SYSTEM OF EQUATIONS

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SYSTEM OF EQUATIONS INEQUALITIES

• CHAPTER 6
• DCT1043

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CONTENT

• 6.1 System of Linear Equations
• 6.11 Solve using inverse matrix
• 6.12 Solve using Cramers Rule
• 6.13 Solve using Gauss Gauss Jordan
• Elimination Method
• 6.2 System of Nonlinear Equations
• 6.3 System of Inequalities

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6.1 System of Linear Equations

• By the end of this topic, you should be able to
• Discuss system of linear equations and the types
  of solution namely unique, inconsistent and
  infinite solutions.
• Write a system of linear equations in matrix form
• Solve a system of linear equation by using
  inverse matrix, Cramers Rule, and Gauss
  Gauss-Jordan Elimination Method.

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What is system?

• is an assemblage of entity/objects, real or
  abstract, comprising a whole with each and every
  component/ element interacting or related to
  another one.
• Solar system, blood system, computer system,
  ext..

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System of Linear Equations
The system of linear equations
where
Can be written in matrix form as
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Augmented Matrix
For the system of linear equations
where
The augmented matrix is given by,
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Types of solution
m Number of Row n Number of
Column Unique only 1 solution (the system is
consistent) Infinite many solution (the system
is consistent) None No solution (the system is
not consistent)
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6.11 Solve using Inverse Matrix

• Only for Square matrix
• The formula given by

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Examples 1 (Solve using Inverse Matrix)
Solve each of the following system of equality by
Inverse Matrix
1
2
3
4
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6.12 Solve Using Cramers Rule

• Only for Square matrix
• The formula given by

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Examples 2 (Solve Using Cramers Rule)
Solve each of the following system of equality by
Cramers Rule
1
2
3
4
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6.13 Solve Using Gauss Gauss-Jordan
Elimination Method

• For any matrix
• Gauss Elimination Method
• Reduce the augmented matrix Ab into row
  echelon form
• Starting with the last nonzero row, use
  back-substitution to find X
• Gauss-Jordan Elimination Method
• Reduce the augmented matrix Ab into reduced
  row echelon form IX

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Examples 3 (Solve Using Gauss Gauss-Jordan
Elimination Method)
Solve each of the following system of equality by
Gauss Gauss-Jordan Elimination Method
1
2
3
4
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Example 4 (Solve system of equation )

• Use inverse matrix, Cramers Rule, and Gauss
  Gauss-Jordan Elimination Method to solve the
  following system of equation. Compare you answer.

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6.2 System of NonLinear Equations

• By the end of this topic, you should be able to
• Solve a System of NonLinear Equations using
  substitution
• Solve a System of NonLinear Equations using
  elimination

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Solve a System of NonLinear Equations

• System of NonLinear Equations contains 1 or more
  nonlinear equation.
• The solution(s) represent the point(s) of
  intersection (if any) of the graphs of the
  equations.
• There is no general methodology
• Substitution, elimination or neither
• If the system contains 2 variables easy to
  graph (lines, quadratic (parabolas), hyperbolas,
  circles ellipse), then graph them.

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Examples 5 (Solve system of NonLinear
Equations)
Solve each of the following system of nonlinear
equality
1
2
3
4
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6.3 System of Inequalities

• By the end of this topic, you should be able to
• Graph an inequality
• Graph a system of Inequalities

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Graph an Inequality

• Replace the inequality symbol by an equal sign
  and graph the resulting equation
• If the inequality is strict, use dashes mark
• If the inequality is non-strict, use a solid mark
• In each of the regions, select a test point P
• If the coordinate of P satisfy the inequality,
  then all the points in that region satisfy the
  inequality. Indicate this by shading the region
• If the coordinate of P do not satisfy the
  inequality, then none of the points in that
  region do.

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Examples 6 (Graph an Inequality)
Graph each of the following Inequality
1
2
3
4
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Graph a system of inequality

• Graph each inequality in the system
• Superimpose all the graphs
• The overlapping regions are the solutions of the
  system.
• If there is no overlapping region, the system has
  no solution.

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Examples 7 (Graph a system of Inequality)
Graph each of the following system of Inequality
1
6
2
3
4
5
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THaNk YoU