3.1 NOTES Solving Systems of Linear Equations Graphically A system of two linear equations in two variables x and y consist of two equations of the following form: Ax + By = C Equation 1 Dx + Ey = F Equation 2 where the solution (x,y) satisfies both equations. Checking Solutions of a Linear System: 3x – 2y = 2 x + 2y = 6 1.) Is (2,2) a solution of the above system of equations ? 2.) Is (0,-­‐1) a solution of the above system of equations ? Solving a System Graphically : Graphically, the solution of the system of equations is the point or points where the two lines intersect. Find the solution of the following system of equations graphically: 3x – 2y = 2 10
x + 2y = 6 8
Verify this answer on your graphing calculator using the intersect function. 6
4
Write each equation in slope-­‐intercept form 2
so that you can enter them into your calculator. 3x – 2y = 2 x + 2y = 6 – 10
–5
5
10
–2
–4
–6
–8
Put the equations into the y= screen. Graph. Use: 2nd CALC 5:intersect – 10
ENTER ENTER ENTER 1.) Solve the following system of equations graphically. Verify your answer on your graphing calculator. 2x – 2y = -­‐8 2x + 2y = 4 Check Algebraically. . 2.) Solve the following system of equations graphically. Verify your answer on your graphing calculator. 3x – 2y = 6 3x – 2y = 2 3.) Solve the following system of equations graphically. Verify your answer on your graphing calculator. 2x – 2y = -­‐8 -­‐2x + 2y = 8 10
8
6
4
2
– 10
–5
5
10
5
10
5
10
–2
–4
–6
–8
– 10
10
8
6
4
2
– 10
–5
–2
–4
–6
–8
– 10
10
8
6
4
2
– 10
–5
–2
–4
–6
–8
– 10
How many solutions for a system of linear equations?? Application : You are checking out cell phone plans and discover that Talk Anytime Wireless charges $50.00 per month for the first phone line and charges $20.00 per additional phone line. Text Away Wireless charges $80.00 per month for the first phone line and $5.00 per additional phone line. Use your graphing calculator to create the graph of a system of equations to determine the number of additional phone lines for which it would be cheaper to use Talk Anytime verses Text Away. Application : James and Zach began saving money from their part-­‐time jobs. James started with $50 in his savings and earns $10 per hour at his job. Zach started with $225 in his savings and earns $7.50 per hour. If both boys save all of their earnings (and we disregard tax) when will they have the same amount of savings? 3.1 HOMEWORK The graph of a system of two linear equations is shown. Circle the phrase that applies. No Solution No Solution No Solution Infinitely many solutions Infinitely many solutions Infinitely many solutions Exactly 1 Solution Exactly 1 Solution Exactly 1 Solution Without the use of a graphing calculator, graph the linear system and tell how many solutions it has. If there is exactly one solution, estimate the solution and check it algebraically. 1) x = 5 x + y = 1 2) y = -­‐5 – x x + 3y = -­‐15 10
10
8
8
6
6
4
4
2
2
– 10
–5
–2
–4
–6
–8
– 10
5
10
– 10
–5
5
–2
–4
–6
–8
– 10
10
3) 3
x + y = 5 4
3x + 4y = 2 – 10
5) – 10
4) -­‐4y = 24x + 4 y= -­‐6x – 1 10
10
8
8
6
6
4
4
2
2
–5
5
10
– 10
–5
–2
–2
–4
–4
–6
–6
–8
–8
– 10
– 10
2x – y = 7 y = 2x + 8 1
x + 7y = 2 3
2
x + 4y = 2 3
6) 10
6
4
4
2
2
5
–4
–6
–8
– 10
8
6
–2
10
10
8
–5
5
10
– 10
–5
5
–2
–4
–6
–8
– 10
10
7) You are choosing between two long distance phone services. Company A charges $.09 per minute plus a $4 monthly fee. Company B charges $.11 per minute with no monthly fee. a.) Let x be the number of minutes you call long distance in one month, and let y be the total cost of long distance phone service. Write and graph two equations representing the cost of each company’s service. (Use your graphing calculator to help you to graph the functions). b.) Use your graphing calculator to find the point where the graphs intersect. Which customers should choose Company A. Which customers should choose company B? 3.2 NOTES Solving Linear Systems Algebraically Solve the following system of equations graphically (verify your answer on your graphing calculator) 10
