Purchase MAT 104 - Chapter 1 Section 1- Linear Equations

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Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 241.1 Linear Equations Chapter 1 Section 1: Linear EquationsIn this section, we will… Solve linear equations Solve equations that lead to linear equations Solve applied problems involving linear equations1.1 Linear Equations: Solve Linear EquationsA linear equation in one variable (say, x) is any equation that can be written in the form: ax + b = c where a, b, c are real numbers and0.a �linear equation:expression:5123( 2)x x+ = +5123( 2)x x+ - +can be solvedcannot be solved (only simplified)What does it mean to solve an equation?1.1 Linear Equations: Solve Linear EquationsEquations with the same solution set are called equivalent equations.The following properties will be used to isolate the variable on one side of the equation. If any quantity is added to (or subtracted from) both sides of an equation, an equivalent equation is formed.If both sides of an equation are multiplied (or divided) by the same non-zero number, an equivalent equation is formed.If , , are real numbers and then: a b c a ba c b ca c b c=+ = +- = -If , , are real numbers , 0 then: a ca b c a b ca c b cbc= ��= �=Addition Prop. of EqualitySubtraction Prop. of EqualityMultiplication Prop. of EqualityDivision Prop. of Equality1.1 Linear Equations: Solve Linear EquationsSolving Linear Equations in One Variable:1. If the equation contains fractions, multiply both sides by the magic number (LCM of the denominators) to clear fractions2. Use the Distributive Property to remove the parentheses (then combine the like-terms on each side of the equation)3. Use the Addition and Subtraction Properties to get all of the variables on one side of the equation together (and all of the numbers on the other side of the equation)4. Use the Multiplication and Division Properties to make the coefficient of the variable equal to 15. Check your potential solution (by substituting your potential solution into the original equation to ensure that it satisfies it) If your potential solution does not check, you need to fix the error – do not leave the incorrect answer!Isolate the variable by “undoing” the operations:{1.1 Linear Equations: Solve Linear EquationsExamples: Solve the equation and check your result.0.5 0.5 0.02x= -checkoptional: we will clear this equation of decimals1.1 Linear Equations: Solve Linear EquationsExamples: Solve the equation and check your result.checkclear the parentheses first3( 2) 2 (5 )x x x- = - + +1.1 Linear Equations: Solve Linear EquationsExamples: Solve the equation and check your result.checkwe will clear this equation of fractions1 23 3 2 x x= -1.1 Linear Equations: Solve Linear EquationsExamples: Solve the equation and check your result.checkwe will clear this equation of fractions2 1 16 33xx++ =1.1 Linear Equations: Solve Linear EquationsExamples: Solve the equation and check your result.checkwe will clear this equation of fractions4 3 1 2 2 5 3t t t- +- = +1.1 Linear Equations: Solve Equations that Lead to Linear Equations Some equations lead to linear equations, once they have been simplified. We will study two types:Type #1: Polynomial Equations that Lead to Linear EquationsExamples: Solve the equation and check your result.check2( 2)( 3) ( 3)x x x+ - = +1.1 Linear Equations: Solve Equations that Lead to Linear Equations Examples: Solve the equation and check your result.check2 3(4 ) 8w w w- = -1.1 Linear Equations: Solve Equations that Lead to Linear Equations Type #2: Rational Equations that Lead to Linear Equations•we will first note any domain restrictions•factor all denominators•multiply both sides by the LCD of all of the rational expressions in the equation•divide out any common factors•solve the resulting linear equation•check your potential solutionWith a proportion, we can simply cross multiply. {Warning: Be sure to check your potential solution. A potential solution that fails to check can be a result of:(1)an incorrect solution that must be fixed or(2)an extraneous solution that must be discardedWhat is a proportion?1.1 Linear Equations: Solve Equations that Lead to Linear Equations Examples: Solve the equation and check your result.checknote:3 1 1 3 6x- =1.1 Linear Equations: Solve Equations that Lead to Linear Equations Examples: Solve the equation and check your result.checknote:2 6 23 3xx x-= -+ +1.1 Linear Equations: Solve Equations that Lead to Linear Equations Examples: Solve the equation and check your result.checknote:3 2 4 01 1x xx x- -- =- -1.1 Linear Equations: Solve Equations that Lead to Linear Equations Examples: Solve the equation and check your result.checknote:8 5 4 3 10 7 5 7y yy y+ -=- +1.1 Linear Equations: Solve Applied Problems Involving Linear EquationsSolving formulas for a specified variable = isolate the specified variableExamples: Solve each formula for the indicated variable.0 for v gt v t=- +(1 ) for A P rt r= +1.1 Linear Equations: Solve Applied Problems Involving Linear EquationsHow to Solve a Word Problem:Step 1: Read the problem until you understand it. • What are we asked to find? • What information is given?• What vocabulary is being used?Step 2: Assign a variable to represent what you are looking for. Express any remaining unknown quantities in terms of this variable.Step 3: Make a list of all known facts and form an equation or inequality to solve. It may help to make a labeled: diagram, table or chart, graphStep 4: SolveStep 5: State the solution in a complete sentence by mirroring the original question. Be sure to include units when necessary.Step 6: Check your result(s) in the words of the problem• Does your solution make sense?1.1 Linear Equations: Solve Applied Problems Involving Linear EquationsExamples: A total of $10,000 is to be divided between Peter and Lois, with Lois to receive $3,000 less than Peter. How much will each receive?1.1 Linear Equations: Solve Applied Problems Involving Linear EquationsExamples: Peter, who is paid time-and-a-half for hours worked in excess of 40 hours, had a gross weekly wages of $442 for 48 hours worked. What is his regular hourly rate?1.1 Linear Equations: Solve Applied Problems Involving Linear EquationsExamples: Going into the final


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