Name:_____________________________________
Chapter 3 Note Packet on Linear Systems
Period:__________
Notes#11: 3-1 Solving Linear Systems by graphing and 3-3 Graphing Linear Inequalities
A system of two linear equations in two variables x and y has the form:
A solution to a system of linear equations in two variables is an ordered pair that satisfies both
equations in the system.
Practice: In 1-2, determine if each ordered pair is a solution to the system::
1. (1, 2)
2 x  3 y  4
.
2x  y  4
2. (7, 6)
Solving a system of linear equations by graphing.
1. Rewrite each equation in slope-intercept form.
2. Determine the slope and y-intercept and use them to graph each equation
3. The solution to the linear system is the ordered pair where the lines intersect.
4. Check your solution algebraically by substitution into both equations. In order to be a solution it must
make both equations true.
Practice: In 1-2, Graph the linear system and state the solution. Then check the solution algebraically.
1.
3x  2 y  4
2x  y  1
2.
x  2y  2
x  4 y  14
Types of Linear Systems
1

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.
Number of Solutions of a Linear System: There are only three possible outcomes.
The relationship between the graph of a linear system and the system’s number of solutions is described
below:
Exactly One Solution
Infinitely Many Solutions
No Solution
Lines intersect at one point.
The point of intersection is the
solution of the system. These
systems are called consistent
and independent.
Lines coincide (overlap) have
infinitely many solutions. Any
ordered pair that is on the
coinciding lines is a solution.
These systems are called
consistent and dependent.
Lines that are parallel have no
solution since they never
intersect. These systems are
just called inconsistent.
Practice: In 1-3, Solve the system by graphing. Then classify the system as consistent and independent, or
consistent and dependent, or inconsistent.
3x  y  12
2 x  3 y  15
20 x  12 y  24
1.
2. 1
3.
3
5x  3 y  6
 x  4 y  2
x  y  3
2
2
Graph to #1
Graph to #2
Graph to #3
2
To Graph Systems of Linear Inequalities.
To graph a system of linear inequalities, follow these steps:
1. Graph each inequality in the system.
2. Identify the region that is common to all the graphs of the inequalities. This region is the graph
of the system. The graph of the system is the region where the shading overlaps.
Practice: In 1- 3, Graph the system of inequalities and state an ordered pair that is a solution, if possible.
1.
y  3x  2
y  x  4
2.
4x  2 y  8
y  2 x  3
y3
y  x4 2
5.) x  5
y  2
y  x 1
4.) 2 x  3 y  6
y  3 x  2  5
y
y
10
10
9
9
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
1
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
3.
1
2
3
4
5
6
7
8
9 10
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
1
2
3
4
5
6
7
8
9 10
-2
-3
-4
-5
-6
-7
-8
-9
-10
HW#11: Pg. 156 #4-14 even, 26, 27, 28 & Pg. 171 #6-24 x3
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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:
y  3
b  a 1
2.)
1.)
2x  3y  3
b  3 a
3.)
y   x  15
4 x  3 y  38
5.)
d  3c  1
d  5c  2
Possible “strange” answers:
2x  8 y  6
7.)
x  4y  8
4.)
2m  3n  4
m  4n  9
6.)
C(n) = -4n – 3
C(n) = 2n + 9
8.)
y  2x  4
6 x  3 y  12
4
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.)
x  2y  7
3x  2 y  5
10.)
11.)
3c  6d  6
5c  2d  14
12.)
Possible “strange” answers:
2x  8 y  6
13.)
x  4y  8
14.)
6 x  y  14
10 x  y  2
3 x  5 y  10
5 x  7 y  10
2x  y  4
6 x  3 y  12
HW#12: Pg. 164 #4-48 x3’s & Review at end of note packet # 1-8
5
Notes#13: Section 3-4 Solving Systems of Linear Equations in Three Variables
A linear equation in three variables x, y and z has the form:
A solution to a system of linear equation in three variables is an ordered triple (x, y, z) that satisfies
