- WE WANT TO GRAPH THE LINEAR EQUATION X + 2Y = 6
USING A TABLE OF VALUES.
NOW ONE TECHNIQUE WOULD BE TO DETERMINE
THE X AND Y INTERCEPTS,
BUT FOR THIS VIDEO WHAT WE'RE GOING TO DO
IS SELECT SPECIFIC VALUES OF X
AND THEN DETERMINE THE CORRESPONDING VALUES OF Y
WHERE EACH X AND Y VALUE WOULD REPRESENT A POINT ON THE LINE.
SO IF WE'RE GOING TO SELECT VALUES OF X AS 7
IN THIS EQUATION
AND THEN DETERMINE Y,
IT'S OFTEN HELPFUL TO SOLVE THE EQUATION FOR Y.
SO LET'S START BY DOING THAT.
TO SOLVE FOR Y WE'D HAVE TO SUBTRACT X
ON BOTH SIDES OF THE EQUATION.
THIS WOULD BE 0.
WE ARE LEFT WITH 2Y = I'M GOING TO WRITE THIS AS -X + 6.
AND THEN THE LAST STEP TO SOLVE FOR Y
WOULD BE TO DIVIDE EVERYTHING BY 2.
SO WE'LL DIVIDE THIS BY 2, THIS BY 2, AND THIS BY 2.
SO NOW WE'LL SIMPLIFY
AND THE EQUATION WE'LL USE TO COMPLETE THE TABLE
IS GOING TO BE Y = INSTEAD OF -X OVER 2
I'M GOING TO WRITE THIS IS AS -1/2 x X
AND THEN 6 DIVIDED BY 2 IS = TO 3.
AND NOW LOOKING AT OUR TABLE OF VALUES,
THE FIRST COLUMN IS GOING TO B X
AND THE SECOND COLUMN WILL BE Y.
NOW IT IS EASIER TO GRAPH POINTS
WHEN THE COORDINATES ARE NOT FRACTIONS.
SO WE WANT TO SELECT VALUES OF X
SO THAT WHEN WE MULTIPLY HERE
THE PRODUCT WILL NOT CONTAIN FRACTIONS.
SO WE WANT TO SELECT X VALUES
THAT WOULD BE MULTIPLES OF THE DENOMINATOR
OR IN THIS CASE MULTIPLES OF 2.
SO WE COULD SELECT 0, 2, AND 4
BECAUSE THESE ARE ALL MULTIPLES OF 2.
LET'S ALSO SELECT -2 AND -4.
NOW, WE'LL SUBSTITUTE THESE VALUES INTO THE EQUATION
AND THEN DETERMINE THE CORRESPONDING VALUE OF Y.
SO WHEN X IS 0 WE'D HAVE -1/2 x 0 + 3,
THAT'S GOING TO BE = TO 3.
SO THE POINT WHERE THE X COORDINATE IS = TO 0
AND THE Y COORDINATE IS 3
WOULD BE A POINT ON THE GIVEN LINE.
NOW, WE'LL REPLACE X WITH 2
SO Y IS = TO -1/2 x 2 OR 2 OVER 1 + 3.
NOTICE HOW THE 2 SIMPLIFIED A 1
SO WE'RE LEFT WITH -1 + 3, THAT'S +2.
SO THE POINT WITH AN X COORDINATE OF 2
AND A Y COORDINATE OF 2 WOULD BE ON THE GIVEN LINE.
AND NOW WE'LL SUBSTITUTE 4 FOR X.
WE HAVE -1/2 x 4 OVER 1 + 3.
AGAIN THE 2 AND THE 4 SIMPLIFY.
2 SIMPLIFIES TO 1, 4 SIMPLIFIES TO 2
SO THIS PRODUCT WOULD BE -2 + 3, WHICH IS 1.
SO THE POINT WHEN THE X COORDINATE IS 4
AND THE Y COORDINATE IS 1 IS ON THE LINE.
NEXT WE HAVE X = -2.
SO WE HAVE -1/2 x -2 OVER 1 + 3.
HERE'S OUR COMMON FACTOR OF 2 HERE AND HERE.
SO THIS ENDS UP BEING -1 x -1, WHICH IS 1, + 3, WHICH IS 4.
SO THE LINE CONTAINS THE POINT
WHERE THE X COORDINATE IS -2 AND THE Y COORDINATE IS 4.
AND THEN FOR THE LAST X VALUE WE HAVE X = -4.
SO WE HAVE -1/2 x -4 OVER 1 + 3.
THIS SIMPLIFIES AGAIN.
THERE'S ONE FACTOR OF 2 IN 2 AND 2 FACTORS OF 2 IN 4.
THIS IS -1 x -2, THAT'S 2 + 3 THAT'S 5.
SO THE LINE CONTAINS THE POINT WITH AN X COORDINATE OF -4
AND A Y COORDINATE OF 5.
LET'S GO AHEAD AND PLOT THESE 5 POINTS
TO VERIFY THEY DO FORM A LINE.
0,3 WOULD BE HERE 2,2 RIGHT 2 UP 2,
4,1 RIGHT 4 UP 1, -2,4 LEFT 2 UP 4
AND -4,5 LEFT 4 AND UP 5.
NOTICE HOW THESE 5 POINTS DO FORM A LINE,
WHICH VERIFIES OUR WORK IS CORRECT.
THIS WOULD BE THE GRAPH OF OUR LINEAR EQUATION.
WHERE THE EQUATION OF THIS LINE IS X + 2Y = 6
OR SOLVE FOR Y WE HAD Y = - 1/2X + 3.
THESE 2 LINES ARE EQUIVALENT,
JUST WRITTEN IN A DIFFERENT FORM.
I HOPE THIS WAS HELPFUL.