Section 3.2 Solving Systems of Equations Algebraically

1. Section 3.2 Solving Systems of Equations Algebraically • Use algebraic computation to get precise solutions • 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

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 you will learn two ways to solve linear systems algebraically (precision).

3. Solving Systems of Linear EquationsUsing the Substitution Method

4. if a=b and b=c then a=cSubstitution Method - Example • You can pick either variable to start – you will get the same (x,y) solution. Itmay take some work to isolate a variable: • Solve for (A)’s yor Solve for (A)’s x

5. if a=b and c=d then a+c=b+dUsing the Elimination(Addition) Method

6. if a=b then ca=cbElimination Method – Even Multiples • When one variable term is an evenmultiple of its matching variable term,you only have to multiply one equation. • Let’s eliminate y: Multiply (A) by -2

7. Elimination Method – No Multiples • When no variable terms are even multiples, you have to multiply bothequations by different numbers. • Let’s eliminate y: Multiply (A) by 7 and (B) by -3

8. Clearing Fractions or Decimals

9. Systems without a Single Point Solution

10. 0 = 4 untrueInconsistent Systems - how can you tell? • An inconsistent system has no solutions. (parallel lines) Substitution Technique Elimination Technique

11. 0 = 0 or n = n Dependent Systems – how can you tell? • A dependent system hasinfinitely many solutions. (it’s the same line!)Substitution Technique Elimination Technique

12. Application Problem (board work) TEMPORARY HELP A law firm had to hire several workers to help finish a large project. From the billing records shown in Illustration 3, determine the daily fee charged by the employment agency for a clerk-typist and for a computer programmer. The daily fees: $105 for a clerk-typist $185 for a programmer

13. Geometry Example (Board work) TRAFFIC SIGNAL In Illustration 7, brace A and brace B are perpendicular. Find the values of x and y. x is 150° y is 30°

14. Preparing for Test 1 • For Both Chapters (1 & 2): • Read over each Chapter Review (p.69 and p.172) • Look through each set of Review Exercises, bypassing the “easy” problems, doing some of the “hard” ones • Solving equations graphically won’t be on the test • The Test: 3:05pm-4:35pm • Be on time to class – no extra time for test if you are late • 90 minutes, 20 questions, 100 total points • Bring your own #2 pencils & calculator, I supply scratch paper • When finished, you may leave the room, but return by 4:45 • If you miss the test, you must take it in the Math Lab before our next class meeting or you will receive a test grade of 0 • All tests count toward your grade

15. Test 1 – the race is on! • Next Class: • Homework 3.1, 3.2 will be due • Test 1 3:05 Sharp(Chapters 1 and 2) • Section 3.3 –Solving Systems of 3 Equations