(i) Intersecting lines
(ii) Parallel lines
(iii) Coincident lines
Arrangements:
(I) Given the straight condition \[2x\text{ }+\text{ }3y\text{ }\text{ }8\text{ }=\text{ }0\] .
To track down one more direct condition in two factors with the end goal that the mathematical portrayal of the pair so framed is converging lines, it ought to fulfill beneath condition;
\[\left( a1/a2 \right)\text{ }\ne \text{ }\left( b1/b2 \right)\]
Consequently, another condition could be \[2x\text{ }\text{ }7y\text{ }+\text{ }9\text{ }=\text{ }0\] , with the end goal that;
\[\left( a1/a2 \right)\text{ }=\text{ }2/2\text{ }=\text{ }1\] and \[\left( b1/b2 \right)\text{ }=\text{ }3/\text{ }-\text{ }7\]
Obviously, you can see another condition fulfills the condition.
(ii) Given the straight condition \[2x\text{ }+\text{ }3y\text{ }\text{ }8\text{ }=\text{ }0\] .
To track down one more straight condition in two factors with the end goal that the mathematical portrayal of the pair so shaped is equal lines, it ought to fulfill beneath condition;
\[\left( a1/a2 \right)\text{ }=\text{ }\left( b1/b2 \right)\text{ }\ne \text{ }\left( c1/c2 \right)\]
Consequently, another condition could be \[6x\text{ }+\text{ }9y\text{ }+\text{ }9\text{ }=\text{ }0\] , with the end goal that;
\[\left( a1/a2 \right)\text{ }=\text{ }2/6\text{ }=\text{ }1/3\]
\[\left( b1/b2 \right)\text{ }=\text{ }3/9=\text{ }1/3\]
\[\left( c1/c2 \right)\text{ }=\text{ }-\text{ }8/9\]
Unmistakably, you can see another condition fulfills the condition.
(iii) Given the straight condition \[2x\text{ }+\text{ }3y\text{ }\text{ }8\text{ }=\text{ }0\] .
To track down one more straight condition in two factors with the end goal that the mathematical portrayal of the pair so shaped is incidental lines, it ought to fulfill underneath condition;
\[\left( a1/a2 \right)\text{ }=\text{ }\left( b1/b2 \right)\text{ }=\text{ }\left( c1/c2 \right)\]
Consequently, another condition could be \[4x\text{ }+\text{ }6y\text{ }\text{ }16\text{ }=\text{ }0\] , with the end goal that;
\[\left( a1/a2 \right)\text{ }=\text{ }2/4\text{ }=\text{ }1/2\] ,
\[\left( b1/b2 \right)\text{ }=\text{ }3/6\text{ }=\text{ }1/2,\]
\[\left( c1/c2 \right)\text{ }=\text{ }-\text{ }8/\text{ }-\text{ }16\text{ }=\text{ }1/2\] Obviously, you can see another condition fulfills the condition.