System of 1 st Degree Equations with 2

System of 1 st Degree Equations with 2 Variables

Solving a system of equations (2 rules) to find the ordered pair (x, y) that is common to both rules • Example: To rent a car you are given 2 choices. Choice A Choice B Costs 0. 30¢ per km. Cost is $30 and then 0. 20¢ per km. Write the rules for each situation Choice A y = 0. 30 x Choice B y = 0. 20 x + 30

Question: How many km will you drive for the cost to be the same? Create 1 Table of Values for both situations # of km 0 100 200 300 400 500 Choice A 0 30 60 90 120 150 Choice B 30 50 70 90 110 130 If you drive 300 km, the cost will be the same for choices A and B

Graph this situation cost • Point of intersection of the 2 lines is called the SOLUTION SET (300, 90) x, y The point of intersection must match your table of values !! 0 1 50 # of km

What and where is the solution set? What? ? • It is the point where the value of x has the same value of y in both situations. Where? ? • It is found in the table of values. Look for the x that has the same y value below it. • It is found on the graph. Look to see where the two lines intersect.

Example #2 You have been given the choice of two options for a babysitting job. Option #1 Option #2 You will be paid $4 an hour. You will be paid $5 initially and then $3 an hour. After how many hours will it not matter which payment option you choose? 1 - write the rules(system of equations) 2 - make table of values 3 - draw the graph 4 - answer the question

Write out the rules for the two options Option #1 y = 4 x Option #2 y = 5 + 3 x Table of Values Hours 0 worked 1 2 3 4 5 6 7 Option 1 0 4 8 12 16 20 24 28 Option 2 5 8 11 14 17 20 23 26 Looking at the table of values, the solution set is found at (5, 20). This means that after working 5 hours, you would make $20 regardless of which option you choose.

Graphing to find solutions set • Pay Solution set (5, 20) 2 1 # hours

Workbook page 122 Activity #1 4306 then do page 124 question #1 a, c, d they give you the rules…you make the table of values and the graph.

Page 124 #1 a x 0 1 2 Y= 2 x +3 3 5 7 Y= -1 x +6 6 5 4 x 0 1 0. 5 Y=4 x +3 3 7 5 Y=8 x+1 1 9 5

Comparison Method Using the algebraic method is much easier and faster than table of values and graphing. Y = 0. 30 x Y = 0. 20 x + 30 We want to know when the y is the same…so we make the equations equal to each other 0. 30 x = 0. 20 x + 30 0. 30 x – 0. 20 x = 0. 20 x + 30 – 0. 20 x 0. 10 x = 30 0. 10 X = 300

Now that you have the x, how do you find the y ? ? ? To find the “y” substitute the answer you got for x into EITHER of the rules…pick the easier one ! Y= 0. 30 x Y = 0. 30 • 300 Y= 90 Now you have both answers for the solution set…. . (300, 90)

Example #2: Your family needs to choose between two vacations. Miami Costs $25 oo for airfare Costs $170 per night New Orleans $2950 airfare $140 per night Question #1 : How many nights would you stay for the cost to be the same? Question #2 : What would the cost be?

Write the system of equations to begin: Ym = 2500 + 170 x Yn = 2950 + 140 x 2500 + 170 x -140 x = 2950 +140 x -140 x 2500 + 30 x = 2950 2500 + 30 x – 2500 = 2950 -2500 Answers: 30 x = 450 30 30 Question 1 X = 15 days Question 2 = $5050 Y = 2500 + 170(15) Y= 2500 + 2550 Y = 5050

If you want to go for more than 15 nights…which city would be the less expensive? Miami for 20 nights Y = 2500 + 170(20) y-= 2500 + 3400 Y= 5900 New Orleans Y=2950 + 140(20) Y= 2950 + 2800 Y=5750

Math 4306 For next Tuesday morning you MUST have the following finished: Page 124 #1 a, b, c, d (use comparison notes) #2 Page 125 #3, 4 Page 127 #7 Be quiet and begin NOW you have 25 minutes to work on this. ‘translate this …” means write the 2 rules !

Workbook 396 -92 page 124 #1 a, b, c, d find solutions set by comparison #2 all Page 125 all page 126 all page 127 all finish for tomorrow Thursday Feb 4

Math 396 -91 Page 125 all page 126 all page 127 all Finish for Tues Feb 9

Workbook 306 -05 page 122 Activity #1 Choice of two Options Page 124 #1 a, b, c, d using comparison method #2 a, b, c, d state the variables write the 2 rules find the answer using comparison Page 125 #3, 4

Math 396 -92 Workbook page 127 #7 Page 129 Activity 3 Page 132 #5, 6 When done: You may begin your honest Page 143 # 2, 3 numbers project. Page 144 #4, 6, 7 Page 145 #9 Page 146 #10, 11, 12 Finish for next wednesday

Page 145 #9 Rule A Rule B Rule C y = 300 y = 10 x y = 5 x + 100 zero direct partial x 0 10 20 30 Y = 10 x 0 100 200 300 150 200 250 Y= 5 x + 100