Sullivan Algebra and Trigonometry: Section 11.1 Systems of Linear Equations

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Sullivan Algebra and Trigonometry Section
11.1Systems of Linear Equations

• Objectives of this Section
• Solve Systems of Equations by Substitution
• Solve Systems of Equations by Elimination
• Identify Inconsistent Systems
• Express the Solutions of a System of Dependent
  Equations

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A system of equations is a collection of two or
more equations, each containing one or more
variables.
To solve a system of equations means to find all
solutions of the system.
When a system of equations has at least one
solution, it is said to be consistent otherwise
it is called inconsistent.
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If the graph of the lines in a system of two
linear equations in two variables intersect, then
the system of equations has one solution, given
by the point of intersection. The system is
consistent and the equations are independent.
y
Solution
x
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If the graph of the lines in a system of two
linear equations in two variables are parallel,
then the system of equations has no solution,
because the lines never intersect. The system is
inconsistent.
y
x
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If the graph of the lines in a system of two
linear equations in two variables are coincident,
then the system of equations has infinitely many
solutions, represented by the totality of points
on the line. The system is consistent and
dependent.
y
x
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Two Algebraic Methods for Solving a System
1. Method of substitution2. Method of
elimination
STEP 1 STEP 2 Solve for x in (2)
Substitute into (1)
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STEP 3 Solve for y
STEP 4 Substitute y 4 into (2)
Solution (3, 4)
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STEP 5 Verify solution
Solution (3, 4)
(1)
(2)
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Multiply (1) by 3
When adding these 2 equations, you get
0x 0y 33
This equation has no solution so the system is
inconsistent. The lines are parallel.
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Consistent Dependent.