Advanced Algebra II
3.1 Solve Linear Systems by Graphing (page 153)
Objective: Solve systems of linear equations by graphing
Vocabulary
Systems of linear equations (linear system)
Ax  By  C
Dx  Ey  F
Solution – the ordered pair (x, y) that satisfies each equation
 Point where the lines intersect
Consistent – system that has at least one solution
Inconsistent – system that has no solutions
Independent – a consistent system that has exactly one solution
Dependent – a consistent system that has infinitely many solutions
Concepts
Exactly One Solution
Infinitely Many Solutions
No Solution
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Advanced Algebra II
Classroom Examples
Graph the linear system and estimate the solution. Then check the solution algebraically.
y  8x  8
1)
2)
3)
3
y   x 8
2
4x  y  8
2 x  3 y  18
4 x  5 y  10
2x  7 y  4
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Advanced Algebra II
Solve the system. Then classify the system as consistent and independent, consistent and dependant, or
inconsistent.
4)
5)
6)
4x  3 y  8
8 x  6 y  16
2x  y  4
2x  y  1
2 x  y  5
y  x  2
7) A vendor sold 200 tickets for an upcoming concert. Floor seats were $36 and stadium seats were $28. The
vendor sold $6080 in tickets. Write a linear system to model the concert information.
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Advanced Algebra II
3.2 Solve Linear Systems Algebraically
(page 160)
Objective: Solve linear systems algebraically
Concepts
Substitution Method
1. Solve one equation for one variable.
2. Substitute the expression from step 1 into the other equation and solve.
3. Substitute the value from step 2 into the equation from step 1 and solve.
Elimination Method (Linear Combinations)
1. Multiply one or both equations by a constant to obtain coefficients that are opposites for one of
the variables.
2. Add the two equations and solve for the remaining variable.
3. Substitute the value from step 2 into one of the original equations and solve.
Classroom Examples
Solve the system using the substitution method.
1)
2 x  5 y  5
x  3 y  3
2)
4 x  3 y  2
x  5 y  9
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Advanced Algebra II
Solve the system using the elimination method.
3)
3x  7 y  10
6x  8 y  8
4)
3x  3 y  15
5x  9 y  3
Solve the system using any method. Then classify the system as consistent and independent, consistent and
dependant, or inconsistent.
5)
7)
x  2y  4
3x  6 y  8
3x  2 y  1
2 x  y  4
6)
8)
12 x  3 y  9
4 x  y  3
8x  2 y  4
2 x  3 y  13
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Advanced Algebra II
9)
11)
2x  3 y  4
6x  9 y  8
x y 4
6 x  6 y  24
10)
12)
3x  6 y  9
4 x  7 y  16
5 x  12 y  20
x  2y  6
13) A hair salon receives a shipment of 84 bottles of hair conditioner to use and sell. There are two types,
Type A is for regular hair and costs $6.50 per bottle, and Type B is for dry hair and costs $8.25 per bottle. If
the total sale for the conditioners is $588, how many of each type were in the shipment.
14) You and your sister combine your weekly overtime earning to buy a gift for your mother. Your overtime
rate is $18 per hour and your sister’s overtime rate is $24 per hour. The gift you are purchasing costs
$288. If you worked two more hours of overtime than your sister, how many overtime hours did each of
you work?
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Advanced Algebra II
3.3 Graph Systems of Linear Inequalities
(page 168)
Objective: Graph systems of linear inequalities
Vocabulary
System of linear inequalities
x y 8
4x  y  6
Solution – ordered pair (x, y) that is a solution for each of the inequalities in the system
Graph of system of linear inequalities – set of all solutions of the system (shaded region)
Concepts
Graphing a System of Linear Inequalities
1. Graph each inequality and use arrows to show which region should be shaded
2. Shade the region that is common to all of the graphs (region where all of the arrows are pointing)
Classroom Examples
Graph the systems of inequalities.
1)
2)
y  2 x  5
y  x3
y  3x  2
y  x  4
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Advanced Algebra II
2x  3y  6
3)
4)
5)
2
y  x4
3
y4
y  x 5
4x  2 y  8
y  2 x  3
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Advanced Algebra II
6)
x  y  3
6 x  y  1
y  x3
7) x  1
y  x  3
y  x 1
8)
1
x2
2
x  2
y
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Advanced Algebra II
Extension: Graph Linear Equations in Three Variables (page 177)
Objective: Graph linear equations in three variables
Concepts
Graphing Linear Equations in Three Variables
1.
2.
3.
4.
Solve for the x-intercept by setting y and z to equal 0
Solve for the y-intercept by setting x and z to equal 0
Solve for the z-intercept by setting x and y to equal 0
Plot and then connect the points.
(x, 0, 0)
(0, y, 0)
(0, 0, z)
Classroom Examples (will be a classroom handout)
Sketch the graph of the following equations.
1) 3x  4 y  6 z  12
2) 4 x  3 y  2 z  12
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Advanced Algebra II
3) 7 x  7 y  2 z  14
4) 2 x  9 y  3z  18
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Advanced Algebra II
3.4 Solve Systems of Linear Inequalities in Three Variables
(page 178)
Objective: Solve systems of equations in three variables
Vocabulary
Linear Equation in three variables - ax  by  cz  d
System of three linear equations
2x  y  z  5
3x  2 y  z  16
4 x  3 y  5z  3
Solution – ordered triple (x, y, z) whose values make each equation true

