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Chapter 3: Systems of Linear Equalities and Inequalities
Chapter 3: Systems of Linear Equations and Inequalities Assignment Sheet
Date Topic Assignment Completed 3.1 Solving Systems of Linear Equations
Graphically Homework 3.1
3.2 Solving Linear Systems Algebraically Homework 3.2 Day 1 3.2 Solving Linear Systems Algebraically Homework 3.2 Day 2 3.3 Graphing Systems of Linear
Inequalities Homework 3.3 # 1-‐6
3.3 Graphing Systems of Linear Inequalities
Finish Homework 3.3
CHAPTER 3 REVIEW PracticeB 3.1-‐3.3 Chapter 3 Exam None
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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:
y = 32x −1
x + 2y = 6 Verify this answer on your graphing calculator using the intersect function. Write each equation in slope-‐intercept form so that you can enter them into your calculator. 3x – 2y = 2 x + 2y = 6 Put the equations into the y= screen. Graph. Use: 2nd CALC 5:intersect ENTER ENTER ENTER
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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
How many solutions for a system of linear equations??
Application : At a local animal shelter, the supplies for each dog generally costs twice as much as the supplies for a cat. We need to care for 164 cats and 24 dogs with a budget of $4240. How much can be spent on each dog?
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?
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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 2) y = -‐5 – x x + y = 1 x + 3y = -‐15
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3)
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34x + y = 5 4) -‐4y = 24x + 4
3x + 4y = 2 y= -‐6x – 1
5) 2x – y = 7 6)
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13x + 7y = 2
y = 2x + 8
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23x + 4y = 2
7) You are choosing between two movie rental services. Company A charges $2.99 per movie plus a $20 monthly fee. Company B charges $4.99 per movie with no monthly fee. How many movies could you rent and get charged the same monthly bill. If you only rent, on average, 8 movies per month, which is the better deal for you? Create a graph below to solve this problem. You can use your graphing calculator to compute the exact point of intersection.
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3.2 NOTES Solving Linear Systems Algebraically Solve the following system of equations graphically (verify your answer on your graphing calculator)
y = 3x − 3y = −x + 5
Now, use the substitution method and the linear combination method to solve. Substitution Linear Combination
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Solve the system of equations graphically and then use the substitution method to solve.
1022
133−=+
−=yxxy
Solve the following system of equations by using the linear combination method.
148522127
=+−−=−
yxyx
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Solve the following system of equations graphically and then use one algebraic method of your choice to verify your answer.
601063053
−=+−=−yx
yx
Solve the following system of equations graphically and then use one algebraic method of your choice to verify your answer.
824936−=−
=−yxyx
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?
D = 30t
D = 40 t − 560
⎛⎝⎜
⎞⎠⎟
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?
Let C = the cost of a camera after the discount
Let x = the original cost of a camera
3. You are starting a business selling boxes of hand-painted 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? (Write an equation to represent Total Expenses and another equation to represent Total Earnings.)
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? (Let C = the cost in $ and x = the number of days commuting.)
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? (Let T = days spent in Tampa and R = days spent in Orlando.)
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3.3 NOTES Graphing Systems of Linear Inequalities
Graph the following system of equations: 2x + 3y = 6 x – 2y = 2 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:
1)
y ≥ −3x −1y < x + 2
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2.) x ≤ 0y ≥ 0x − y ≥ −2
3.) −x < yx + 3y < 9x ≥ 2
4.) x + 2y ≤ 102x + y ≤ 82x − 5y < 20
Write the system of inequalities that correspond with the shaded region.
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3.3 HOMEWORK Graph the system of linear inequalities.
1)
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y > −2y ≤ 1
2)
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y > −5xx ≤ 5y
3)
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x − y > 72x + y < 8
4)
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y < 4x > −3y > x
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5)
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2x − 3y > −65x − 3y < 3x + 3y > −3
6)
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y < 5y > −62x + y ≥ −1y ≤ x + 3
Challenge. Write a system of linear inequalities for the region.
