Algebra 2
Chapter 3
Mr. Barr
Chapter 3: Systems of Equations 3.1 Solving Systems by Graphing
Objective: Solve a system of equations in 2 variables by graphing
What is a system of equations
A system of equations is when you have two or more equations using the same variables.
The solution to the system is the point that satisfies ALL of the equations. This point will be an ordered pair.
When graphing, you will encounter three possibilities:
- The point where the lines intersect is your solution.
- These lines never intersect! Since the lines never cross, there is NO SOLUTION!
- These lines are the same!
- Since the lines are on top of each other, there are INFINITELY MANY SOLUTIONS!
Examples:
1)2x + y = 4 2) y = 2x – 3 3) 3x – y = 8
x – y = 2 -2x + y = 1 2y = 6x - 16
Using the Graphing Calculator:
1. Enter your equation in the Y = screen. ( equations have to be solved for y )
2. Hit GRAPH.
3. 2nd,TRACE (CALC), option#5 intersect
4. Move spider close to the intersection using left and right arrows.
5. Hit ENTER 3 times. You’ll see the x- and y-values of the intersection point at the bottom of the screen.
Examples: Solve using a graphing calculator
4) 5) 6)
3.1 Assignment
7) 8) 9)
10) 11) 12)
Chapter 3: Systems of Equations 3.2 Solving Systems of Equations Algebraically
Using Substitution
Steps: 1. Solve for one variable in one equation.
2. Substitute for that variable in the other equation.
3. Solve.
4. Replace to find the other variable.
5. Check your solution.
Which answer checks correctly?
a) (2, 2)b) (5, 3)c) (3, 5) d) (3, -5)
3x – y = 4
x = 4y - 17
Examples: solve the system using substitution
1) x + y = 5 2) 3y + x =7
y = 3 + x 4x – 2y = 0
If you solved the first equation in the system below for x what would be substituted into the bottom equation?
2x + 4y = 4a) -4y + 4b) -2y + 2c) -2x + 4d) -2y + 22
3x + 2y = 22
Think About it!
3) x = 3 - y 4) 2x + y = 4 What does it mean if the result is true?
x + y = 7 4x + 2y = 8a) The lines intersect b) The lines are parallel
c) The lines are coinciding
Substitution Assignment: Solve each system using substitution
Using Elimination
Steps:
- Put equations into standard form
- Determine which variable to eliminate
- Add the equations
- Plug value back in to find the other variable
5. Check your solution
Examples: Solve each system using elimination
1)x + y = 5 2) 4x + y = 73) -y = -7 + 2x
3x – y = 7 -4x + 2y = 2 4x + y = 5
Try these!
4) 5)
Changing the coefficient: Sometimes you have to multiply one or both of the equations to obtain
opposite coefficients
6) 2x + 2y = 67) x + 4y = 78) 3x + 4y = -1
3x – y = 5 4x – 3y = 94x – 3y = 7
Elimination Assignment: Solve each system of equations by using elimination
Chapter 3: Systems of Equations 3.3 Solving Systems of Inequalities by Graphing
To solve a system of inequalities, graph the inequalities in the same coordinate plane. The solution of the system is the region shaded for all the inequalities.
Examples: Solve each system of inequalities by graphing
Try these!
3.3 Assignment Part 1
‘
Find the vertices of an enclosed region
Sometimes the graph of a system of inequalities produces an enclosed region in the form of a polygon. You can find the vertices of the region by a combination of the methods used earlier in this chapter: graphing, substitution, and/or elimination.
Examples:
1)Find the vertices of the triangle formed by 5x + 4y < 20, y < 2x + 3 and x – 3y < 4.
Graph the inequalities: Find the vertices graphically or algebraically:
2)
3)3x + y 6
2x – y -1
x -2
y 4
4) 2x + y 6
x + y 2
1 x 2
y 3
3.3 Assignment Part 2
Chapter 3: Systems of Equations 3.4 Optimization with Linear Programming
3.4 in class practice
Solving Linear Programming Problems
When solving Linear Programming problems, use the following procedure:
1)Define the variables
2)Write a system of inequalities
3)Graph the system of inequalities
4)Find the coordinates of the vertices of the feasible region
5)Write an expression to be maximized or minimized
6)Substitute the coordinates of the vertices in the expression
7)Select the greatest or least result to answer the problem
Example: A painter has exactly 32 units of yellow dye and 54 units of green dye. He plans to mix as many gallons as possible of color A and color B. Each gallon of color A requires 4 units of yellow dye and 1 unit of green dye. Each gallon of color B requires 1 unit of yellow dye and 6 units of green dye. Find the maximim number of gallons he can mix.
