Please open your laptops, log in to the MyMathLab course web site, and open Daily Quiz 18. You will have 10 minutes for today’s quiz. The second problem on today’s quiz is a graphing problem similar to those you did on the HW 18 worksheet. Do this second problem on the grid on the back of your answer sheet. Make sure you write the function equation from the online problem on your answer sheet. That problem will be graded by hand and the score added to your online score after class.

Please CLOSE YOUR LAPTOPS, and turn off and put away your cell phones, and get out your note- taking materials.

Weekly Quiz 5 tomorrow on HW 18, 19, 20, 21 This quiz is worth 10 points, will have 8 questions and a 30-minute time limit, and will be given after the lecture on section 4.5. The practice quiz has 12 questions and a 50- minute time limit. You can take the practice quiz as many times as you want, and only your best score will count towards your overall course grade. The practice quiz is a required 4-point assignment, and is due at the start of class tomorrow, along with HW 21.

Section 4.3 Solving Systems of Equations by Elimination

In addition to the graphing and substitution methods you learned in the last two sections, 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.)

Steps in solving a system of linear equations by the addition method, also known as the 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 either x or y are opposites. 3)Add the equations to eliminate the chosen variable. 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 SOLUTION: Multiply both sides of the first equation by 5 and the second equation by 3. (NOTE: It’s easier to eliminate y, since the coefficients already have opposite signs so we don’t need to multiply by any negatives. We would get the same solution, however, if we choose to multiply the first equation by 2 and the second one by -3 to eliminate the x terms.) 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) Example

Combine the two resulting equations (eliminating the variable y). 30x – 15y = x + 15y = x = -42 x = -1 (divide both sides by 42) Example (cont.)

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 (add 6 to both sides and simplify) y = -1 (divide both sides by -3) Our computations have produced the point (-1,-1). Example (cont.)

Check the point in the original equations. First equation, 6x – 3y = -3 6(-1) – 3(-1) = -3 true Second equation, 4x + 5y = -9 4(-1) + 5(-1) = -9 true The solution of the system is (-1, -1). Example (cont.)

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? ANSWER: Solve first eqn for y. What would be your first step in using the elimination method on this problem? ANSWER: Multiply first eqn by 5. Which method is easier to use on this problem? ANSWER: Probably elimination (fewer steps.) (2, -9)

Solve the following system of equations using the elimination method. First multiply both sides of each equation by a number that will clear the fractions out of the equation. Example

Multiply both sides of the first equation by the LCD of all the fraction denominators. LCD of 3, 4, and 2 is 12. Example (cont.) (multiply both sides by 12) (simplify both sides)

Now look at the two equations. 8x + 3y = -18 4x – 2y = x + 4y = 32 Can we get by with one multiplication 7y = 14 to eliminate a variable? y = 14/7 = 2 Which variable should we eliminate, and how? Example (cont.) Second equation: (multiply both sides by 8) (simplify both sides) -2*[4x – 2y = -16] Add the equations Solve for y

Substitute the value for y into one of the original equations. (For ease, I’ll use one that has been cleared of fractions.) 8x + 3y = -18 8x+ 3(2) = -18 8x+ 6 = -18 8x = = -24 x = -3 Our computations have produced the point (-3, 2). Example (cont.)

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.) Example (cont.) First equation, true Second equation, true The solution is the point (-3, 2).

Problem from 4.2 lecture: 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? (9,-7)

There are three types of answers you will encounter in the Section 4.3 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 statement such as “0 = 0” )

What if you end up with “y = 0” when solving a system? Example: 2x + 5y = 5 -2x + 2y = -5 7y = 0 y = 0/7 = 0 Now plug y = 0 back into one of the two original equations : 2x + 5·0 = 5  2x + 0 = 5  2x =5  x = 5/2 So the solution is (5/2, 0), a single ordered pair. Now check that ordered pair in the second equation: -2x + 2y = -5 -2·5/2 + 2·0 = = -5 checks!

Recap of methods for solving a system of linear equations: 1.Graphing method: Not usually the method of choice for getting an exact solution. 2.Substitution method: Works especially well when one equation is already solved for one of the two variables. 3.Elimination (addition) method: Works well for equations in standard form and for equations with fractional coefficients.

To solve each of the following systems: Which would be the easiest method to use? What would be the best first step to take? 1.4x + 7y = 8 and x = 2y – 1 2.2x + 3y = 10 and 2x – 3y = x + 3y = 10 and 5x + 6y = 7 4.4x + 7y = 8 and 3x + 19y = -12 Answer: Substitution. (Substitute 2y-1 in for x in 1 st equation.) Answer: Elimination. (Just add the two equations together as they are, and y will be eliminated.) Answer: Elimination. (Multiply 1 st equation by -2, then add to second equation to eliminate y.) Answer: Elimination. (We’ll have to do two multiplications no matter which variable we eliminate, but we can eliminate x using smaller numbers, so multiply first eqn. by 3 and second eqn. by -4, then add the resulting eqns. together to eliminate x.)

REMINDERS: The assignment on today’s material (HW 21) is due at the start of the next class session. Practice Weekly Quiz 5 is also due at the start of class tomorrow. Lab hours in 203: Mondays through Thursdays 8:00 a.m. to 7:30 p.m. Please remember to sign in on the Math 110 clipboard by the front door of the lab

You may now OPEN your LAPTOPS and begin working on the homework assignment. We expect all students to stay in the classroom to work on your homework till the end of the 55- minute class period. If you have already finished the homework assignment for today’s section, you should work ahead on the next one or work on the next practice quiz/test.