Unit 3 Systems of Equations

Section 3.1 Solving Linear Systems by Graphing Objective(s): Solve a system of linear equations in two variables using graphing.

Essential Question: Explain how to tell from a graph of a system of linear equations if it will have one solution, no solution, or infinitely many solutions.

Homework: Assignment 3.1. #1 15 in the homework packet. Notes:VocabularyA system of equations in two variables x and y consists of two equations.A solution of a system of two equations in two variables is an ordered pair that makes both equations true.Checking to See if an Ordered Pair is a Solution1. Substitute the ordered pair into both equations.2. If the ordered pair is a solution to both equations, then it is a solution to the system of equations.Determine whether the ordered pair is a solution of the system of linear equations.

Example 1:, Example 2:,

Solving a System of Equations by Graphing1. Graph both equations of the same set of axes.2. Find the point of intersection.

Ex: Solve the system by graphing:

Step One: Graph both equations on the same coordinate plane.

Step Two: Find the coordinates of the point of intersection of the two lines.

The lines appear to intersect at the point .Solve the system by graphing.

Example 3:Example 4:


Special Cases If the graphs of the equations in a system are parallel (do not intersect), then the system has NO SOLUTION. Systems with no solution are called inconsistent. If the graphs of the equations in a system are the same line (coincident), then the system has INFINITELY MANY SOLUTIONS. Systems with infinite solutions are called dependent. Systems with solutions are called consistent.Tell if the system is inconsistent, the equations are dependent, or consistent.


Besides graphing, what is another way to tell if lines are parallel?Parallel lines have the _________________ slope and _______________ y-intercepts.How can you tell the slope of a line if the equation is in standard form?

Does the system have one solution, no solution, or an infinite number of solutions?


Section 3.2 Solving Linear Systems Algebraically Objective(s): Solve a system of linear equations in two variables using substitution and linear combinations. Solve application problems involving systems of equations.

Essential Question: When choosing a method for solving a system of equations, when would you use linear combinations, and when would you use substitution?

Homework: Assignment 3.2. #16 27 in the homework packet. Notes:Solving a System of Linear Equations by Substitution:

Ex: Solve the system by substitution.Unit 3 Systems of EquationsName: _______________________

Period: ______________________

Reflection:1

Reflection:5

Step One: Since x and -4y 8 are EQUAL, one can replace the other. Substitute -4y 8 into the first equation replacing x. Solve.

Step Two: Substitute the value from Step One into either of the original equations and solve for the remaining variable.Step Three: Write your answer as an ordered pair and check in both of the original equations.

Solution:

Solve the system of equations.


Some systems of equations do not have an equation that can be solved nicely for one of the variables. If this occurs, we can solve the system using a new method.Solving a System of Equations by Linear Combinations (Elimination):

Ex: Solve the system by linear combinations.Step One: Write the two equations in standard form. (These two are already in standard form)

Step Two: Multiply one or both of the equations by a constant to obtain coefficients that are opposites for one of the variables.

Step Three: Add the two equations from Step Two. One of the variable terms should be eliminated. Solve for the remaining variable.

Step Four: Substitute the value from Step Three into either one of the original equations to solve for the other variable.

Step Five: Write your answer as an ordered pair and check in the original system.

Solution:

We can multiply the first equation by to obtain a y-coefficient of 6 in the first equation (the opposite of )

Solve the system of equations.

Example 3:Example 4:Multiply the _______ equation by _________

Example 5:Example 6:Multiply the _______ equation by _________Multiply the first equation by _________Multiply the second equation by ______

Choosing an Appropriate Method: Substitution is the method of choice when one of the equations is easily solvable (or already solved) for one of the variables. If this is not the case, use linear combinations to solve the system.Ex: Which method would be BEST for solving the following system of equations?

a) Linear Combinationsb) Substitution

Special Cases: As we know from solving systems of equations by graphing, systems of equations can have exactly one solution, infinitely many solutions, or no solution. System of Equations with Infinitely Many Solutions:

Ex: Solve the system using the method of your choice.Note: Because the equations are in standard form, and are not easily solvable for one of the variables, we will use linear combinations.Step One: Done. The equations are in standard form.

