SYSTEMS OF LINEAR EQUATIONS WARM UP TAKE 10

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INTRODUCTION TO SYSTEM OF LINEAR EQUATIONS • A SYSTEM OF LINEAR EQUATIONS CONSISTS OF

• THE SOLUTION TO A SYSTEM OF EQUATION IS THE POINT OR POINTS

LET’S LOOK AT AN EXAMPLE- IS THE GIVEN ORDERED PAIR A SOLUTION TO THE

THERE ARE 3 WAYS YOU CAN SOLVE FOR A SYSTEM OF LINEAR EQUATIONS: 1.

WHEN SOLVING FOR A SYSTEM OF LINEAR EQUATIONS YOU CAN HAVE 3 DIFFERENT TYPES

TO SOLVE A SYSTEM OF LINEAR EQUATION BY GRAPHING I FIRST NEED TO •

• LOOKING AT THE ABOVE GRAPHS AND SOLUTIONS, I CAN TELL THAT THE

SOLVING SYSTEMS BY SUBSTITUTION: 1. SOLVE ONE EQUATION FOR EITHER VARIABLE (X= OR Y=)

2 X-3 Y=-24 X+6 Y=18 • SOLVE ONE EQUATION FOR EITHER VARIABLE (X= OR

SUBSTITUTE THE EXPRESSION FROM STEP ONE INTO THE SECOND EQUATION 2 X-3 Y=-24 X=18

SOLVE THE SECOND EQUATION FOR THE GIVEN VARIABLE 2 (18 -6 Y)-3 Y=-24 36

PLUG YOUR SOLUTION BACK INTO THE FIRST EQUATION X+6 Y=18 X+6(4)=18 X+24=18 X=-6 Y=4

WRITE YOUR SOLUTION AS AN ORDERED PAIR (-6, 4)

GO BACK AND CHECK YOUR SOLUTION IN BOTH EQUATIONS. 2 X-3 Y=-24 2(-6)-3(4)=-24 -12

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SYSTEMS OF LINEAR EQUATIONS

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INTRODUCTION TO SYSTEM OF LINEAR EQUATIONS • A SYSTEM OF LINEAR EQUATIONS CONSISTS OF 2 OR MORE LINEAR EQUATIONS THAT USE THE SAME VARIABLES. • EX: 2 X+3 Y=10 OR 5 A+2 B=14 • 4 X-7 Y=7 A+6 B=14

• THE SOLUTION TO A SYSTEM OF EQUATION IS THE POINT OR POINTS THAT MAKE BOTH OR ALL OF THE EQUATIONS TRUE. • REMEMBER THAT A POINT IS REPRESENTED BY AN ORDERED PAIR (X, Y)

LET’S LOOK AT AN EXAMPLE- IS THE GIVEN ORDERED PAIR A SOLUTION TO THE GIVEN SYSTEM? • No YES

THERE ARE 3 WAYS YOU CAN SOLVE FOR A SYSTEM OF LINEAR EQUATIONS: 1. GRAPHING 2. SUBSTITUTION 3. ELIMINATION

WHEN SOLVING FOR A SYSTEM OF LINEAR EQUATIONS YOU CAN HAVE 3 DIFFERENT TYPES OF SOLUTIONS: 1. ONE SOLUTION 2. NO SOLUTION 3. INFINITELY MANY SOLUTIONS

TO SOLVE A SYSTEM OF LINEAR EQUATION BY GRAPHING I FIRST NEED TO • CONVERT TO SLOPE INTERCEPT FORM • STATE M AND B FOR BOTH EQUATIONS • FIND POINT OF INTERSECTION • STATE THE ANSWER AS AN ORDERED PAIR

• LOOKING AT THE ABOVE GRAPHS AND SOLUTIONS, I CAN TELL THAT THE SYSTEM WILL HAVE AN INFINITE NUMBER OF SOLUTIONS WHEN THE LINES HAVE THE ______________________. THE SYSTEM WILL HAVE NO SOLUTIONS WHEN ______________________. THE SYSTEM OF EQUATIONS WILL HAVE ONE SOLUTION WHEN______________________ _.

• LOOKING AT THE ABOVE GRAPHS AND SOLUTIONS, I CAN TELL THAT THE SYSTEM WILL HAVE AN INFINITE NUMBER OF SOLUTIONS WHEN THE LINES HAVE THE SAME Y-INTERCEPT B, AND THE SAME SLOPE M. THE SYSTEM WILL HAVE NO SOLUTIONS WHEN THE LINES HAVE THE SAME SLOPE M, BUT DIFFERENT Y-INTERCEPTS B. THE SYSTEMS WILL HAVE ONE SOLUTION WHEN THE LINES HAVE DIFFERENT SLOPES M.

SOLVING SYSTEMS BY SUBSTITUTION: 1. SOLVE ONE EQUATION FOR EITHER VARIABLE (X= OR Y=) 2. SUBSTITUTE THE EXPRESSION FROM STEP ONE INTO THE SECOND EQUATION 3. SOLVE THE SECOND EQUATION FOR THE GIVEN VARIABLE 4. PLUG YOUR SOLUTION BACK INTO THE FIRST EQUATION

2 X-3 Y=-24 X+6 Y=18 • SOLVE ONE EQUATION FOR EITHER VARIABLE (X= OR Y=) • WHICH EQUATION IS THE EASIEST TO SOLVE FOR ONE VARIABLE?

2 X-3 Y=-24 X+6 Y=18 • SOLVE ONE EQUATION FOR EITHER VARIABLE (X= OR Y=) -6 y • X+6 Y=18 X=18 -6 y

SUBSTITUTE THE EXPRESSION FROM STEP ONE INTO THE SECOND EQUATION 2 X-3 Y=-24 X=18 -6 Y 2 (18 -6 y)-3 y=-24

SOLVE THE SECOND EQUATION FOR THE GIVEN VARIABLE 2 (18 -6 Y)-3 Y=-24 36 -12 y-3 y=-24 36 -15 y=-24 -15 y=-60 Y=4

PLUG YOUR SOLUTION BACK INTO THE FIRST EQUATION X+6 Y=18 X+6(4)=18 X+24=18 X=-6 Y=4

WRITE YOUR SOLUTION AS AN ORDERED PAIR (-6, 4)

GO BACK AND CHECK YOUR SOLUTION IN BOTH EQUATIONS. 2 X-3 Y=-24 2(-6)-3(4)=-24 -12 -12=-24 X+6 Y=18 (-6, 4) -24=-24 ✔ � -6+6(4)=18 -6+24=18 18=18 ✔ �