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## Area 3.2 Explaining Frameworks of Mathematical statements Logarithmically

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**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**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).**Solving Systems of Linear EquationsUsing the Substitution**Method**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**if a=b and c=d then a+c=b+dUsing the Elimination(Addition)**Method**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**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**0 = 4 untrueInconsistent Systems - how can you tell?**• An inconsistent system has no solutions. (parallel lines) Substitution Technique Elimination Technique**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**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**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°**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**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