Which of the following pairs of linear equations are consistent/inconsistent?

Question.

Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically :

(i) $x+y=5.2 x+2 y=10$

(ii) $x-y=8,3 x-3 y=16$

(iii) $2 x+y-6=0,4 x-2 y-4=0$

(iv) $2 x-2 y-2=0,4 x-4 y-5=0$


Solution:

(i) $x+y=5$ ...(i)

$2 x+2 y=10 \quad \ldots$ ...(ii)

$\frac{a_{1}}{a_{2}}=\frac{1}{2}, \frac{b_{1}}{b_{2}}=\frac{1}{2}, \frac{c_{1}}{c_{2}}=\frac{-5}{-10}=\frac{1}{2}$

i.c., $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}$

4. Which of the following pairs

Hence, the pair of linear equations is consistent

(i) and (ii) are same equations and hence the graph

is coincident straight line.

Which of the following pairs

(ii) $x-y=8$ ....(i)

$3 x-3 y=16$ ....(ii)

$\frac{a_{1}}{a_{2}}=\frac{1}{3}, \frac{b_{1}}{b_{2}}=\frac{-1}{-3}=\frac{1}{3}, \frac{c_{1}}{c_{2}}=\frac{8}{16}=\frac{1}{2}$

$\Rightarrow \frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{z}}$

Therefore, lines have no solution
Hence, inconsistent

(iii) $2 x+y=6$ ....(i)

$4 x-2 y=4$ ....(ii)

$\frac{a_{1}}{a_{2}}=\frac{2}{4}=\frac{1}{2}, \frac{b_{1}}{b_{2}}=\frac{1}{-2}=\frac{-1}{2}, \frac{c_{1}}{c_{2}}=\frac{6}{4}=\frac{3}{2}$

$\Rightarrow \frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$

Therefore, lines have unique solution

Hence, consistent

from (i)
from (ii)

Which of the following pairs of linear

from graph $x=2, y=2$

(iv) $2 x-2 y=2$ ...(i)

$4 x-4 y=5$ ....(ii)

$\frac{a_{1}}{a_{2}}=\frac{2}{4}=\frac{1}{2}, \frac{b_{1}}{b_{2}}=\frac{-2}{-4}=\frac{1}{2}, \frac{c_{1}}{c_{2}}=\frac{2}{5}$

$\Rightarrow \frac{\mathrm{a}_{1}}{\mathrm{a}_{2}}=\frac{\mathrm{b}_{1}}{\mathrm{~b}_{2}} \neq \frac{\mathrm{c}_{1}}{\mathrm{c}_{2}}$

Therefore, lines have no solution.
Hence, Inconsistent.

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