all three equations in the system.
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.
2 x  y  5 z  12
4 x  6 y  z  34
1. (1, 4, 2): 5 x  4 y  2 z  17
2. (7, −1, 0): 2 x  5 y  8 z  9
3x  2 y  z  7
5 x  2 y  4 z  33
Solving 3-Variable Systems: Use the Elimination Method
1. Label your equations as A, B, and C. Now choose a variable to eliminate: x, y or z.
2. Choose two pairs of equations to work with. (A and C or A and B, etc). Use the elimination
method twice to eliminate the variable that you have chosen.
3. You should now have two equations with the same two variables leftover. Solve this system using
elimination.
4. Substitute the values found in Step 3 into one of the original equations and solve for the remaining
variable.
 If you obtain a false equation, such as 1 = 2, in any of the steps, then the system has no
solution.
 If you do not obtain a false equation, but obtain an identity such as 0 = 0, then the system
has infinitely many solutions.
6
Example:
x  2y  z  0
x  y  2z  7
2 x  y  3z  5
Step1:
Label equations as A, B, C
Choose a variable to
eliminate. Let’s choose x.
Step 3: Use elimination to solve the two boxed
equations from Step 2. Solve for both y and z.
Step 2: Choose two pairs of equations to work with.
Re-write these equations and use elimination to
remove the x’s.
A
B
A
C
-1(A)
B
-2(A)
C
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.
3x  2 y  4 z  11
x yz 2
1. 2 x  y  3z  4
2. 3x  3 y  3z  14
5 x  3 y  5 z  1
x  2y  z  4
7
Substitution Method for 3-Variable Systems
1. Rewrite one of the three given equations by solving for one variable in terms of the other two
variables.
2. Substitute the expression found in step 1 into the other two equations given and simplify. This will
give you a system of two equations with only two variables.
3. Solve the new linear system that only has two equations with two variables. (See methods above
on how to solve a system of linear equations in two variables)
5. Substitute the values found in Step 3 into the equation formed in step 1 and solve for the remaining
variable.
 If you obtain a false equation, such as 1 = 2, in any of the steps, then the system has no
solution.
 If you do not obtain a false equation, but obtain an identity such as 0 = 0, then the system
has infinitely many solutions.
Example:
2 x  y  z  9
x  3y  z  1
3 x  y  2 z  12
Step1:
Label equations as A, B, C.
Choose a variable to isolate
in either equation A, B, or C.
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
Let’s choose to isolate x in
equation B:
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.
8
Practice: In 1-2 Solve the system using the substitution method
2 x  y  6 z  1
x  y  2 z  5
1. 3x  2 y  5 z  16
2. x  2 y  z  8
7 x  3 y  4 z  11
2 x  3 y  z  13
HW#13: Pg. 182 #3-18 x3 & Review at end of packet # 9-17 all
9
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
1. y  2 x  6  3
2. y   x  1  5
3
For questions 3-4, solve the system by graphing.
1
 3 x  y  0
3 x  y  10
3. 
4. 
 y  2x  5
1 x  4 y  9
 6
For questions 5-6, solve the system by substitution.
5 x  y  5
5 x  y  7
5. 
6. 
9 x  4 y  20
 y  6  5x
For questions 7-8, solve the system by elimination (linear combination) method.
x  y  1
7 x  12 y  22
7. 
8. 
9 x  8 y  0
5 x  8 y  14
For questions 9-10, graph each system of inequalities.
x  3
3x  y  4


9.  y  10
10.  y  2 x  1
x  y  7
 x  y  4


For questions 11-12, solve each system of equations by elimination or substitution.
2 x  4 y  z  18
2 x  y  2


11. 6 x  3 y  2 z  2
12.  x  3 y  4 z  26
5 x  6 y  z  17
4 x  y  6 z  37


For questions 13-15, simplify each expression.
 

15.  8 x3 yz 2   2 x    6 x 4 yz    z 
13. 2 x2 y
5
4 x3 y 2 z
2
3
2
 1

14.  m2n3   m4n 
3
 4

2
For questions 16-17, solve each equation.
16. 3x  1  7
17. 4 x  5  9  1
10