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Int. Alg. Notes
Test #3 Review
Page 1 of 9
Section 4.1: Systems of Linear Equations in Two Variables
A system of linear equations is a grouping of two or more linear equations, each of which contains one or
more variables.
A solution of a system of linear equations consists of values for the variables that are solutions to ALL of the
equations in the system.
Geometric/Visual Interpretation of a System of Two Linear Equations in Two Variables:
INTERSECT: The lines intersect at one point, and thus the system has exactly one solution. This type of
system is called consistent and the equations are called independent.
PARALLEL: The lines never intersect (i.e., they are parallel to one another), and thus the system has no
solutions. This type of system is called inconsistent.
COINCIDENT: The lines lie on top of each other, and thus the system has infinitely many solutions. This type
of system is called consistent and the equations are called dependent.
Solving a System of Two Linear Equations by Graphing

Graph both the lines.

Read the coordinates of the intersection point off the graph.

Check to see if those coordinates are the solution.
Solving a System of Two Linear Equations Using Substitution

Solve one of the equations for one of the unknowns.

Substitute the expression for that unknown into the other equation.

Solve the resulting equation in one unknown.

Substitute that solution into the first equation to solve for the remaining variable.

Check your answer.
Solving a System of Two Linear Equations Using Elimination

Multiply one or both equations by nonzero constants so that the coefficients of one of the variables are
additive inverses.

Add the two equations to obtain a new equation in one unknown.

Solve the resulting equation in one unknown.

Substitute that solution into either of the original equations and solve for the remaining variable.

Check your answer.
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Test #3 Review
Page 2 of 9
Section 4.2: Problem Solving: Systems of Two Linear Equations Containing Two Unknowns
Big Skill: You should be able to write out a model for a real-world situation involving two equations in two
unknowns, then solve that system to get an answer.
Section 4.3: Systems of Linear Equations in Three Variables
Example of a system of three linear equations in three unknowns:
y  z  2
 x 

 x  2 y  3z  12
2 x  2 y  z  9

A linear equation with three variables describes a two-dimensional plane embedded in three dimensions:
When two planes intersect, the intersection is usually a line and when three planes intersect, the intersection is
usually a point:
y
x
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Test #3 Review
Page 3 of 9
Geometric/Visual Interpretation of a System of Three Linear Equations in Three Variables:
Exactly one solution: A consistent system with independent equations where the planes intersect at a single
point.
No solution: An inconsistent system where the planes are either all parallel, or intersect along parallel lines.
Inconsistent systems yield a false equation (like 0 = 3) after trying to solve them.
Infinitely many solutions: A consistent system with dependent equations where the planes all intersect along
the same line, or are all coincident. Dependent systems yield one or more equations of 0 = 0 after applying
Gaussian elimination.
Example of a system of three linear equations in three unknowns that is in triangular form:
x  2 y  z  1

y  2z  5


z  3

Notice: the name triangular form comes from the “blank” triangular space in the lower left corner due to no x
or y variables. The goal of solving a system of linear equations using elimination is to get the system into
triangular form, because a triangular form system is really easy to solve using back-substitution.
Key fact behind the technique of elimination:

Multiplying an equation in a system by a constant, or adding two equations in a system together results
in a new system of equations called a transformed system, and the solution to the transformed system is the
same as the solution to the original system.
Steps for Solving a System of Three Linear Equations in Three Unknowns Using Elimination

Eliminate the x variable from the second and third equations using elimination.

Eliminate the y variable from the third equation using elimination.

Solve for z in the third equation.

Substitute z into the second equation to find the solution for y, then substitute y and z into the first
equation to find the solution for x.
Rules for showing your work:

Draw an arrow from one transformed system to the next, and write on the arrow what you did to
transform the system.

Any equation that is unchanged gets copied from one system to the next.
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Test #3 Review
Page 4 of 9
Example:
y  z  2
 x 

 x  2 y  3 z  12
2 x  2 y  z  9

 (1)(eqn. #1) is placed in row#1
 2 x  2 y  2 z  4

y  4 z  14


 4y 
z  5

 (4)(eqn. #2) is placed in row#2
y  z 
2
 x 

 x  2 y  3 z  12
 2 x  2 y  z  9

 (eqn. #1)  (eqn. #2) is placed in row#2
 2 x  2 y  2 z  4

4 y  16 z  56


 4y 
z  5

 (eqn. #2)  (eqn. #3) is placed in row#3
y 
z 
2
 x 

y  4 z  14

 2x  2 y 
z  9

 2 x  2 y  2 z  4

4 y  16 z  56


 17 z  51

 (eqn. #3)  ( 17) is placed in row#3
 (2)(eqn. #1) is placed in row#1
 2 x  2 y  2 z  4

y  4 z  14

 2x  2 y 
z  9

 (eqn. #1)  (eqn. #3) is placed in row#3
 2 x  2 y  2 z  4

y  4 z  14


 4y 
z  5

 2 x  2 y  2 z  4

4 y  16 z  56


z  3

Substitute into equation #2:
4 y  16  3  56  y  2
Substitute into equation #1:
2 x  2  2   2  3  4  x  1
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Test #3 Review
Page 5 of 9
Section 4.4: Using Matrices to Solve Systems
Example of a system of three equations in three unknowns (that happens to be in triangular form) and
that system’s augmented matrix:
 1 2 1 1 
x  2 y  z  1



y  2 z  5  0 1 2 5


 0 0 1 3
z  3

Steps for Solving a System of Three Linear Equations in Three Unknowns Using an Augmented Matrix

Eliminate the x coefficients from the second and third rows using elimination.

