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# Is a dependent pair of linear equations always consistent why or why not?

Last updated date: 19th Jul 2024
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Hint: In words, it is where the two graphs intersect, what they have in common. So if an ordered pair is a solution to one equation, but not the other, then it is NOT a solution to the system.
A consistent system is a system that has at least one solution.
An inconsistent system is a system that has no solution.

Complete step-by-step solution:
To solve this type of word problems of linear equations,
Let us consider
${a_1}x + {b_1}y + {c_1} = 0$and ${a_2}x + {b_2}y + {c_2} = 0$ be two linear equations.
And here if,
$\Rightarrow \dfrac{{{a_1}}}{{{a_2}}} \ne \dfrac{{{b_1}}}{{{b_2}}}$ then the equation is always consistent and every consistent solution has always at least one solution.
The dependent pair of linear equations are always consistent because the other name of the dependent pair if equations is only coincidental lines which are consistent and the intersecting lines also consistent so the dependent pair of equations are always consistent.
Therefore, every dependent pair of linear equations are always consistent.

Note: $\Rightarrow$ If ${a_1}x + {b_1}y + {c_1} = 0$ and ${a_2}x + {b_2}y + {c_2} = 0$ be two linear equations.
But, $\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} \ne \dfrac{{{c_1}}}{{{c_2}}}$ this type of condition happens then, the equation is inconsistent and that equation has no solution.