Chapter 2 Solving Linear Equations Mathematically Speaking Can you identify what happens in each step 15x 13y 4 3x 2y 15x 13y 12x 8y 15x 12x 13y 8y 15 12 x 13 8 y 3x 5y Can you identify what has happened in each step Given 15x 13y 4 3x 2y 15x 13y 12x 8y Distributive 15x 12x 13y 8y Commutative Factor 15 12 x 13 8 y Addition 3x 5y Identify the steps used to solve the equation m 4 29 m 4 29 4 4 m 25 Given Inverse Evaluate Identify the steps used to solve the equation 3x 4 19 4 4 3x 15 3 3 x 5 Given Inverse Evaluate Inverse Evaluate Identify the steps used to solve the equation 5x 4 2 x 4 18 5x 4 2x 8 18 5x 4 2x 10 3x 14 14 x 3 Identify the steps used to solve the equation 5x 4 2 x 4 18 5x 4 2x 8 18 5x 4 2x 10 3x 14 14 x 3 Given Distributive Addition Inverse Ops Like terms Inverse Ops Identify the steps used to solve the equation 5x 3 2x 7x 8 9x 3x 3 16x 8 11 19x 11 x 19 11 x 19 Identify the steps used to solve the equation 5x 3 2x 7x 8 9x Given 3x 3 16x 8 Like Terms 11 19x Inverse Ops 11 Inverse Ops x 19 11 Symmetric x 19 property So what is the definition Which of these equations are linear Not Linear Linear x y 5 2x 3y 4 7x 3y 14 y 2x 2 3 y 4 x2 y 5 x 5 y xy 5 x2 y2 9 y x2 The 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 What 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 There is no possible solution INCONSISTENT With linear equations this means there is no point of intersection 2 2 One linear equation in one variable One Solution 3x 4 19 4 4 3x 15 3 3 x 5 Infinite Solutions IDENTITY 14 5x 4 x 4x 8 18 14 5x 4 5x 8 18 5x 10 5x 10 10 10 No Solution INCONSISTENT 7x 3 1x 2x 8 8x 6x 3 6x 8 3 8 3 8 2 3 Several linear equations in one variable Systems of Equations Solving systems of equations with two or more linear equations Substitution Elimination Cramer s Rule Graphical Representation The 3 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 There 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 5 3x 4y 13 Substitute the first equation into the second 3x 4 2x 5 13 Solve for the variable 3x 8x 20 13 11x 33 x 3 Substitute back into one of the original equations y 2 3 5 1 Final Answer 3 1 Elimination 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 manipulated Some systems both equations must be manipulated Setup to Eliminate Given 2x 4y 8 3x 4y 2 The y terms are opposites they will eliminate Add the two equations 5x 10 x 2 Substitute into an original equation 3 2 4y 2 6 4y 2 4y 4 y 1 Final Answer 2 1 Manipulate ONE eqn to Eliminate Given 2x 2y 8 3x 4y 2 Multiply the first equation by 2 to elim y terms 4x 4y 16 3x 4y 2 Add the two equations 1x 14 x 14 Substitute into an original equation 3 14 4y 2 42 4y 2 4y 40 y 10 Final Answer 14 10 Manipulate BOTH eqns to Eliminate Given 2x 3y 4 3x 4y 2 Multiply the first equation by 3 the second equation by 2 to elim x terms 6x 9y 12 6x 8y 4 Add the two equations y 8 Substitute into an original equation 2x 3 8 4 2x 24 4 2x 20 x 10 Final Answer 10 8 Identity Example 2x 3y 12 y 2 3 x 4 Using substitution 2x 3 2 3 x 4 12 2x 2x 12 12 12 12 Identity Inconsistent Example 3x 4y 18 3x 4y 9 Use Elimination by multiplying Eqn 2 by 1 3x 4y 18 3x 4y 9 0 9 False Inconsistent 3 Equations 3 Variables required Eqn1 3y 2z 6 Eqn2 2x z 5 Eqn3 x 2y 8 Solve Eqn2 for z z 2x 5 Now substitute into Eqn1 3y 2 2x 5 6 3y 4x 10 6 3 Equation continued NEW 4x 3y 16 Eqn3 x 2y 8 Now one can either substitute or eliminate NEW 4x 3y 16 Eqn3 4 4x 8y 32 5y 16 y 16 5 And still continued Now having a value for y one can substitute into x 2 16 5 8 x 8 32 5 40 5 32 5 x 72 5 This can now be substituted into our Eqn2 solved for z z 2 72 5 5 z 144 5 5 144 5 25 5 z 119 5 Final Answer 72 5 16 5 119 5 Matrices Cramer s Rule Dimensions row x columns a c b d Determinant ad bc e f Cramer s Rule set up e x f b d determinant y a b e f determinant Example 2x 3y 5 4x 5y 7 2 4 3 5 5 7 The determinant is 10 12 2 x 5 7 3 5 2 y 2 4 5 7 2 Solve for x and y 5 3 2 x setup y setup 7 5 4 2 25 21 2 4 2 x 2 5 7 2 14 20 2 6 2 y 3 Final answer 2 3 You cannot use Cramer s Rule if the difference of the products is 0 Verbal Models Verbal Models are math problems written in word form General Rule Like reading English Left to Right Special Cases Change in order terms some time called turnaround words Cliff Notes Math Word Problems 2004 Convert into Math Two plus some number A number decreased by three Nine into thirty six Seven cubed Eight times a number Ten more than five is what number 2 x x 3 36 9 7 3 73 8x 5 10 x MORE Convert into Math Twenty five percent of what number is twentytwo The quantity of three times …