8
6
4
y = 3x − 3
y = −x + 5
Now, use the substitution method and the linear combination method to solve. Substitution Linear Combination 2
– 10
–5
5
–2
–4
–6
–8
– 10
10
Solve the system of equations graphically and then use the substitution method to solve. 10
y = 3 x − 13
2 x + 2 y = −10
8
6
4
2
– 10
–5
5
10
–2
–4
Solve the following system of equations by using the linear combination method. –6
–8
– 10
7 x − 12 y = −22
− 5 x + 8 y = 14 Solve the following system of equations graphically and then use one algebraic method of your choice to verify your answer. 10
3 x − 5 y = 30
8 − 6 x + 10 y = −60
6
4 2
– 10
–5
5
10
–2
–4
–6
–8
– 10
Solve the following system of equations graphically and then use one algebraic method of your choice to verify your answer. 6x − 3y = 9
4 x − 2 y = −8
10
8
6
4
2
– 10
–5
5
–2
–4
–6
–8
– 10
10
Application: Cross Training You want to burn 380 calories during 40 minutes of exercise. You burn about 8 calories per minute skateboarding and 12 calories per minute running. How long should you spend doing each activity? (Hint: use two separate equations for time and calories) 3.2 Homework (Day 1) Solve the following problems using the substitution method. 1) 5x + 3y = 4 2) -­‐2x + y = 6 3) – y = -­‐3x + 4 y = -­‐5x + 16 4x – 2y =5 -­‐9x+3y = -­‐12 Solve the following problems using the linear combination method. 4) y = 6x + 2 5) -­‐9x + 6y = 0 6) -­‐15x – 2y = -­‐31 -­‐18x +3y = 4 -­‐12x + 8y = 0 4x + 6y – 11 = 0 3.2 Homework (Day 2)
1.
A bus station 15 miles from the airport runs a shuttle service to and from the airport. The 9:00
a.m. bus leaves for the airport traveling 30 mph. The 9:05 a.m. bus leaves for the airport traveling 40
mph. Write a system of linear equations to represent distance as a function of time for each bus. How
far from the airport will the 9:05 a.m. bus catch up to the 9:00 a.m. bus?
2.
The school yearbook staff is purchasing a digital camera. Recently the staff received two ads in
the mail. The ad for store #1 states that all digital cameras are 15% off. The ad for store #2 gives a
$300 coupon to use when purchasing any digital camera. Assume that the lowest priced digital camera
is $700. When could you get the same deal at either store?
3.
You are starting a business selling boxes of hand-painter greeting cards. To get started, you
spend $36 on paint and paintbrushes that you need. You buy boxes of plain cards for $3.50 per box,
paint the cards, and then sell them for $5 per box. How many boxes must you sell for your earnings to
equal your expenses? What will your earnings and expenses equal when you break even?
4.
You commute to center city 5 days per week on a SEPTA train. You can purchase a monthly
pass for $140 per month or purchase a round trip ticket each day that you commute for $9.50 per ticket.
What is the number of days that you must ride to begin saving money by using the monthly pass?
5.
A soccer league offers two options for membership plans. Option A: an initial fee of $40 and
then you pay $5 for each game that you play. Option B: you have no initial fee but must pay $10 for
each game that you play. After how many games will the total cost of the two options be the same?
6.
For spring break some college students are planning a 7 day trip to Florida. They estimate that it
will cost $275 per day in Tampa at an all inclusive and $400 per day in Orlando for a similar all
inclusive hotel. The total budget per student for the 7 day trip is $2300. How many days should they
spend in each location to meet the limitations of their budget?