Step 1: Define the variablesStep 4: Find the coordinates of the vertices of the feasible region
Step 2: Write a system of inequalities
Step 5: Write and expression to be maximized or minimized
Step 3: Graph the system if inequalitiesStep 6: Substitute the coordinates of the vertices in the epxression
Step 7: Select the greatest or least result to answer the problem
3.4 Assignment
1)You need to buy some filing cabinets. You know that Cabinet X costs $10 per unit, requires six square feet of floor space, and holds eight cubic feet of files. Cabinet Y costs $20 per unit, requires eight square feet of floor space, and holds twelve cubic feet of files. You have been given $140 for this purchase, though you don't have to spend that much. The office has room for no more than 72 square feet of cabinets. How many of which model should you buy, in order to maximize storage volume?
2)Fly-High Airlines sells business class and tourist class seats for its charter flights. To charter a plane at least 5 business class tickets must be sold and at least 9 tourist class tickets must be sold. The plane does not hold more than 30 passengers. Fly-High makes $40 profit for each business class ticket sold and $45 profit for each tourist class ticket sold. In order for Fly-High Airlines to maximize its profits, how many tourist class seats should they sell?
3)The Plexus Dance Theatre Company will appear at the University of Georgia. According to school policy, no more than 2000 general admission tickets can be sold and no more than 4000 student tickets can be sold. It costs $0.50 per ticket to advertise the dance company to the students and $1 per ticket to advertise to the general public. The dance company has an advertising budget of $3000 for this show. Find the maximum profit the company can make if it charges $4 for a student ticket and $7 for a general admission ticket. How many student tickets should they sell?
4)Funtime Airways flies from Palau to Nauru weekly if at least 12 first class tickets and at least 16 tourist class tickets are sold. The plane can not carry more than 50 passengers. Funtime Airways makes $26 profit for each tourist class seat sold and $24 profit for each first class seat sold. In order for Funtime Airways to maximize its profits, how many of each type of seat should they sell? What is the maximum profit?
5)Reynaldo Electronica manufactures radios and tape players. The manufacturing plant has the capacity to manufacture at most 600 radios and 500 tape players. It costs the company $10 to make a radio and $12 to make a tape player. The company can spend $8400 to make these products. Reynaldo Electronica makes a profit of $19 on each radio and $12 on each tape player. To maximize profits, how many of each product should they manufacture?
6)Bob builds tool sheds. He uses 10 sheets of dry wall and 15 studs for a small shed and 15 sheets of dry wall and 45 studs for a large shed. He has available 60 sheets of dry wall and 135 studs. If Bob makes $390 profit on a small shed and $520 on a large shed, how many of each type of building should Bob build to maximize his profit?
7)A company makes a product in two different factories. At factory X it takes 30 hours to produce the product and at factory Y it takes 20 hours. The costs of producing these items are $50 at factory X and $60 at factory Y. The company’s labor force can provide 6000 hours of labor each week and resources are $12,000 each week. How should the company allocate its labor and resources to maximize the number of products produced?
Chapter 3: Systems of Equations 3.5 Systems of Equations in Three Variables
Use the methods used for solving systems of linear equations in two variables to solve systems of equations in three variables. A system of three equations in three variables can have a unique solution, infinitely many solutions, or no solution. A solution is an ordered triple.
Steps:1) Put all equations in standard form
2)Use elimination to make a system of two equations in two variables.
3)Solve the system of two equations
4)Substitute to find the third variable
5)Write the answer as on ordered triple
Examples: Solve the system of equations:
1)
2)
3.5 Assignment Part 1:
Solve each system of equations:
Real – World Problems
Steps: 1) Define the variables
2)Translate the information in the problems into three equations
3)Solve the system of equations
4)Interpret and label the solution
Example:
The Laredo Sports Shop sold 10 balls, 3 bats, and 2 bases for $99 on Monday. On Tuesday they sold 4 balls, 8 bats, and 2 bases for $78. On Wednesday they sold 2 balls, 3 bats, and 1 base for $33.60. What are the prices for 1 ball, 1 bat, and 1 base?
Define the variables:Translate the information into three equations:
Let x =
Let y =
Let z =
Solve the system:
Interpret the solution:
3.5 Assignment Part 2
1) Carly is training for a triathlon. In her training routine each week, she runs 7 times as far as she swims, and she bikes 3 times as far as she runs. On week she trained a total of 232 miles. How far did she run that week?
2)At the arcade, Ryan, Sara, and Tim played video racing games, pinball, and air hockey. Ryan spent $6 for 6 racing games, and 5 games of air hockey. Sara spent $12 for 3 racing games, 4 pinball games, and 5 games of air hockey. Tim spent $12.25 for 2 racing games, 7 pinball games, and 4 games of air hockey. How much did each of the games cost?
3)A natural food store makes its own brand of trail mix out of dried apples, raisins, and peanuts. One pound of the mixture costs $3.18. It contains twice as much peanuts by weight as apples. On pound of dried apples costs $4.48., a pound of raisins $2.40, and a pound of peanuts $3.34. How many ounces of each ingredient are contained in 1 pound of the trail mix?
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