Solve Linear Systems by Graphing

A “System” is a set of equations. A Linear System is two linear equations (two lines) Solving a Linear System The point of intersection of the lines is called the “solution” The solution is where the two lines cross It is the point at which both functions have the same input with the same output (x,y)

There are 3 ways to solve a system: Graphing This method is only successful if you draw an accurate graph. It is often not very precise. We will learn this method today. We will learn 2 more precise methods later in the unit

Graph to solve the linear system. y = 2x - 2 y = - 𝟒 𝟑 x + 8 Remember slope intercept form y = mx + b b = y-intercept Plot point (0,b) to start graph Where do the lines intersect? m = slope follow rise to plot more points (3, 4) is the solution to this system of linear equations. run

Graph to solve the linear system. y = -2x + 6 y = - 𝟏 𝟐 x - 3 To find the solution, graph both lines and find where they intersect (6, -6) is the solution to this system of linear equations.

System from a word problem Selling Hats Aprende Student Council would like to sell hats as a school fundraiser. There are two companies that they are considering. Hats R Us charges a $60 set up fee and $3 per hat. Crazy 4 Caps charges a $10 set up fee and $5 per hat. Which company is the better deal? Start by writing an equation for each company: Use slope intercept form y = mx + b Remember that the ‘set up fee’ will be the y intercept / starting value (the b) And the repeated charged per each hat will be the slope / rate of change (the m) Hats R Us ___________ C = 3x + 60 Crazy 4 Caps___________ C = 5x + 10

Graph the equations by first making a table of values for each company then plot the points. Hats R Us # of Hats (x) Total Cost (C) 60 5 75 10 90 15 105 20 120 25 135 30 150 35 165 40 180 45 195 50 210 Crazy 4 Caps # of Hats (x) Total Cost (C) 10 5 35 60 15 85 20 110 25 135 30 160 185 40 210 45 235 50 260 When do the two companies have the same price?

Graph the equations by first making a table of values for each company then plot the points. Hats R Us # of Hats (x) Total Cost (C) 60 5 75 10 90 15 105 20 120 25 135 30 150 35 165 40 180 45 195 50 210 Crazy 4 Caps # of Hats (x) Total Cost (C) 10 5 35 60 15 85 20 110 25 135 30 160 185 40 210 45 235 50 260 The two companies have the same price when the lines intersect and when the values in the table have the same output for the same input. (25,135) At 25 hats, both companies cost $135.

Hats R Us & Crazy 4 Caps… Under what circumstances is buying hats from Crazy 4 Caps cheaper than buying hats from Hats R Us? Under what circumstances is buying hats from Hats R Us cheaper than buying hats from Crazy 4 Caps? Suppose the Aprende Student Council decides to use Hats R Us to purchase 57 hats. What is the total cost? C = 60 + 3(57) C = 60 + 171 C = $231 $231

Another Example Keeping Safe The managers of a shopping center want to upgrade their security system. Two providers bid for the job. Super Locks will charge $3,975 to install the equipment and the $6.00 per day to monitor the system and respond to the alerts. Fail Safe will charge $995 to install the equipment and then $17.95 per day to monitor the system and respond to alerts. Both companies are reliable and capable, so the choice comes down to cost.

The cost of the security services from Super Locks and Fail Safe depends on the number of days the company provides service. Write an equation that can be used to calculate the cost for “x” number of days for each company. Use the format: y = mx + b Instead of using “y”, use “C” for cost. The one time charge is b, the y-intercept. That’s how much you would pay with zero days of monitoring. The repeating charge is the slope. That’s the daily monitoring fee. Super Locks is $3,978 to install and $6.00 per day to monitor. Fail Safe is $995 to install and $17.95 per day to monitor.

Write the Equations Super Locks: Fail Safe: C = 6x + 3978 Fail Safe: C = 17.95x + 995 Super Locks is $3,978 to install and $6.00 per day to monitor. Fail Safe is $995 to install and $17.95 per day to monitor. Which company is the better deal? What does the best deal depend on? TIME

Graph the two equations to find out when each company is the best price. To graph each equation, substitute a number of days in for x and solve for C. This will make a table of values. Plot each ordered pair. Super Locks Fail Safe Days (x) Cost (C) 3978 995 100 4578 2790 200 5178 4585 300 5778 6380 400 6378 8175 500 6978 9970 Fail Safe Super Locks Which company is the better deal? Or, when is EACH company the better deal? How can you tell?

Use the graph to answer the following questions… For what number of days will the costs for the two companies be the same? What is the cost? For what number of days will Super Locks cost less than Fail Safe? For what number of days will Super Locks cost less than $6,000? What is the cost of one year of service from Fail Safe? Fail Safe Super Locks

Make a graph by plotting each plant’s ordered pairs (W, H). Practice: 14 12 10 8 6 4 2 Plant A and Plant B are on different watering schedules. This affects their rate of growth. Compare the growth of the two plants to determine when their heights will be the same. Let W = number of weeks Let H = height of the plant after W weeks H E I G T 0 1 2 3 4 5 Plant A Plant B Weeks (W) Height (H) 4 2 1 6 8 10 3 14 Weeks At which week do the plants have the same height? What is their height?