3.3 NOTES
Graphing Systems of Linear Inequalities
10
Graph the following system of equations: 8
2x + 3y = 6 6
x – 2y = 2 4
2
– 10
–5
5
10
–2
–4
–6
–8
– 10
Decide whether the ordered pair is a solution to the inequality listed in the table. If so, write yes. If not, write no. (0,0) (2,3) (3,-­‐1) (4,0) (-­‐3,4) (-­‐5,0) (6,0) (3,-­‐2) (1,5) 2x + 3y ≤ 6 x − 2y ≤ 2 Plot the points on your graph using the following symbols: Solution to both inequalities: * Solution to neither inequality: x Solution to exactly one inequality: o Graph the following systems of inequalities: 10
y ≥ −3x − 1
8
y<x+2
1) 6
4
2
– 10
–5
5
10
–2
–4
–6
–8
– 10
10
8
6
x≤0
2.) y ≥ 0
x − y ≥ −2
10
8
4
2
– 10
–6
–8
– 10
3.) 2
–5
5
10
−x < y
x + 3y < 9 x≥2
–2
–4
–6
–8
10
– 10
8
6
4
2
4.) 10
–4
4
5
–2
6
– 10
–5
x + 2y ≤ 10
2x + y ≤ 8 2x − 5y < 20
– 10
–5
5
–2
–4
–6
–8
– 10
10
Write the system of inequalities that correspond with the shaded region. 3.3 HOMEWORK Graph the system of linear inequalities. y > −2
y ≤1
1) 10
8
6
4
2
– 5
– 10
5
10
–2
–4
–6
–8
– 10
3) – 10
x−y>7
2x + y < 8
y > −5x
x ≤ 5y
2) 10
8
6
4
2
– 10
–5
5
–4
–6
–8
– 10
y<4
x > −3 y>x
4) 10
8
8
6
6
4
4
2
2
5
10
10
–2
10
–5
– 10
–5
5
–2
–2
–4
–4
–6
–6
–8
–8
– 10
– 10
10
2x − 3y > −6
5) 5x − 3y < 3 x + 3y > −3
– 10
6) y<5
y > −6
2x + y ≥ −1
y≤ x+3
10
8
6
4
2
–5
5
10
– 10
–5
–2
–4
–6
–8
– 10
Challenge. Write a system of linear inequalities for the region. 10
8
6
4
2
5
10
–2
–4
–6
–8
– 10
3.4 NOTES Linear Programming optimization: finding the maximum or minimum value of some quantity
linear programming: the process of maximizing or minimizing a linear objective function subject to
constraints that are linear inequalities
constraints: graphing restrictions
feasible region: the graph of the system of constraints
constraints.
x ≥0
y ≥0
2x + y ≤ 8
x + 3y ≤ 9
minimum value: ____________
maximum value: ____________
Ex: Find
the
minimum
value and
the
maximum
value of
the
objective
function
C = 5x –
2y subject
to the
following
Ex: FU N DRAISER Your class plans to raise money by selling T-shirts and baseball caps. The plan
is to buy the T-shirts for $8 and sell them for $12 and to buy the caps for $4 and sell them for $7.
The planning committee estimates that you will not sell more than 120 items. Your class can
afford to spend as much as $800 to buy the articles. The constraints on your fun-raising activity
are given by the system of inequalities below. Your class can only sell combinations of T-shirts
and caps indicated by points that are solutions to the system.
c + t ≤ 120
4c + 8t ≤ 800
c ≥0
t ≥0
a) Your club wants to maximize profit. Write the profit functions p in terms of c and t.
b) At which point do you think the maximum value of p will occur?
c) Which combination of baseball caps and T-shirts maximizes profit? What is the maximum
profit?
Ex: PIN ATAS Piñatas are made to sell at a craft fair. It takes 2 hours to make a mini piñata and
3 hours to make a regular-sized piñata. The owner of the craft booth will make a profit of $12 for
each mini piñata sold and $24 for each regular-sized piñata sold. If the craft booth owner has no
more than 30 hours available to make piñatas and wants to have at least 12 piñatas to sell, how
many of each size piñata should be made to maximize profit?