– WE WANT TO SOLVE THE SYSTEM

OF EQUATIONS USING THE SUBSTITUTION METHOD, WHICH IS AN ALGEBRAIC METHOD FOR

SOLVING A SYSTEM OF EQUATIONS. TO SOLVE A SYSTEM OF EQUATIONS

WE WANT TO FIND AN ORDERED PAIR OR AN X AND Y VALUE THAT WOULD

SATISFY BOTH OF THE EQUATIONS, BUT WE DO HAVE TO KEEP IN MIND THAT IT IS POSSIBLE THAT THERE

MAY BE NO SOLUTION OR EVEN AN INFINITE NUMBER

OF SOLUTIONS TO A SYSTEM OF EQUATIONS. TO KEEP THINGS ORGANIZED

LET’S CALL THIS EQUATION ONE AND WE’LL CALL

THIS EQUATION TWO. THE FIRST STEP

IN THE SUBSTITUTION METHOD IS TO SOLVE ONE OF THE EQUATIONS

FOR ONE OF THE VARIABLES. LUCKILY, IF WE TAKE A LOOK AT

EQUATION TWO IT’S ALREADY SOLVED FOR Y. WE HAVE Y=5/2X – 5. THE NEXT STEP

IN THE SUBSTITUTION METHOD IS TO USE THIS EQUATION HERE

THAT’S SOLVED Y AND PERFORM SUBSTITUTION

INTO EQUATION ONE, MEANING IF WE KNOW THAT

Y=5/2X – 5, WE CAN REPLACE Y IN THE FIRST

EQUATION WITH 5/2X – 5. DOING THIS WILL GIVE US AN

EQUATION WITH ONE VARIABLE. SO BY PERFORMING THIS

SUBSTITUTION, EQUATION 1 WOULD BE 5X – 2 x,

INSTEAD OF Y, WE’LL HAVE 2 x 5/2X – 5=10. AGAIN, NOTICE HOW WE HAVE AN

EQUATION WITH ONE VARIABLE, IN THIS CASE X. SO NOW WE’LL SOLVE THIS EQUATION

FOR X AND THEN PERFORM BACK

SUBSTITUTION TO DETERMINE THE VALUE OF Y. SO THE FIRST STEP HERE IS GOING

TO BE TO CLEAR THE PARENTHESIS. SO WE’LL DISTRIBUTE. SO WE’LL HAVE 5X. FOR THIS FIRST PRODUCT WE COULD

THINK OF THIS AS -2/1. NOTICE HOW THE 2s WOULD SIMPLIFY

OUT SO WE’D HAVE -5X. AND THEN FOR SECOND PRODUCT

WE CAN THINK OF THIS AS -2 x -5=+10. SO WE’D HAVE +10=10. NOTICE HOW WE DO HAVE LIKE TERMS

HERE AND 5X – 5X=0, SO WE’RE LEFT WITH 10=10. AND NOTICE HOW THERE IS NO

LONGER ANY VARIABLE TERMS. WE’RE LEFT WITH 10=10. WHEN THE VARIABLES SIMPLIFY OUT AND WE’RE LEFT WITH A TRUE

STATEMENT OR TRUE EQUATION, 10 IS ALWAYS EQUAL TO 10, THIS TELLS US WE HAVE AN

INFINITE NUMBER OF SOLUTIONS. IF THE VARIABLES SIMPLIFY OUT AND WE’RE LEFT WITH A FALSE

STATEMENT OR FALSE EQUATION, WE WOULD HAVE NO SOLUTIONS. IF YOU REMEMBER WHEN WE WERE

SOLVING EQUATIONS GRAPHICALLY, IF A SYSTEM HAD AN INFINITE

NUMBER OF SOLUTIONS THE TWO EQUATIONS PRODUCED

THE SAME LINE. SO LET’S GO AHEAD

AND TAKE A MOMENT AND LOOK AT THE GRAPH OF THESE

TWO EQUATIONS. WELL EQUATION ONE AND EQUATION

TWO DO PRODUCE THE SAME LINE, AND THEREFORE THEY HAVE

AN INFINITE NUMBER OF POINTS IN COMMON, WHICH DOES VERIFY THAT WE HAVE AN INFINITE NUMBER

OF SOLUTIONS.

NICE