- WE WANT TO SOLVE THE SYSTEM OF LINEAR EQUATIONS
BY GRAPHING.
TO SOLVE A SYSTEM OF EQUATIONS,
WE WANT TO FIND THE X AND Y VALUE
THAT SATISFY BOTH OF THE EQUATIONS,
NOT JUST ONE OF THEM.
AND TO DO THIS USING THE GRAPHING METHOD,
WE'LL GRAPH BOTH LINES ON THE SAME COORDINATE PLANE
AND THEN ANALYZE THE GRAPH TO DETERMINE THE SOLUTION.
LET'S REVIEW WHAT COULD HAPPEN
WHEN WE GRAPH TWO LINES ON THE SAME COORDINATE PLANE.
WELL, THE FIRST POSSIBILITY IS THE TWO LINES INTERSECT
AS WE SEE HERE.
AND SINCE THERE'S ONE POINT OF INTERSECTION,
WE HAVE ONE SOLUTION.
AND THE SOLUTION WOULD BE THE COORDINATES
OF THE POINT OF INTERSECTION.
THIS TYPE OF SYSTEM IS CONSISTENT,
AND THE EQUATIONS ARE INDEPENDENT.
IF WE GRAPH THE TWO LINES AND THEY'RE PARALLEL,
THERE'S NO POINTS OF INTERSECTION,
THEREFORE THERE'S NO SOLUTION.
AND THE SYSTEM IS INCONSISTENT,
AND THE EQUATIONS ARE INDEPENDENT.
THE LAST POSSIBILITY IS THE TWO LINES COINCIDE
OR THEY'RE ON TOP OF ONE ANOTHER,
SO THEY HAVE AN INFINITE NUMBER OF POINTS IN COMMON,
THEREFORE THERE'S AN INFINITE NUMBER OF SOLUTIONS.
AND WHEN THIS HAPPENS, THE SYSTEM IS CONSISTENT,
AND THE EQUATIONS ARE DEPENDENT.
SO GOING BACK TO OUR EXAMPLE,
LET'S GRAPH THE FIRST LINE IN BLUE.
WE HAVE X + Y = 4.
LET'S GO AHEAD AND MAKE A TABLE OF VALUES FOR THAT.
LET'S X = 0 AND MAYBE X = 2.
WELL, IF X IS ZERO, Y WOULD HAVE TO BE 4.
0 + 4 IS FOUR.
AND IF X IS 2, Y WOULD HAVE TO BE 2
BECAUSE 2 + 2 IS ALSO FOUR.
SO LET'S GO AHEAD AND PLOT THESE TWO POINTS.
HERE'S THE POINT (0,4).
HERE'S THE POINT (2,2).
LET'S GO AHEAD AND GRAPH THE FIRST LINE.
THEN THE SECOND LINE, WE HAVE 4X - 3Y = 9.
FOR THIS LINE, LET'S GO AHEAD
AND PUT IT IN IT SLOPE-INTERCEPT FORM
OR THE FORM Y = MX + B.
SO WE'LL TAKE THIS EQUATION HERE AND SOLVE FOR Y.
SO IF WE HAVE 4X - 3Y = 9, TO SOLVE FOR Y,
WELL, FIRST SUBTRACT 4X ON BOTH SIDES.
SO WE'D HAVE -3Y EQUALS-- WE'LL PUT THE X-TERM FIRST,
SO -4X + 9.
AND NOW WE'LL DIVIDE EVERYTHING BY -3.
SO WE'D HAVE THE EQUATION Y EQUALS--THIS WOULD BE +4/3 X.
AND THIS WOULD BE -3.
SO NOW WE'LL PLOT THE Y-INTERCEPT OF -3
AND THEN USE THE SLOPE OF 4/3
TO FIND ANOTHER POINT ON THE LINE.
SO HERE'S THE Y-INTERCEPT.
NOW WE'LL GO UP FOUR AND RIGHT THREE.
WE GO UP FOUR AND RIGHT THREE.
AGAIN, IT LOOKS LIKE WE JUST FOUND
THE POINT OF INTERSECTION HERE.
LET'S GO AHEAD AND SKETCH THE RED LINE.
AGAIN, BECAUSE THERE'S ONE POINT OF INTERSECTION,
WE HAVE ONE SOLUTION WHICH WOULD BE THE COORDINATES
OF THIS POINT OF INTERSECTION.
SO WE HAVE AN X-COORDINATE OF THREE
AND A Y-COORDINATE OF ONE.
SO THIS WOULD BE OUR SOLUTION, BUT IF WE WANTED TO,
WE COULD ALSO GIVE THE SOLUTION AS X = 3 AND Y = 1.
IF WE WANT TO CHECK THIS, WE CAN.
NOTICE IF WE SUBSTITUTE THREE FOR X AND ONE FOR Y,
IT SATISFIES EQUATION ONE.
AND THEN FOR EQUATION TWO,
WE'D HAVE 4 x 3--THAT'S 12-- MINUS 3 x 1 OR -3.
AND 12 - 3 IS EQUAL TO 9.
AND BECAUSE THIS SYSTEM HAS A SOLUTION,
WE CAN SAY IT'S CONSISTENT.
AND THEN BECAUSE THE EQUATIONS ARE DIFFERENT,
WE CAN SAY THE EQUATIONS ARE INDEPENDENT.
OKAY. I HOPE YOU FOUND THIS HELPFUL.