- WE WANT TO SOLVE THE SYSTEM OF EQUATIONS
USING THE SUBSTITUTION METHOD.
TO SOLVE A SYSTEM OF EQUATIONS MEANS TO DETERMINE
THE ORDERED PAIR, OR X AND Y VALUE,
THAT'S GOING TO SATISFY BOTH OF THE EQUATIONS.
WE LEARNED WHEN SOLVING SYSTEMS GRAPHICALLY
THAT IT'S POSSIBLE TO HAVE ONE UNIQUE SOLUTION
CONSISTING OF ONE X VALUE AND ONE Y VALUE.
BUT IT'S ALSO POSSIBLE TO HAVE NO SOLUTIONS,
OR EVEN AN INFINITE NUMBER OF SOLUTIONS.
LET'S REVIEW THE STEPS FOR SOLVING A SYSTEM
USING THE SUBSTITUTION METHOD, AND THEN WE'LL SOLVE THIS.
THE FIRST STEP IN THE SUBSTITUTION METHOD
IS TO SOLVE FOR A VARIABLE IN ONE OF THE EQUATIONS.
STEP TWO, WE SUBSTITUTE IN THE OTHER EQUATION
FOR THE VARIABLE ISOLATED.
AND THEN STEP THREE, WE SOLVE THE EQUATION
THAT WILL NOW CONTAIN ONE VARIABLE.
STEP FOUR, PERFORM BACK SUBSTITUTION
TO SOLVE FOR THE OTHER VARIATION.
AND THEN WE SHOULD CHECK OUR SOLUTION.
NOW, LOOKING AT OUR EXAMPLE,
NOTICE HOW THE SECOND EQUATION IS ALREADY SOLVED FOR Y.
SO WE CAN USE THIS AND PERFORM SUBSTITUTION,
AND DO THE FIRST EQUATION.
IF WE KNOW Y = 2X,
WE CAN REPLACE Y IN THE FIRST EQUATION WITH 2X.
AND THIS WILL GIVE US AN EQUATION WITH ONE VARIABLE.
SO LET'S CALL THIS EQUATION ONE AND THIS EQUATION TWO,
JUST TO KEEP THINGS ORGANIZED.
SO USING EQUATION TWO, WE'LL SUBSTITUTE INTO EQUATION ONE.
SO EQUATION ONE WOULD BE X + INSTEAD OF 2 x Y
WE'LL HAVE 2 x 2X.
AGAIN, Y = 2X.
AND THEN WE'LL HAVE = -20.
NOTICE HOW THE EQUATION NOW ONLY CONTAINS ONE VARIABLE,
SO WE CAN NOW SOLVE THIS EQUATION
TO DETERMINE THE VALUE OF X.
SO WE'LL HAVE X + 4X = -20.
AND X + 4X OR 1X + 4X = 5X.
DIVIDE BOTH SIDES BY 5, AND WE HAVE X = -4.
NOW WE NEED TO BE CAREFUL HERE, BECAUSE WE'RE NOT DONE.
REMEMBER THE SOLUTION CONSISTS OF ONE X VALUE AND ONE Y VALUE.
SO WE KNOW X = -4,
BUT NOW WE NEED TO PERFORM BACK SUBSTITUTION
INTO ONE OF THESE TWO EQUATIONS TO DETERMINE THE VALUE OF Y.
LET'S GO AHEAD AND USE EQUATION TWO TO DETERMINE THE VALUE OF Y.
WE KNOW Y = 2 x X, AND WE KNOW X = -4.
SO Y = 2 x -4, WHICH IS -8.
SO OUR SOLUTION IS X = -4 AND Y = -8,
WHICH CAN BE EXPRESSED AS AN ORDERED PAIR,
WHERE THE FIRST COORDINATE WOULD BE -4,
AND THE SECOND COORDINATE WOULD BE -8.
NOW, BECAUSE THIS IS A SOLUTION OF A SYSTEM,
IT SHOULD SATISFY BOTH OF THESE EQUATIONS.
AND LET'S GO AHEAD AND CHECK THAT.
WE'LL SUBSTITUTE -4 FOR X AND -8 FOR Y INTO EQUATION ONE
AND EQUATION TWO.
SO USING EQUATION ONE WE WOULD HAVE -4 + 2 x -8 = -20.
LET'S MAKE SURE THIS IS TRUE.
WE'D HAVE -4 + -16, AND -4 + -16 = -20,
SO THIS CHECKS.
WE ALSO NEED TO VERIFY THIS ORDERED PAIR
WOULD SATISFY EQUATION TWO, WHERE Y = 2X.
SO WE'D HAVE -8 = 2 x -4, WHICH ALSO CHECKS.
SO THIS VERIFIES THAT OUR SOLUTION IS CORRECT.
AND JUST TO MAKE THE CONNECTION
TO HOW WE SOLVE THE SYSTEM OF EQUATIONS GRAPHICALLY,
REMEMBER IF WE WERE TO GRAPH THESE TWO LINES
ON THE SAME COORDINATE PLANE,
THIS SHOULD BE OUR POINT OF INTERSECTION.
LET'S GO AHEAD AND TAKE A LOOK AT THAT.
HERE'S THE GRAPH OF EQUATION ONE.
HERE'S THE GRAPH OF EQUATION TWO.
AND NOTICE THAT OUR POINT OF INTERSECTION HERE
DOES HAVE THE COORDINATES OF (-4,-8),
WHICH AGAIN, IS THE SOLUTION TO OUR SYSTEM OF EQUATIONS.
WE'LL TAKE A LOOK AT ANOTHER EXAMPLE IN THE NEXT VIDEO.