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Grade ScaleAny questions on the 4.1A homework problems?PowerPoint PresentationSection 4.1 BSlide 5Slide 6Slide 7Slide 8Slide 9Slide 10Problem from today’s homework:Problem from yesterday’s homework:Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19There are three types of answers you will encounter in the Section 4.1 homework problems, corresponding to the three different ways two lines can intersect:Slide 21Slide 22Slide 23Slide 24Recap of methods for solving a system of linear equations:To solve each of the following systems: Which would be the easiest method to use? What would be the best first step to take?Reminder:Slide 28Grade ScaleTest 1 Results:•Average class score after partial credit: __________•Commonly missed questions: #_________________•If you got less than 70% on Test 1, make sure to go over your quiz with me or a TA sometime today or tomorrow to help you prepare for tomorrow’s test.Any questions on the 4.1A homework problems?Now pleaseCLOSE YOUR LAPTOPSand turn off and put away your cell phones.Sample Problems Page Link(Dr. Bruce Johnston)Section 4.1 BSolving Systems of Equations by Elimination•In addition to the graphing and substitution methods you learned in the last section, a third method that can be used to solve systems of equations is called the addition or elimination method. (This method is probably the one you will use most often, so pay special attention to these next few slides!)•To use this method, you multiply one or both equations by numbers that will allow you to add the two equations together and eliminate one of the variables.•From that point on, the rest of the solution follows the same steps as the substitution method (solving for the one remaining variable, then plugging that back in to the original equations to get the value of the other variable.)Solving a system of linear equations by the addition or elimination method:1) Rewrite each equation in standard form, eliminating fraction coefficients.2) If necessary, multiply one or both equations by a number so that the coefficients of a chosen variable are opposites.3) Add the equations.4) Find the value of the remaining variable by solving the equation from step 3.5) Find the value of the second variable by substituting the value found in step 4 into either original equation.6) Check the proposed solution in the original equations. (ALWAYS do this, since it’s very easy to make one of those annoying arithmetic mistakes in these kinds of problems!)Solve the following system of equations using the elimination method.6x – 3y = -3 and 4x + 5y = -9•Multiply both sides of the first equation by 5 and the second equation by 3.First equation,5(6x – 3y) = 5(-3) 30x – 15y = -15 (use the distributive property)Second equation,3(4x + 5y) = 3(-9) 12x + 15y = -27 (use the distributive property)•Combine the two resulting equations (eliminating the variable y). 30x – 15y = -15 12x + 15y = -27 42x = -42 x = -1 (divide both sides by 42)•Substitute the value for x into one of the original equations. 6x – 3y = -3 6(-1) – 3y = -3 (replace the x value in the first equation) -6 – 3y = -3 (simplify the left side) -3y = -3 + 6 = 3 (add 6 to both sides and simplify) y = -1 (divide both sides by -3)•Our computations have produced the point (-1,-1).•Check the point in the original equations.First equation, 6x – 3y = -3 6(-1) – 3(-1) = -3 trueSecond equation, 4x + 5y = -94(-1) + 5(-1) = -9 true•The solution of the system is (-1, -1).Problem from today’s homework:Problem from yesterday’s homework:Note: This problem can also be solved by the elimination method.•What would be the first step in solving this problem with the substitution method?•What would be your first step in using the elimination method on this problem?•Which method is easier to use on this problem?Solve the following system of equations using the elimination method.24121234132yxyx• First multiply both sides of the equations by a number that will clear the fractions out of the equations.•Multiply both sides of each equation by 12. (Note: you don’t have to multiply each equation by the same number, but in this case it will be convenient to do so.)First equation,234132 yx2312413212 yx(multiply both sides by 12)1838  yx(simplify both sides)•Combine the two equations.8x + 3y = -186x – 3y = -2414x = -42 x = -3 (divide both sides by 14)Second equation,24121 yx 212412112  yx(multiply both sides by 12)(simplify both sides)2436  yx•Substitute the value for x into one of the original equations. (For ease, I’ll use one that has been cleared of fractions.) 8x + 3y = -188(-3) + 3y = -18 -24 + 3y = -18 3y = -18 + 24 = 6 y = 2•Our computations have produced the point (-3, 2).•Check the point in the original equations. (Note: Here you should use the original equations before any modifications, even though they involve fractions, to detect any computational errors that you might have made.)First equation,234132 yx23)2(41)3(3223212 trueSecond equation,24121 yx2)2(41)3(2122123true• The solution is the point (-3, 2).Problem from today’s homework:Problem from yesterday’s homework:Note: This problem can also be solved by the elimination method.•What would be the first step in solving this problem with the substitution method?•What would be your first step in using the elimination method on this problem?•Which method is easier to use on this problem?There are three types of answers you will encounter in the Section 4.1 homework problems, corresponding to the three different ways two lines can intersect:1. Intersection in a single point (Answer is an ordered pair.) (The two lines have different slopes)2. No common intersection (parallel lines) (Answer: N for No Solution) (The lines have the same slope, different y-intercepts; all variables drop out by elimination, leaving a false statement such as “0 = 3”)3. The two equations represent the same line, so the Intersection is all the points on the line (Answer: I for Infinitely many solutions) (Lines have same slope AND y-intercept; all variables drop out by elimination, leaving a true


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UW Stout MATH 110 - Lecture Notes

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