - \dfrac{1}{2}$
∴$\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}}$
Thus, this pair of linear equations is consistent and has infinite solutions and represents overlapping lines.
\[\dfrac{4}{3}x + 2y = 8;2x + 3y = 12\]
We have to write it as, $\dfrac{4}{3}x + 2y - 8 = 0,2x + 3y - 12 = 0$
Linear equations are always in the form of ${a_1}x + {b_1}y + {c_1} = 0,{a_2}x + {b_2}y + {c_2} = 0$,
So to check whether the linear equation is consistent. We need to compare the variables of the given equation with the mentioned condition $\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}}or\dfrac{{{a_1}}}{{{a_2}}} \ne \dfrac{{{b_1}}}{{{b_2}}}$
So, $\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{\dfrac{4}{3}}}{2} = \dfrac{4}{6} = \dfrac{2}{3},\dfrac{{{b_1}}}{{{b_2}}} = \dfrac{2}{3},\dfrac{{{c_1}}}{{{c_2}}} = \dfrac{{ - 8}}{{ - 12}} = \dfrac{2}{3}$
 ∴$\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}}$
This pair of linear equations is consistent and has infinite solutions and represents overlapping lines.

Note: For knowing whether a linear equation is consistent or not, check if the equation satisfy the condition of $\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}}or\dfrac{{{a_1}}}{{{a_2}}} \ne \dfrac{{{b_1}}}{{{b_2}}}$. If it satisfies any of the conditions then it is consistent.
If $\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}}$then the linear equation is consistent and has infinite solutions and will represent overlapping lines.
If $\dfrac{{{a_1}}}{{{a_2}}} \ne \dfrac{{{b_1}}}{{{b_2}}}$, then the linear equation has unique solution and represents a pair of intersecting lines.
If $\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} \ne \dfrac{{{c_1}}}{{{c_2}}}$, then the linear equation will be inconsistent and do not have any solution and represents a pair of parallel lines.