Solving Systems of Equations and Inequalities

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Solving Systems of Equations and Inequalities

• Section 2.1 Two variable linear
  equations
• Section 2.2 Three variable linear
  equations
• Section 2.3 Matrices Resolution of linear
  systems
• Section 2.5 Determinants Inverses of Matrices
• Section 2.6 Solving Systems of Linear
  Inequalities
• Section 2.7 Linear
  Programming

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Section 2.1 Two Variable Linear Equations

• A set of two or more linear equations that each
  contain two variables.
• These equations represent lines that will
  intersect, overlap, or be parallel to each other.

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Section 2.1 Two Variable Linear Equations

• When these lines intersect or overlap each other
  they are said to be consistent. This means they
  have at least one point of intersection. If
  there is only one point of intersection, the
  lines are independent. If there is more than one
  point of intersection, the lines are dependent.
• When two lines do not intersect, they are said to
  be inconsistent because the lines are parallel.

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Section 2.1 Two Variable Linear Equations

• Check your understanding
• Given the following two linear equations,
  determine the consistency dependence of the
  system. 2x y 5 4x 2y 8
• Consistent, independent
• Consistent, dependent
• Inconsistent

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Section 2.1 Two Variable Linear Equations

• Check your understanding
• Given the following two linear equations,
  determine the consistency dependence of the
  system. 2x y 5 4x 2y 8
• Consistent, independent
• Consistent, dependent
• Inconsistent

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Section 2.1 Two Variable Linear Equations

• Check your understanding
• Given the following two linear equations,
  determine the consistency dependence of the
  system. 3x 2y 5 6x 2y 8
• Consistent, independent
• Consistent, dependent
• Inconsistent

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Section 2.1 Two Variable Linear Equations

• Check your understanding
• Given the following two linear equations,
  determine the consistency dependence of the
  system. 3x 2y 5 6x 2y 8
• Consistent, independent
• Consistent, dependent
• Inconsistent

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Section 2.1 Two Variable Linear Equations

• Systems of two linear equations are usually
  solved using either the elimination method or the
  substitution method.
• The elimination method uses a process of removing
  one of the two variables simultaneously from both
  equations through addition of equal but opposite
  coefficients.

In blue we have y 3x - 2, and in red we have y
-x 2.
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Section 2.1 Two Variable Linear Equations
In blue we have y 3x - 2, and in red we have y
-x 2.
Blue 3x y 2 Red x y 2
4x 4
With the variable y having equal but opposite
values we can add the two equations together and
eliminate the y variable. We get 4x 4. This
allows x 1, and by inserting x 1 back into
the equation we can see that y 1.
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Section 2.1 Two Variable Linear Equations

• The substitution method uses a process of
  rewriting one of the two equations by isolating
  one of the variables and then substituting the
  equation into the other equation.

In blue we have y 3x - 2, and in red we have y
-x 2.
With both equations in the form of y equals, we
can substitute the 2nd equation into the first
and we get -x 2 3x 2. Solving for x we
get -4x -4 and x 1.
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Section 2.1 Two Variable Linear Equations

• Check your understanding
• Given the following two linear equations, use the
  substitution method to start the solution of the
  system.
• 5x 3y 22 6x 2y 20
• Y 10 6x
• Y 10 3x
• Y 10 3x

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Section 2.1 Two Variable Linear Equations

• Check your understanding
• Given the following two linear equations, use the
  substitution method to start the solution of the
  system.
• 5x 3y 22 6x 2y 20
• Y 10 6x
• Y 10 3x
• Y 10 3x

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Section 2.2 Three variable linear systems

• Three linear equation systems are solved using
  the elimination and substitution methods.
• The technique requires the isolation of one of
  the variables amongst the three equations.
• This gets repeated for a 2nd variable and the
  remaining variable is then determined.
• Afterwards, the other variables are determined.

