- WE WANT TO GRAPH THE LINE Y = 2/3X - 4
USING A TABLE OF VALUES
WHICH MEANS WE WANT TO SELECT VALUES OF X,
SUBSTITUTE THEM INTO THE EQUATION,
THEN DETERMINE THEIR CORRESPONDING Y VALUE.
EVERY TIME WE FIND AN X AND Y VALUE THAT SATISFY THE EQUATION,
IT GIVES US THE COORDINATES OF A POINT THAT WOULD BE ON THE LINE.
SO IN OUR TABLE, LET THE FIRST COLUMN BE OUR X VALUES
AND WE'LL LET THE SECOND COLUMN BE OUR Y VALUES.
NOTICE IN OUR EQUATION WE HAVE A FRACTION,
WE HAVE Y = 2/3 x X - 4.
SINCE IT IS EASIER TO PLOT POINTS ON THE COORDINATE PLANE
THAT DON'T CONTAIN FRACTIONS,
WE WANT TO SELECT AN X VALUE HERE
SO THIS PRODUCT WOULD NOT CONTAIN A FRACTION.
IF WE SELECT X VALUES THAT ARE MULTIPLES OF THE DENOMINATOR,
THIS PRODUCT WOULD NOT CONTAIN FRACTIONS.
IN THIS CASE WE WANT TO SELECT X VALUES THAT ARE MULTIPLES OF 3.
FOR EXAMPLE, 0, 3 AND 6 ARE MULTIPLES OF 3
AND THEREFORE WILL PRODUCE INTEGER VALUES FOR Y.
LET'S ALSO SELECT -3 AND -6. THESE ARE ALSO MULTIPLES OF 3.
NOW LET'S SEE HOW THIS WORKS OUT.
WE'LL FIRST SUBSTITUTE 0 FOR X, SO WE'D HAVE 2/3 x 0 - 4.
WELL THAT'S GOING TO BE = -4,
WHICH MEANS OUR LINE WOULD CONTAIN THE POINT
WITH AN X COORDINATE OF 0 AND A Y COORDINATE OF -4.
NOW WE'LL SUBSTITUTE 3 FOR X, SO WE'D HAVE 2/3 x 3,
BUT I'M GOING TO PUT 3/1 - 4
AND AGAIN NOTICE HOW THIS 3 AND THIS 3 SIMPLIFY TO 1
SO WE HAVE 2 - 4 = -2.
SO OUR LINE MUST CONTAIN A POINT
WITH AN X COORDINATE OF +3 AND A Y COORDINATE OF -2
AND NOW WE'LL SUBSTITUTE 6 FOR X
SO WE'LL HAVE 2/3 x 6 OR 6/1 - 4.
AGAIN NOTICE HOW THE 3 AND THE 6 SIMPLIFY,
THERE'S 1, 3 AND 3, AND TWO 3s AND 6.
SO THIS PRODUCT IS NOW JUST 4, SO 4 - 4 = 0.
SO THE LINE MUST CONTAIN THE POINT WITH AN X COORDINATE OF 6
AND A Y COORDINATE OF 0.
WHEN X = -3, WE'D HAVE 2/3 x -3/1 - 4
AND AGAIN NOTICE HOW WE HAVE A COMMON FACTOR OF 3 HERE.
THIS SIMPLIFIES THE 1, THIS SIMPLIFIES TO -1.
SO WE HAVE 2 x -1 THAT'S -2 - 4, SO WE HAVE -6.
SO THE LINE CONTAINS THE POINT WITH X COORDINATE -3,
Y COORDINATE -6.
THEN THE LAST X VALUE WE SELECTED WAS -6
SO WE'LL HAVE 2/3 x -6/1 - 4.
AGAIN, WE HAVE A COMMON FACTOR OF 3, THIS SIMPLIFIES TO 1,
THIS SIMPLIFIES TO -2, SO NOW WE HAVE -4 - 4 = -8.
AGAIN THIS WAS 2 x -2 IT'S -4 - 4, -8.
SO THE LINE MUST ALSO CONTAIN THE POINT WITH X COORDINATE -6,
Y COORDINATE -8.
NOW IF WE PLOT THESE FIVE POINTS,
IT SHOULD FORM A SINGLE LINE.
LET'S MAKE SURE THAT'S TRUE.
SO 0, -4 WOULD BE OUR Y INTERCEPT AT THIS POINT HERE.
3, -2, WE'D GO RIGHT THREE UNITS AND DOWN TWO UNITS,
6, 0 WOULD BE ON THE X AXIS HERE
AND THEN WE HAVE THE POINT -3, -6
SO LEFT 3 UNITS, DOWN 6 UNITS.
AND THE LAST POINT, -6, -8 LEFT 6, DOWN 8,
WE SHOULD BE RIGHT ABOUT HERE.
AND NOTICE HOW THESE 5 POINTS DO FORM A SINGLE LINE
WHICH DOES VERIFY THAT OUR WORK WAS CORRECT.
SO THE LINE Y = 2/3X - 4 WOULD BE THIS LINE HERE.
AND I DO WANT TO EMPHASIZE THAT THE REASON
WE DIDN'T HAVE ANY FRACTIONS HERE FOR OUR COORDINATES,
IS BECAUSE WE SELECTED X VALUES
THAT WERE MULTIPLES OF THE DENOMINATOR.
IN THIS CASE MULTIPLES OF 3.
SO FOR EXAMPLE,
IF WE WERE TRYING TO SKETCH THE LINE Y = 2/5X - 4,
WE'D WANT TO SELECT X VALUES THAT WERE MULTIPLES OF 5
TO AVOID FRACTIONAL VALUES FOR Y.