Step Two: Multiply the first equation by 4 and the second equation by to eliminate the y terms.

Step Three: Note: Both of the variables were eliminated!

If both of the variables are eliminated, and we end up with a true statement (i.e. ), then the equation has INFINITELY MANY SOLUTIONS. Note: If we were to graph these two equations, the two lines would be the same line.Solve the system of equations.

Example 7:

Systems of Equations with No Solution:

Ex: Solve the system using the method of your choice.Note: Because the first equation is easily solvable for y, we will use substitution.

Step One:

Step Two:Note: The variable was eliminated!

If the variable is eliminated, and we end up with a false statement (i.e. ), then the equation has NO SOLUTION.Note: If we were to graph these two equations, the two lines would be parallel.Solve the system of equations.

Example 8:

Solve by any method.


Problem-Solving Plan:Step One: Write a verbal model.Step Two: Assign labels.Step Three: Write an algebraic model.Step Four: Solve the algebraic model using one of the methods for solving a system of equations.Step Five: Answer the question asked and label the answer appropriately.Application Problems with Systems of EquationsEx: A sporting goods store receives a shipment of 124 golf bags. The shipment includes two types of bags, full-size and collapsible. The full-size bags cost $38.50 each. The collapsible bags cost $22.50 each. The bill for the shipment is $3430. How many of each type of golf bag are in the shipment?

Step One: (# of Full-Size Bags) + (# of Collapsible Bags) = (Total # of Golf Bags in the Shipment)

(Rate) (# of Full-Size Bags) + (Rate)(# of Collapsible Bags) = (Cost of Shipment)Step Two: # of Full-Size Bags = F# of Collapsible Bags = CTotal # of Bags = 124Rate of Full-Size Bags = 38.50Rate of Collapsible Bags = 22.50Cost of Shipment = 3430

Step Three:

Step Four:We will use substitution.

Step Five: There are 40 full-size bags and 84 collapsible bags in the shipment.

Example 11:A health store wants to make trail mix with raisins and granola. The owner mixes granola, which costs $4 per pound, and raisins, which cost $2 per pound, together to make 25 lbs of trail mix. How many pounds of raisins should he include if he wants the mixture to cost him a total of $80?

Step One:pounds of ____________ + pounds of ____________ = ____________

(Rate) (____________) + (Rate)( ____________) = ______________________

Step Two: lbs of granola = Glbs of raisins= RTotal lbs of trail mix = _____Rate of granola = ________Rate of raisins = ________Cost of trail mix = ________

Step Three: (equations)

Step Four: (solve)

Step Five: (answer)

Sample CCSD Common Exam Practice Question(s):1. The equations for two lines are given below:

What is the x-coordinate of the point of intersection of the two lines?A. 2B. 1C. -1D. -22. What is the x-coordinate of the solution of the system of equations?

A. 3B. 1C. 1D. 33. A coin bank contains only dimes and nickels. The bank contains 46 coins. When 5 dimes and 2 nickels are removed, the total value of the coins is $3.40. How many nickels did the coin bank start with?A. 12B. 22C. 24D. 34

Section 3.3 Graphing and Solving Systems of Linear Inequalities Objective(s): Graph the solution set of a system of linear inequalities.

Essential Question: Describe the procedure for solving a system of linear inequalities.

Homework: Assignment 3.3. #28 33 in the homework packet. Notes:VocabularyA solution of a system of linear inequalities is an ordered pair that is a solution of each inequality in the system.A system of linear inequalities is a set of two or more linear inequalitiesTesting if an Ordered Pair is a Solution to a System of Linear Inequalities

Ex: Use the system of linear inequalities Is (0, -6) a solution?Test the point in both inequalities. It is a solution if and only if it satisfies both inequalities.

