1. Section 3.2 Systems of Equations in Two Variables  Exact solutions by using algebraic computation  The Substitution Method (One Equation into Another)  The Elimination Method (Adding Equations)  How to identify Consistent Systems (one solution – lines cross) Inconsistent Systems (no solution – parallel lines) Dependent Systems (infinitely many solutions – same line)  Comparing the Methods 3.21

2. Definition Simultaneous Linear Equations Consider the pair of equations together 4x + y = 10 -2x + 3y = -12 Each line has infinitely many pairs (x, y) that satisfy it. But taken together, only one pair (3, -2) satisfies both. Finding this pair is called solving the system. In 3.1, you learned to solve a system of two equations in two variables by graphing (approximation). In this section we will learn to solve linear systems algebraically (precision). 3.22

3. Solving Systems of Linear Equations Using the Substitution Method 3.23

4. Substitution Method - Example  You can pick either variable to start, you will get the same (x,y) solution. It may take some work to isolate a variable:  Solve for (A)’s yorSolve for (A)’s x 3.24

5. Solving Systems of Linear Equations Using the Elimination (Addition) Method 3.25

6. Elimination Method – multiply 1  You can pick either equation to multiply. Sometimes you have to multiply both. It may take some work to match up terms:  Multiply A by -2to eliminate y 3.26

7. Elimination Method – multiply both  When multiplying both equations, pick the LCD of both coefficients of the same variable, and insure there are unlike signs:  Eliminate x: Multiply A by 5 and B by -2 (GCD = 10) 3.27

8. Special Cases 3.28

9. Inconsistent Systems - how can you tell?  An inconsistent system has no solutions. (parallel lines) Substitution Technique Elimination Technique 3.29

10. Dependent Systems – how can you tell?  A dependent system has infinitely many solutions. (same line) Substitution Technique Elimination Technique 3.210

11. Next  Section 3.3 – Applications: Systems of 2 Equations Section 3.3 3.211