Systems of Equation!!

Let’s go to the Carnival! When two (or more) linear equations relate the same two (or more) quantities, the equations form a system of linear equations. What does this mean? Let’s go to the Carnival!

Rides !! Let's go! There are two ticket plans: Pay $35 to get into the park and $3.00 per ride Or Pay $45 to get into the park and $2.00 per ride.

x represents the number of rides in both cases There are two ticket plans: Pay $35 to get in and $3.00 per ride Pay $45 to get in and $2.00 per ride. What quantities do we have here? What quantities do we have here? Cost to get in C $35 Cost to get in C $45 Cost per ride C $3.00 Cost per ride C $2.00 Number of Rides V x Number of Rides V x Total Cost V y Total Cost V y x represents the number of rides in both cases y represents the total cost in both cases

Turn to page 4 in your packet and do Quick Check 1

What quantities…?? y = 3x + 35 y = 2x + 45 What were our quantities here? What were our quantities here? Cost to get in C = b $35 Cost to get in C = b $45 Cost per ride C/R = m $3.00 Cost per ride C/R = m $2.00 Number of Rides V x Number of Rides V x Total Cost V y Total Cost V y What is our equation for this data? What is our equation for this data? y = 3x + 35 y = 2x + 45 We have two equations relating the same two quantities. What quantities…??

Turn to page 4 in your packet and do Quick Check 2

Let’s graph these equations… y = 3x + 35 y = 2x + 45 John bought the pink plan, and Sally bought the green plan. How many rides do they have to go on to spend the same amount of money for the day? Let’s graph these equations…

What will the graph show us? $ The total cost is the dependent variable. y 90 80 The y-axis keeps track of the amount of money it costs. 70 60 50 40 30 20 The x-axis keeps track of the number of rides. 10 X 2 4 6 8 10 12 14 16 18 20 # rides The number of rides is the independent variable

Y y = 3x + 35 This line shows all the points (x, y) = (# rides, cost) that make the pink equation true. $ y = 2x + 45 y = 3x + 35 90 y = 2x +45 80 This line shows all the points (x, y) = (# rides, cost) that make the green equation true 70 60 50 40 30 20 10 X 2 4 6 8 10 12 14 16 18 20 # rides

Right !!! It’s the one point that makes both equations true. Y $ What does this big blue dot show us?? y = 3x + 35 90 y = 2x +45 80 70 Right !!! It’s the one point that makes both equations true. 60 50 40 30 20 (10, 65) = (ten rides, $65) 10 X 2 4 6 8 10 12 14 16 18 20 # rides

Turn to page 4 in your packet and do Quick Check 3

Let’s check… y = 3x + 35 y = 2x +45 65 = 3(10) + 35 65 = 2(10) + 45 (10, 65) = (ten rides, $65) Our graph shows that these two equations share the point (10, 65). So x = 10 and y = 65 should make both equations true. ? ? 65 = 3(10) + 35 65 = 2(10) + 45 65 = 30 + 35 True!! 65 = 20 + 45 True!!

Turn to page 4 in your packet and do Quick Check 4

How do we do this… Algebraically ??

y = 3x + 35 y = 2x + 45 We already know… We want…. y = y John bought the pink plan, and Sally bought the green plan. How many rides do they have to go on to spend the same amount of money for the day? y = 3x + 35 y = 2x + 45 We already know… Number of Rides = x Number of Rides = x Total Cost = y Total Cost = y We want…. Total Cost = Total Cost or y = y But that means… 3x + 35 = 2x + 45

10 rides ! That’s just what our graph said!!! 3x + 35 = 2x +45 Subtract 2x from both sides. x + 35 = 45 Simplify x + 35 – 35 = 45 – 35 Subtract 35 from both sides. x = 10 Simplify That’s just what our graph said!!! 10 rides !

Turn to page 5 in your packet and do Quick Check 5

x = 10 rides YES!!! y = 3x + 35 y = 2x +45 y = 3(10) + 35 Will the y’s agree?? YES!!! What do you think?? x = 10 rides y = 3x + 35 y = 2x +45 y = 3(10) + 35 y = 2(10) + 45 y = 30 + 35 y = 20 + 45 y = 65 y = 65 y = $65

Turn to page 5 in your packet and do Quick Check 6

When two (or more) linear equations relate the same two (or more) quantities, the equations form a system of linear equations. What does this mean?

The solution to a system of linear equations is the one point that makes both equations true. That is, the solution is the point of intersection of the graphs of the two linear equations.

Y $ y = 3x + 35 90 y = 2x +45 80 70 60 50 40 Got it?? 30 20 10 X 2 4 6 8 10 12 14 16 18 20 # rides