1Chapter 2Solving Linear EquationsMathematically Speaking15x + 13y – 4(3x+2y)15x + 13y – 12x - 8y15x – 12x + 13y - 8y(15 – 12)x + (13 – 8)y3x + 5yCan you identify what happens in each step?Can you identify what has happened in each step?15x + 13y – 4(3x+2y)15x + 13y – 12x - 8y15x – 12x + 13y - 8y(15 – 12)x + (13 – 8)y3x + 5y        2Identify the steps used to solve the equation, m + 4 = 29.m+4=29- 4=-4m =25 GivenInverse +  -EvaluateIdentify the steps used to solve the equation.3x + 4 = 19- 4 = - 43x = 15÷ 3= ÷ 3x = 5 GivenInverse + −EvaluateInverse * ÷EvaluateIdentify the steps used to solve the equation.5x – 4 = 2(x – 4) + 185x – 4 = 2x – 8 + 185x – 4 = 2x + 103x = 14314=x3Identify the steps used to solve the equation.5x – 4 = 2(x – 4) + 185x – 4 = 2x – 8 + 185x – 4 = 2x + 103x = 14314=x•GivenDistributiveAdditionInverse Ops Like termsInverse OpsIdentify the steps used to solve the equation.-5x + 3 + 2x = 7x – 8 + 9x-3x +3 = 16x -811 = 19xx=19111911=xIdentify the steps used to solve the equation.-5x + 3 + 2x = 7x – 8 + 9x-3x +3 = 16x -811 = 19xx=19111911=xLike TermsInverse OpsInverse OpsSymmetric property Given4So what is the definition? Which of these equations are linear?x+y = 52x+ 3y = 47x-3y = 14y = 2x-2y=4 x2+ y = 5x = 5xy = 5x2+y2= 9y = x23yLinearNot LinearThe degree must be one.2.1 What is a solution?What happens when one solves an equation?You might say “One gets an answer.”What is the format of that answer?What happens when one solves an equation?1. The solutions is a Unique solution.2. The solution is Infinite solutions. 3. The is no possible solution.5What happens when one solves an equation?1. The solution is a Unique solution.•There is only ONE numerical answer to solve the equation.2. The solution is Infinite solutions. •IDENTITY. The equations are mathematically equivalent.3. The is no possible solution.•INCONSISTENT. With linear equations this means there is no point of intersection.2.2 One linear equation in one variableOne Solution.3x + 4 = 19- 4 = - 43x = 15÷ 3= ÷ 3x = 56Infinite Solutions. IDENTITY14 + 5x – 4 = (x + 4x) + 1814 + 5x – 4 = 5x – 8 + 185x + 10 = 5x + 1010 = 10No Solution. INCONSISTENT-7x + 3 + 1x = 2x – 8 - 8x-6x +3 = -6x -83 = -883−≠2.3 Several linear equations in one variable7Systems of EquationsSolving systems of equations with two or more linear equationsSubstitutionEliminationGraphical RepresentationThe3 possible solutions still occur.1. The solution is a Unique solution.•This one solution is in the form of a point. (e.g. (x,y), (x,y,z) ) 2. The solution is Infinite solutions. •IDENTITY. The lines are the same line.3. The is no possible solution.•INCONSISTENT. The lines are parallel (2-D) or skew (3-D).Substitution – use substitution when…One of the equations is already solved for a variable.y = 2x – 53x + 4y = 13Substitute the first equation into the second3x + 4(2x – 5) = 13Solve for the variable3x + 8x – 20 = 1311x = 33x = 3Substitute back into one of the original equationsy = 2(3) – 5 = 1 8Elimination – use elimination when substitution is not set up.Elimination ELIMINATES a variable through manipulating the equations.Some equations are setup to eliminate.Some systems only one equation must be manipulatedSome systems both equations must be manipulatedSetup to EliminateGiven2x – 4y = 83x + 4y = 2The y terms are opposites, they will eliminateAdd the two equations5x = 10  x = 2Substitute into an original equation3(2) + 4y = 2  6 + 4y = 2 4y = -4  y = -1 Manipulate ONE eqn. to EliminateGiven2x + 2y = 83x + 4y = 2Multiply the first equation by – 2 to elim. y terms-4x – 4y = -163x + 4y = 2Add the two equations-1x = -14  x = 14Substitute into an original equation3(14) + 4y = 2  42 + 4y = 2 4y = -40  y = -10 9Manipulate BOTH eqns. to EliminateGiven2x + 3y = 43x + 4y = 2Multiply the first equation by 3 & the second equation by -2 to elim. x terms6x + 9y = 12-6x - 8y = -4Add the two equationsy = 8Substitute into an original equation2x + 3(8) = 4  2x + 24 = 4 2x = -20  x = -10  Identity Example2x + 3y = 12y = -2/3 x + 4Using substitution2x + 3(-2/3 x + 4) = 122x – 2x + 12 = 1212 = 12IdentityInconsistent Example3x – 4y = 183x – 4y = 9Use Elimination by multiplying Eqn 2 by -1.3x – 4y = 18-3x + 4y = 180 = 18 FalseInconsistent103 Equations: 3 Variables requiredEqn1: 3y – 2z = 6Eqn2: 2x + z = 5Eqn3: x + 2y = 8Solve Eqn2 for zz = -2x + 5Now substitute into Eqn13y – 2(-2x+5) = 63y + 4x – 10 = 63 Equation continued…NEW: 4x + 3y = 16Eqn3: x + 2y = 8Now one can either substitute or eliminateNEW: 4x + 3y = 16Eqn3(*-4): -4x - 8y = -32-5y = -16y = 16/5Now having a value for y, one can substitute into x + 2(-16/5) = 8x = 8 - 32/5x = 8/5This can now be substituted into our Eqn2 solved for zz = - 2(8/5) + 5 z = -16/5 + 5z = 9/5Final Answer(8/5, -16/5, 9/5)And still continued…11Matrices: Cramer’s RuleDimensions: row x columnsDeterminanta bc dad - bcExample2x + 3y = 54x + 5y = 7The determinant is 10-12 = -22 34 5575 37 52 54 7x setup y setup-2 -2Solve for x and y…4/-2x = -2-6/-2y = 35 37 52 54 7x setup y setup-2 -222125−−22014−−Final answer (-2,3)12You cannot use Cramer’s Rule if the difference of the products is 0.Verbal ModelsVerbal Models are math problems written in word formGeneral Rule: Like reading English -Left to RightSpecial Cases: Change in order terms some time called “turnaround” words (Cliff Notes: Math Word Problems, 2004)Convert into Math…Two plus some numberA number decreased by threeNine into thirty-sixSeven cubedEight times a numberTen more than five is what number2+xx-336 / 97^3 738x5 + 10 = x !13MORE Convert into Math…Twenty-five percent of what number is twenty-two?The quantity of three times a number divided by seven equals nine.The sum of two consecutive integer is 23. .25 * x = 22(3x)/7 = 9x + (x+1) = 23Work Problem.I can mow the yard in 5 hours. My husband can mow the yard in 2 hours. If we mowed together how