- WE WANT TO SOLVE THE SYSTEM OF EQUATIONS
BY USING THE SUBSTITUTION METHOD.
THE SUBSTITUTION METHOD IS AN ALGEBRAIC METHOD
FOR SOLVING A SYSTEM OF EQUATIONS.
THE SOLUTION TO A SYSTEM OF EQUATIONS IS AN ORDERED PAIR
OR AN X AND Y VALUE THAT WOULD SATISFY BOTH OF THE EQUATIONS.
AND ONE MORE THING WE SHOULD KEEP IN MIND IS THAT
WHEN SOLVING A SYSTEM OF EQUATIONS
IT'S POSSIBLE TO HAVE ONE UNIQUE SOLUTION
CONSISTING OF AN X AND Y VALUE,
BUT IT'S ALSO POSSIBLE TO HAVE NO SOLUTIONS
OR EVEN AN INFINITE NUMBER OF SOLUTIONS.
THE FIRST STEP IN THE SUBSTITUTION METHOD
IS TO SOLVE ONE OF THESE EQUATIONS
FOR ONE OF THE VARIABLES,
BUT NOTICE HOW THE SECOND EQUATION IS ALREADY SOLVED
FOR Y.
WE'RE GIVEN THAT Y = -X + 4.
SO WE'RE GOING TO USE THIS SECOND EQUATION
AND PERFORM SUBSTITUTION INTO THE FIRST EQUATION FOR Y.
AGAIN, SINCE Y = -X + 4,
WE CAN REPLACE THE Y IN THE FIRST EQUATION WITH -X + 4.
AND TO KEEP THIS ORGANIZED LET'S CALL THIS EQUATION ONE
AND WE'LL CALL THIS EQUATION TWO.
SO USING EQUATION TWO WE'LL SUBSTITUTE INTO EQUATION ONE.
SO WE'LL HAVE X MINUS--
INSTEAD OF 3 x Y WE'LL NOW HAVE 3 x -X +4 = -24.
NOTICE HOW BY DOING THIS WE HAVE ONE EQUATION WITH ONE VARIABLE
AND NOW WE CAN SOLVE FOR X.
SO THE FIRST STEP HERE IS TO CLEAR THE PARENTHESIS
BY DISTRIBUTING.
SO WE'LL HAVE X--YOU CAN THINK OF THIS AS DISTRIBUTING A -3.
SO -3 x -X = +3X.
SO WE HAVE +3X,
AND THEN SINCE - 3 x +4 = -12 WE'LL WRITE -12.
X + 3X IS THE SAME AS 1X + 3X. SO HERE WE HAVE 4X - 12 = -24.
NEXT STEP HERE WE'LL ADD 12 TO BOTH SIDES.
SO WE HAVE 4X = -12, DIVIDE BOTH SIDES BY 4
AND WE HAVE X = -3.
NOW WE NEED TO BE CAREFUL HERE BECAUSE WE'RE NOT DONE.
REMEMBER THE SOLUTION CONSISTS OF AN ORDERED PAIR
OR AN X AND Y VALUE,
BUT NOW THAT WE KNOW THAT X MUST BE EQUAL TO -3
WE CAN PERFORM SUBSTITUTION INTO EQUATION TWO
TO DETERMINE THE VALUE OF Y.
USING EQUATION TWO WE KNOW THAT Y MUST EQUAL -X, BUT X IS -3.
SO WE'LL HAVE -(-3) + 4.
WELL, THE OPPOSITE OF -3 OR -(-3) IS +3.
SO WE HAVE Y = 7.
SO OUR SOLUTION WOULD BE THE ORDERED PAIR, (-3,7).
OR IF WE WANTED TO WE COULD LIST THIS AS X = -3 AND Y = +7.
SO WE HAVE ONE UNIQUE SOLUTION
AND THEREFORE WE CAN SAY THAT THIS SYSTEM IS CONSISTENT
AND THE EQUATIONS ARE INDEPENDENT.
BUT THE WHOLE IDEA HERE IS THAT X = -3 AND Y = +7
WILL SATISFY BOTH OF THESE EQUATIONS.
SO LET'S GO AHEAD AND JUST CHECK THAT
AND THEN WE'LL ALSO CHECK IT GRAPHICALLY.
SO USING EQUATION ONE, WITH X = -3 AND Y = 7,
WE WOULD HAVE -3 - 3 X 7 = -24.
THIS WOULD BE -3 - 21 = -24, WHICH IS TRUE.
SO THIS SHOWS IT SATISFIES THE FIRST EQUATION
AND THEN WE'LL CHECK EQUATION AS WELL.
WE HAVE Y = -X + 4, WHERE Y = 7 AND X = -3.
AND THIS WOULD BE 7 = 3 + 4 AND 7 = 7.
SO THAT'S CHECKS AS WELL.
SO THIS DOES VERIFY THAT OUR SOLUTION IS CORRECT.
REMEMBER WHEN WE SOLVE THE EQUATIONS GRAPHICALLY,
WE GRAPH BOTH LINES ON THE SAME COORDINATE PLANE
AND THIS WAS THE POINT OF OUR INTERSECTION.
SO LET'S GO AHEAD AND CHECK THIS GRAPHICALLY AS WELL.
THE BLUE LINE IS A GRAPH OF EQUATION ONE
AND THE RED LINE IS A GRAPH OF EQUATION TWO.
NOTICE THAT THEIR POINT OF INTERSECTION HERE
HAS AN X COORDINATE OF -3 AND A Y COORDINATE OF +7,
WHICH, AGAIN, VERIFIES OUR SOLUTION.
OKAY, I HOPE YOU FOUND THIS HELPFUL.