Exactly One Solution – planes intersect in a single point (x, y, z)

Infinitely Many Solutions – planes intersect in a line or are on same plane

No Solutions – planes have no common point of intersection
***For examples look at page 178 in textbook
Concepts
The Elimination Method for a Three-Variable System
1. Pick two equations and use elimination to get rid of one variable. Do this step twice eliminating the
same variable each time.
2. Solve the new linear system for both of its variables.
3. Substitute the values from step 2 into one of the original equations and solve for the remaining
variable.
*If you obtain a false equation, such as 0=1, in any of the steps, then the system has no solution.
**If you do not obtain an identity such as 0=0, then the system has infinitely many solutions.
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Advanced Algebra II
Classroom Examples
Solve the following systems.
4 x  2 y  3z  1
1) 2 x  3 y  5 z  14
6 x  y  4 z  1
2 x  y  6 z  4
2) 6 x  4 y  5 z  7
4 x  2 y  5 z  9
x yz 3
3) 4 x  4 y  4 z  7
3x  y  2 z  5
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Advanced Algebra II
x y z  4
4)
x yz  4
3x  3 y  z  12
3x  y  2 z  10
5) 6 x  2 y  z  2
x  4 y  3z  7
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Advanced Algebra II
x yz  2
6) 2 x  2 y  2 z  6
5 x  y  3z  8
x yz 3
7)
x yz 3
2x  2 y  z  6
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Advanced Algebra II
3.8 Use Inverse Matrices to Solve Linear Systems
(page 210)
Objective: Solve linear systems using inverse matrices
Vocabulary
Identity Matrix – an identity matrix is a square matrix with 1’s on the main diagonal and 0’s in all other entries.
Inverse Matrices – Two square (n x n) matrices A and B are inverse matrices if AB  I and BA  I An n x n
matrix has an inverse if and only if det A  0 . The symbol for the inverse of A is A1 .
Find the inverse of the following matrices.
1)
 3 8
A

 2 5
 2 1 2 


2) B  5 3 0


 4 3 8 
Use Inverse Matrix to Solve Linear System
1. Write system as matrix equation AX  B , Matrix A is coefficient matrix, X is the matris of variables,
and B is the matrix of constants.
2. Find the inverse of matric A
3. Multiply each side of AX  B by A1 on the left to find he solution X  A1B
Classroom Examples
Use an inverse matrix to solve the following linear system.
3)
4)
2 x  3 y  19
x  4 y  7
2 x  3 y  11
5 x  y  19
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Advanced Algebra II
Solve the following systems using inverse matrices.
4 x  y  z  20
5)
6 x  z  27
 x  4 y  5 z  23
2 x  y  z  2w  6
6)
3x  4 y  w  1
x  5 y  2 z  6w  3
5x  2 y  z  w  3
2 a  3b  c  4 d  e  2 f  g  6
4 a  7b  3c  d  4e  7 f  2 g  15
a  b  c  d  e  f  g  4
7) 2 a  b  3c  5d  e  4 f  5 g  54
 a  2b  2c  2 d  3e  2 f  4 g  4
3a  5b  2c  3d  2e  3 f  3 g  17
3a  2b  3c  4 d  2e  f  g  45
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