Name:______Period:______
Chapter 3 Note Packet on Linear Systems
Notes#11: 3-1 Solving Linear Systems by graphing and 3-3 Graphing Linear Inequalities
Practice: In 1-2, determine if each ordered pair is a solution to the system:: .
1. (1, 2) 2. (7, 6)
Solving a system of linear equations by graphing.
Practice: In 1-2, Graph the linear system and state the solution. Then check the solution algebraically.
1. 2.
Types of Linear Systems
· Linear systems that have a solution or many solutions are said to be consistent.
o Linear systems that have only one solution are also called independent.
o Linear systems that have many solutions are also called dependent.
· Linear systems that have no solution are said to be inconsistent.
Practice: In 1-3, Solve the system by graphing. Then classify the system as consistent and independent, or consistent and dependent, or inconsistent.
1. 2. 3.
Graph to #1 Graph to #2 Graph to #3
To Graph Systems of Linear Inequalities.
Practice: In 1- 3, Graph the system of inequalities and state an ordered pair that is a solution, if possible.
1. 2. 3.
4.)/ 5.)
HW#11: Pg. 156 #4-14 even, 26, 27, 28 & Pg. 171 #6-24 x3
Notes#12 Section 3-2 Solving Linear Systems Algebraically
The solution to a system of equations represents the ______where they ______.
It would be great if we could find a way to do this without having to graph the lines!
A. Substitution Method
· If in function notation, rewrite as x’s and y’s. (The function expression will be your y)
· Choose one equation - get ______alone
· Substitute (______) this variable with the new expression in the other equation – USE PARENTHESES!!
· Solve
· Substitute back in the first equation to solve for the other variable
· Express your answer as a point (____, ____)
Solve using the substitution method:
1.)3.) / 2.)
4.)
5.) / 6.) C(n) = -4n – 3
C(n) = 2n + 9
Possible “strange” answers:
7.) / 8.)
B. Addition/Elimination Method
· Line up the two equations in Standard Form.
· Choose either the x’s or y’s to cancel – you want the ______(the number out front) to be equal but ______sign. If this doesn’t match yet, ______one/both equations by a number so that the coefficients do match.
· Add the equations together – watch one variable disappear.
· Solve and substitute back in the first equation to solve for the other variable
· Express your answer as a point (____, ____)
Solve using the addition method:
9.)11.) / 10.)
12.)
Possible “strange” answers:
13.) / 14.)
HW#12: Pg. 164 #4-48 x3’s & Review at end of note packet # 1-8
Notes#13: Section 3-4 Solving Systems of Linear Equations in Three Variables
Verifying a Solution:
Practice: In 1-2, determine whether the given ordered triple is a solution of the system. Label the point as (x, y, z) and plug the values into all three equations. In order for the point to be a solution, it must be true for all three equations.
1. (1, 4, 2): 2. (7, −1, 0):
Solving 3-Variable Systems: Use the Elimination Method
Example:/ Step1:
Label equations as A, B, C
Choose a variable to eliminate. Let’s choose x. / Step 2: Choose two pairs of equations to work with. Re-write these equations and use elimination to remove the x’s.
A A
B C
-1(A) -2(A)
B C
Step 3: Use elimination to solve the two boxed equations from Step 2. Solve for both y and z. / Step 4: Plug y and z into equation A, B, or C to solve for x. Write your final answer as a point
(x, y, z). Watch out for “strange” answers.
Practice: In 1-2, Solve each system using the elimination method. Follow the steps outlined above.
1. 2.
Substitution Method for 3-Variable Systems
Example:/ Step1:
Label equations as A, B, C.
Choose a variable to isolate in either equation A, B, or C.
Let’s choose to isolate x in equation B: / Step 2: Substitute this expression for x back into the other two equations (in this case, equations A and C). Simplify and leave in standard form.
A C
Step 3: Use elimination to solve the two boxed equations from Step 2. Solve for both y and z. / Step 4: Plug the values y and z from Step 3 into your x equation from Step 1. Write your final answer as a point (x, y, z). Watch out for “strange” answers.
Practice: In 1-2 Solve the system using the substitution method
1. 2.
HW#13: Pg. 182 #3-18 x3 & Review at end of packet # 9-17 all
Review for Chapter 3
Please complete each problem on a separate sheet of paper. Show all of your work!
NO WORK = NO CREDIT!
For questions 1-2, find the vertex of the graph of the following functions, determine whether the graph of the function opens up or down, state whether the graph is wider, narrower, or the same width as the graph of y = |x|. Then graph the function.
1. 2.
For questions 3-4, solve the system by graphing.
3. 4.
For questions 5-6, solve the system by substitution.
5. 6.
For questions 7-8, solve the system by elimination (linear combination) method.
7. 8.
For questions 9-10, graph each system of inequalities.
9. 10.
For questions 11-12, solve each system of equations by elimination or substitution.
11. 12.
For questions 13-15, simplify each expression.
13. 14.
15.
For questions 16-17, solve each equation.
16. 17.
2