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Question 36593: Why is it true that any two points satisfying a linear equation will give you the same graph for the line represented by the equation?
Please Help
Answer by venugopalramana(3286)   (Show Source):
You can put this solution on YOUR website! Why is it true that any two points satisfying a linear equation will give you the same graph for the line represented by the equation?
Please Help...WELL THE ANSWER IS IN YOUR QUESTION ITSELF!!LET ME ELABORATE...
FIRSTLY WE SHOULD UNDERSTAND WHAT IS A GRAPH,WHAT IS ITS EQUATION AND VICEVERSA..
AND WHAT IS MEANT BY SATISFYING AN EQUATION.
1.WE SAY A POINT SATISFIES AN EQUATION WHEN ITS COORDINATES ARE SUBSTITUTED IN THE EQUATION MAKES ITS RHS EQUAL ITS LHS.
2.WE SAY A POINT DOES NOT SATISFY AN EQUATION WHEN ITS COORDINATES ARE SUBSTITUTED IN THE EQUATION DOES NOT MAKE ITS RHS EQUAL ITS LHS.
NEXT....
1.WE SAY A GRAPH IS REPRESENTED BY AN EQUATION OR THE GIVEN EQN.IS FOR A PARTICULAR GRAPH,WHEN THE POINTS OBTAINED AS ABOVE FROM THE GIVEN EQN.,ARE PLOTTED AND JOINED
BY A LINE ,THEY FORM THE GRAPH UNDER CONSIDERATION
THESE APPEAR TO BE ELEMENTARY ,BUT THEY STIPULATE 4 IMPORTANT CONSIDERATION AT THE SAME TIME WHICH WILL ANSWER YOUR QUESTION...THEY ARE
GIVEN AS MENTIONED ABOVE AN EQUATION AND ITS GRAPH.....THEN....
1.IF A POINT SATISFIES THAT EQN. THEN IT LIES ON THE GRAPH.
2.IF A POINT DOES NOT SATISFY THAT EQN. THEN IT DOES NOT LIE ON THE GRAPH.
3.IF A POINT IS ON THE GRAPH,THEN IT SATISFIES THE EQN.
4.IF A POINT IS NOT ON THE GRAPH,THEN IT DOES NOT SATISFY THE EQN.
WITH THE ABOVE UNDERSTANDING LET US ANSWER YOUR DOUBT.
THE EQN. IS A LINEAR EQN.YOU WILL SOON LEARN THAT GRAPH OF A LINEAR EQN.IS A STRAIGHT LINE.
SO ONCE IT IS GIVEN THAT THE EQN.IS LINEAR IN X AND Y IT AUTOMATICALLY IMPLIES THAT THE GRAPH IS A STRAIGHT LINE IN PLANE GEOMETRY.
NOW WE KNOW THAT WE NEED ONLY 2 POINTS TO FIX A STRAIGHT LINE.
HENCE ONCE THE LINE IS FIXED BY THE 2 GIVEN POINTS SATISFYING THE GIVEN EQN.ANY OTHER POINT ON THE LINE SHOULD SATISFY THE EQN. AS ELABORATED ABOVE IN THE 4 STIPULATIONS.THAT IS THEY GIVE THE SAME STRAIGHT LINE WHATEVER 2 POINTS YOU TAKE.HENCE ANY 2 POINTS SATISFYING A LINEAR EQN. WILL GIVE YOU THE SAME GRAPH
FOR THE LINE REPRESENTED BY THE EQN. SINCE
THE GRAPH IS A STRAIGHT LINE FOR LINEAR EQN. AND ANY 2 POINTS ON THE LINE UNIQUELY DETERMINE THE LINE.
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