If the number of solutions of $3x-y=2$ and $9x-3y=6$ equations are m, then find $\dfrac{1}{m}$. .
Answer
Verified
Compare the given equations with the general equation of linear equations. Check them with the conditions of consistency for linear equations.
“Complete step-by-step answer:” Let us consider the general linear equation ax + by + c = 0 and another equation mx + ny + d = 0. ax + by + c = 0 mx + ny + d = 0 Compare both the equation with the conditions of consistency for linear equations; (i) System of linear equations is consistent with unique solution if $\dfrac{a}{m}\ne \dfrac{b}{n}$ (ii) System of linear equation is consistent with infinitely many solutions if $\dfrac{a}{m}=\dfrac{b}{n}=\dfrac{c}{d}$ (iii) System of linear equation is inconsistent i.e., it has no solution if $\dfrac{a}{m}=\dfrac{b}{n}\ne \dfrac{c}{d}$ Let us consider 3x – y = 2, compare it with general equation, ax + by + c = 0 $\therefore $ a = 3, b = -1, c = -2 Compare ax – 3y = 6 with general equation mx + ny + d = 0. $\therefore $m = 9, n = -3, d = -6 Now check with all three conditions. $\begin{align} & \dfrac{a}{m}\ne \dfrac{b}{n}\Rightarrow \dfrac{3}{9}=\dfrac{a}{m} \\ & \therefore \dfrac{a}{m}=\dfrac{1}{3} \\ & \dfrac{b}{n}=\dfrac{-1}{-3}=\dfrac{1}{3} \\ \end{align}$ Where shows $\dfrac{a}{m}=\dfrac{b}{n}$ $\therefore $Condition not satisfied. (ii) $\dfrac{a}{m}=\dfrac{b}{n}=\dfrac{c}{d}$ $\dfrac{a}{m}=\dfrac{3}{9}=\dfrac{1}{3}$ $\begin{align} & \dfrac{b}{n}=\dfrac{-1}{-3}=\dfrac{1}{3} \\ & \dfrac{c}{d}=\dfrac{-2}{-6}=\dfrac{1}{3} \\ \end{align}$ $\therefore $This condition is satisfied. $(iii)\text{ }\dfrac{a}{m}=\dfrac{b}{n}\ne \dfrac{c}{d}$ We got $\dfrac{a}{m}=\dfrac{b}{n}=\dfrac{c}{d}$, so condition not satisfied. So in this case, condition 2 is true i.e., $\dfrac{a}{m}=\dfrac{b}{n}=\dfrac{c}{d}$; Hence, it has an infinite number of solutions. So m = infinity, hence $\dfrac{1}{m}=0$.
Note: Substitute values of a, b, c, m, n and d on each condition of consistency. If a system has at least 1 solution, it is consistent. If a consistent system has exactly 1 solution, it is independent. If a consistent system has an infinite number of solutions, it is dependent.
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