Eliminate the y coefficient from the third row using elimination.

Make the coefficient of z in the third equation equal to one by dividing by the appropriate number.

Convert the triangular matrix back into a system of equations.

Substitute z into the second equation to find the solution for y, then substitute y and z into the first
equation to find the solution for x.
Rules for showing your work:

Draw an arrow from one transformed matrix to the next, and write on the arrow what you did to
transform the matrix.

Any row that is unchanged gets copied from one system to the next.
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Test #3 Review
Page 6 of 9
Example:
y  z  2
 x 

 x  2 y  3 z  12
2 x  2 y  z  9

 convert to augmented matrix
2 2 2 4 


 0 1 4 14 
 0 4 1 5
 (4)(row #2) is placed in row#2
 1 1 1 2 


1 2 3 12 
 0 2 1 9 
 (1)(row #1) is placed in row#1
2 2 2
4


 0 4 16 56 
 0 4 1 5
 (row #2)  (row #3) is placed in row#3
1 1 1 2 


 1 2 3 12 
 2 2 1 9 
 (row #1)  (row #2) is placed in row#2
2 2 2
4


 0 4 16 56 
 0 0 17 51
 (row #3)  (17) is placed in row#3
 1 1 1 2 


 0 1 4 14 
 2 2 1 9 
 (2)(row #1) is placed in row#1
 2 2 2
4


 0 4 16 56 
 0 0
1
3
 convert back to a system of equations
2 2 2 4 


 0 1 4 14 
 2 2 1 9 
 (row #1)  (row #3) is placed in row#3
 2 x  2 y  2 z  4

4 y  16 z  56


z  3

2 2 2

 0 1 4
 2 4 1
4

14 
5
Substitute into equation #2:
4 y  16  3  56  y  2
Substitute into equation #1:
2 x  2  2   2  3  4  x  1
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Test #3 Review
Page 7 of 9
Section 4.5: Determinants and Cramer's Rule
Definition: Determinant of a 2  2 matrix
a b
a b 
Suppose that a, b, c, and d are real numbers. The determinant of the 2  2 matrix 
, written as
,

c d
c d 
a b
is
 ad  bc .
c d
Cramer’s Rule for solving a system of two linear equations in two unknowns
ax  by  s
The solution to the system of equations 
is given by
 cx  dy  t
s
x
b
a s
Dy
t d
c t
D
 x and y 

a b
a b
D
D
c d
c d
Provided that D 
a b
c d
 ad  bc  0
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Test #3 Review
Definition: Determinant of a 3  3 matrix
 a1,1 a1,2

The determinant of the 3  3 matrix  a2,1 a2,2
 a3,1 a3,2
Page 8 of 9
a1,3 
a1,1

a2,3  , written as a2,1
a3,3 
a3,1
a1,2
a1,3
a2,2
a2,3 , is calculated using the
a3,2
a3,3
determinants of 2  2 matrices as follows:
a1,1
a1,2
a1,3
a2,1
a2,2
a2,3  a1,1
a3,1
a3,2
a3,3
a2,2
a2,3
a3,2
a3,3
 a1,2
a2,1
a2,3
a3,1
a3,3
 a1,3
a2,1
a2,2
a3,1
a3,2
.
Notice a pattern: the 2  2 determinants are what remains after the row and column corresponding to the matrix
element in front of the 2  2 determinant is blocked out.
Cramer’s Rule for solving a system of three linear equations in three unknowns
 a1 x  b1 y  c1 z  d1

The solution to the system of equations a2 x  b2 y  c2 z  d 2
a x  b y  c z  d
3
3
3
 3
with
a1 b1 c1
d1 b1 c1
a1 d1 c1
a1 b1 d1
D  a2 b2 c2  0 , Dx  d 2 b2 c2 , Dy  a2 d 2 c2 , and Dz  a2 b2 d 2
a3 b3 c3
d3 b3 c3
a3 d3 c3
a3 b3 d3
is given by
D
D
D
x  x , y  y , and z  z .
D
D
D
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Test #3 Review
Page 9 of 9
Section 4.6: Systems of Linear Inequalities
Solving a System of Linear Inequalities:

Graph each inequality separately

Choose the region that is the intersection of all the inequalities.
Example:
 2x

The graph of the solution of the system 3x
5 x


4y

8

2y

4 is:

3y

6
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.