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Section 2.2 Three variable linear systems

• Sample
• X 2y 2z 10
• 2x y 2z 6
• X 3y 2z 1
• Solve for x, y, z.

Isolate the z value first. Combine line 1 2
and then combine line 3 2.
X 2y 2z 10 ? X 2y 2z 10 2x y 2z
6? -2x y 2z -6
-x 3y 4
X 3y 2z 1 ? X 3y 2z 1 2x y
2z 6 ? -2x y 2z -6
-x - 2y -5
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Section 2.2 Three variable linear systems

• Sample
• X 2y 2z 10
• 2x y 2z 6
• X 3y 2z 1
• Solve for x, y, z.

Isolate the x value next. Combine line 1 2
answer and the line 3 2 answer.
X 2y 2z 11 ? X 2y 2z 11 2x y 2z
6 ? -2x y 2z -6
-x 3y 5
X 3y 2z 1 ? X 3y 2z 1 2x y
2z 6 ? -2x y 2z -6
-x - 2y -5
-x 3y 5 ? -x 3y 5 -x -2y -5 ? x 2y
5 5y 10 Y 2, x 1
Solve for z by reinserting y x values. 1
2(2) 2z 10 Z 2.5
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Section 2.3 - Matrices

• Matrix A rectangle array of terms (elements)
  arranged in columns and rows. A matrix with m
  rows and n columns is called an m x n matrix,
  (read m by n matrix).
• Matrices are also used to determine solutions for
  multiple variable linear equations. This
  technique can be used as an alternative to
  elimination or substitution methods.

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Section 2.3 - Matrices

• a11 a12 a13
• a21 a22 a23
• a31 a32 a33

3 x 3 Matrix The first number indicates the row
(horizontal) and the second number indicates the
column number (vertical).
Equal Matrices Two matrices are equal if and
only they have the same dimensions and are equal
element by element.
This expression states that Y 2x 6 and x
2y. Using the substitution method, we see that
Y 2(2y) 6 and so y 2, x 4.
2x 6 2y
Y X

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Section 2.3 - Matrices
Addition of Matrices The sum of two m x n
matrices is a m x n matrix in which the elements
are the sum of the corresponding elements of the
given matrices.

-6 7 -1 4 -3 10
Solve for A B.

-2 0 1 0 5 -8
B
A
-2 (-6) 0 7 1 (-1) 0 4 5 (-3)
-8 10
-8 7 0 4 2 2

A B

A B
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Section 2.3 - Matrices
Subtraction of Matrices The difference of two m
x n matrices is equal to the sum A (-B) where
(-B) is the additive inverse of B.

-6 7 -1 4 -3 10
Solve for A - B.

-2 0 1 0 5 -8
B
A
-2 - (-6) 0 - 7 1 - (-1) 0 - 4 5 - (-3)
-8 - 10
4 -7 2 -4 8 -18

A - B

A - B
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Section 2.3 - Matrices
Scalar Product The product of a scalar k and an
m x n matrix A is an m x n matrix denoted by kA.
Each element of kA equals k times the
corresponding element of A.

Solve for kA.
5

-2 0 1 0 5 -8
k
A
5(-2) 5(0 ) 5(1) 5(0) 5(5) 5(-8)
-10 0 5 0 25 -40

kA

kA
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Section 2.5 Determinants and Inverses

• A determinant is a square array of numbers
  (written within a pair of vertical lines) which
  represents a certain sum of products.
• Calculating a 2 2 Determinant
• In general, we find the value of a 2 2
  determinant with elements a, b, c, d as follows
• We multiply the diagonals (top left bottom
  right first), (bottom left x top right) then
  subtract the first product minus the second.

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Cramers Rule
Cramers Rule begins with the solving of the
determinant for the system followed by the
determinants for each of the variables within the
system.
The determinant for each of the variables is
calculated by first substituting the solution
column values for the variable column values and
then worked on as a 2 x 2 matrix.
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Cramers Rule (continued)
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Cramers Rule (conclusion)