So (0, -6) is NOT a solution.

Test if the ordered pair is a solution to the system of inequalities. Example 1:(7, 5)Example 2:(0, 0)

Graphing a System of Linear Inequalities

Ex: Graph the system Step One: Graph each line on the same coordinate plane.Step Two: Determine whether to use solid or dashed lines. (Recall: Use solid lines for and , and use dashed lines for < and >.)Step Three: Lightly shade the appropriate half-planes for each inequality. Step Four: The solution to the system is the overlapping region formed by the shading in Step Three. Shade darkly this region and erase the regions that have NO overlapping. Step Five: Choose a point in the shaded region (not on either line) and test it in the original system of inequalities.

Choose (0, 2).

Graph the solution to the system of linear inequalities.






Sample CCSD Common Exam Practice Question(s):1. Which graph shows the solution to the system of inequalities below?

Section 3.4 Linear Programming Objective(s): Solve application problems involving linear programming techniques.

Essential Question: Are the vertices of a feasible region the only possible points that satisfy an objective function? Explain your answer.

Homework: Assignment 3.4. #34 40 in the homework packet. Notes:VocabularyA solution of a system of linear inequalities is an ordered pair that is a solution of each inequality in the system.The graph of the system of constraints in linear programming is called the feasible region.Linear programming is the process of optimizing a linear objective function subject to a system of linear inequalities.Solving a Linear Programming ProblemEx: Find the minimum and maximum value of the function P = 2x + 3y subject to the constraints

Note: In a linear programming problem, the maximum or minimum always occurs at one of the vertices of the feasible region.Step One: Graph the feasible region. (Use the constraints.)Step Two: Find the coordinates of the vertices by solving

3 systems of equations.

andThe vertices are (0, 7), (0, 2), and (5, 2).

Step Three: Evaluate the objective function for each of the vertices.xyP = 2x + 3y

07P = 2(0) + 3(7) = 21

02P = 2(0) + 3(2) = 6

52P = 2(5) + 3(2) = 16

Solution: The maximum value of the function P is 21. It occurs when x = 0 and y = 7. The minimum value of the function P is 6. It occurs when x = 0 and y = 2.Example 1:A company makes S pairs of skis and B snowboards under the following constraints:

Find the maximum profit for the company if they

sell the skis for $70 per pair and the snowboards

for $50 each. Note: The horizontal axis is the S-axis, and the

vertical axis is the B-axis.Step One: Graph the feasible region. Step Two: Find the coordinates of the vertices by solving 6 systems of equations.

( , )( , )( , )

( , )( , )( , )

Step Three: Evaluate the objective function for each of the vertices.SBP =

Solution: The maximum value of the function P is __________. It occurs when S = __________ and B = __________.Example 2:A company makes T tape players and C CD players under the following constraints:

Find the maximum profit if the company sells the tape players for $28 each and the CD players for $33 each.

Step One: Graph the feasible region.

Step Two: Find the coordinates of the vertices by solving 3 systems of equations.

( , )( , )( , )

Step Three: Evaluate the objective function for each of the vertices.TCP =

Solution: The maximum value of the function P is __________. It occurs when T = __________ and C = __________.Sample CCSD Common Exam Practice Question(s):The area of a parking lot is 600 square meters. A car requires 6 square meters and a bus requires 30 square meters of space. The lot can handle a maximum of 60 vehicles. Let b represent the number of buses and c represent the number of cars. The diagram below represents the feasible region based on the constraints of the number of vehicles that can be parked in the lot.

To park in the lot, a bus costs $8 and a car costs $3. How many of each type of vehicle can be parked in the lot to maximize the amount of money collected?A. 0 buses and 60 carsB. 10 buses and 50 carsC. 20 buses and 0 carsD. 30 buses and 30 carsReflection:19

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Unit 3 Systems of Equations