document.write( "Question 1153197: Bacteria grow in colonies over time. Which table shows growth of bacteria colonies that can be modeled by a linear graph?
\n" ); document.write( "A.Time (in hours) 1,2,3,4
\n" ); document.write( "Number ofcolonies 1,4,9,16
\n" ); document.write( "B. Time (in hours) 1,2,3,4
\n" ); document.write( "Numbers of colonies 5,10,15,20
\n" ); document.write( "C.Time (in hours) 1,2,3,4
\n" ); document.write( "Numbers of colonies 1,16,81,256
\n" ); document.write( "D. Time (in hours) 1,2,3,4
\n" ); document.write( "Number of colonies 5,25,125,625
\n" ); document.write( "

Algebra.Com's Answer #775380 by MathLover1(20850)\"\" \"About 
You can put this solution on YOUR website!
\r
\n" ); document.write( "\n" ); document.write( " growth of bacteria colonies that can be modeled by a linear graph?\r
\n" ); document.write( "\n" ); document.write( "first determine if \"3\" points lie in a line\r
\n" ); document.write( "\n" ); document.write( "A.Time (in hours) \"1\",\"2\",\"3\",\"4\"
\n" ); document.write( "Number of colonies \"1\",\"4\",\"9\",\"16\"\r
\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "\n" ); document.write( "
Solved by pluggable solver: To determine if 3 points lie in a line
The 3 points lie on a same plane. For all points to lie on a line they should satisfy the equation of a line. Hence any two points taken on a line should calculate to the same slope of a line.
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " In order to prove the 3 points to lie on a line, as there exists a unique line containing three points and every line has a unique slope.
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " Hence it will be sufficient to prove that the slope calculated taking 2 points at a time should be equal.
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " Slope of line taking points (X1,Y1) and (X2,Y2) is
\n" ); document.write( "
\n" ); document.write( " \"slope+=+%28Y2-Y1%29%2F%28X2-X1%29\"
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " \"slope+=+%28%284-1%29%2F%282-1%29%29+=+3\" ........................(1)
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " Slope of line taking points (X3,Y3) and (X1,Y1) is
\n" ); document.write( "
\n" ); document.write( " \"slope+=+%28Y3-Y1%29%2F%28X3-X1%29\"
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " \"slope+=+%28%289-1%29%2F%283-1%29%29+=+4\" ........................(2)
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " From conditions (1) and (2)
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " The 3 points do not a same line.
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " For all points to lie on a line they should satisfy the equation of a line. Hence any two points taken on a line should calculate to the same slope of a line.
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " Here the slopes are unequal hence the points do not lie on same line.
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " To read more on equations of a line refer to articles on wikipedia
\n" ); document.write( "
\n" ); document.write( "

\n" ); document.write( "\n" ); document.write( "The points do not a same line.=>not your answer\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "B. Time (in hours) \"1\",\"2\",\"3\",\"4\"
\n" ); document.write( "Numbers of colonies \"5\",\"10\",\"15\",\"20\"\r
\n" ); document.write( "\n" ); document.write( "It is proved that the \"points\" \"lie\" on \"same\" line\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "C.Time (in hours) \"1\",\"2\",\"3\",\"4\"
\n" ); document.write( "Numbers of colonies \"1\",\"16\",\"81\",\"256\"\r
\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "\n" ); document.write( "
Solved by pluggable solver: To determine if 3 points lie in a line
The 3 points lie on a same plane. For all points to lie on a line they should satisfy the equation of a line. Hence any two points taken on a line should calculate to the same slope of a line.
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " In order to prove the 3 points to lie on a line, as there exists a unique line containing three points and every line has a unique slope.
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " Hence it will be sufficient to prove that the slope calculated taking 2 points at a time should be equal.
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " Slope of line taking points (X1,Y1) and (X2,Y2) is
\n" ); document.write( "
\n" ); document.write( " \"slope+=+%28Y2-Y1%29%2F%28X2-X1%29\"
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " \"slope+=+%28%2816-1%29%2F%282-1%29%29+=+15\" ........................(1)
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " Slope of line taking points (X3,Y3) and (X1,Y1) is
\n" ); document.write( "
\n" ); document.write( " \"slope+=+%28Y3-Y1%29%2F%28X3-X1%29\"
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " \"slope+=+%28%2881-1%29%2F%283-1%29%29+=+40\" ........................(2)
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " From conditions (1) and (2)
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " The 3 points do not a same line.
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " For all points to lie on a line they should satisfy the equation of a line. Hence any two points taken on a line should calculate to the same slope of a line.
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " Here the slopes are unequal hence the points do not lie on same line.
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " To read more on equations of a line refer to articles on wikipedia
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "The points do not a same line.=>not your answer\r
\n" ); document.write( "\n" ); document.write( "D. Time (in hours) \"1\",\"2\",\"3\",\"4\"
\n" ); document.write( "Number of colonies \"5\",\"25\",\"125\",\"625\"\r
\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "\n" ); document.write( "
Solved by pluggable solver: To determine if 3 points lie in a line
The 3 points lie on a same plane. For all points to lie on a line they should satisfy the equation of a line. Hence any two points taken on a line should calculate to the same slope of a line.
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " In order to prove the 3 points to lie on a line, as there exists a unique line containing three points and every line has a unique slope.
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " Hence it will be sufficient to prove that the slope calculated taking 2 points at a time should be equal.
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " Slope of line taking points (X1,Y1) and (X2,Y2) is
\n" ); document.write( "
\n" ); document.write( " \"slope+=+%28Y2-Y1%29%2F%28X2-X1%29\"
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " \"slope+=+%28%2825-5%29%2F%282-1%29%29+=+20\" ........................(1)
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " Slope of line taking points (X3,Y3) and (X1,Y1) is
\n" ); document.write( "
\n" ); document.write( " \"slope+=+%28Y3-Y1%29%2F%28X3-X1%29\"
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " \"slope+=+%28%28125-5%29%2F%283-1%29%29+=+60\" ........................(2)
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " From conditions (1) and (2)
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " The 3 points do not a same line.
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " For all points to lie on a line they should satisfy the equation of a line. Hence any two points taken on a line should calculate to the same slope of a line.
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " Here the slopes are unequal hence the points do not lie on same line.
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " To read more on equations of a line refer to articles on wikipedia
\n" ); document.write( "
\n" ); document.write( "

\n" ); document.write( "\n" ); document.write( "The points do not a same line.=>not your answer\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "so, answer is B:\r
\n" ); document.write( "\n" ); document.write( "let find equation using point-slope formula:\r
\n" ); document.write( "\n" ); document.write( "\"y-y%5B1%5D=m%28x-x%5B1%5D%29\"\r
\n" ); document.write( "\n" ); document.write( "use two points to find a slope\r
\n" ); document.write( "\n" ); document.write( "\"m=%2810-5%29%2F%282-1%29=5\"\r
\n" ); document.write( "\n" ); document.write( "\"y-5=5%28x-1%29\"\r
\n" ); document.write( "\n" ); document.write( "\"y-5=5x-5\"\r
\n" ); document.write( "\n" ); document.write( "\"y=5x-5%2B5\"\r
\n" ); document.write( "\n" ); document.write( "\"y=5x\"\r
\n" ); document.write( "\n" ); document.write( "check the graph:\r
\n" ); document.write( "\n" ); document.write( "\r
\n" ); document.write( "\n" ); document.write( "
\n" ); document.write( "
\n" );