Transcript Angles, Degrees, and Special Triangles

Linear & Nonlinear Systems of
Equations
MATH 109 - Precalculus
S. Rook
Overview
• Section 7.1 in the textbook:
– Solving systems of equations by substitution
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Solving Systems of Equations by
Substitution
Verifying a Solution to a System
• Graphically, when asked to
solve a system of equations,
we are looking for the
point(s) of intersection of
the functions
• Any coordinate (x, y) that satisfies BOTH equations is
known as the solution of the system
• Two common methods to solve a system:
– Substitution (this section)
– Elimination (next section)
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Solving Systems of Equations by
Substitution
• Useful when the coefficient of a variable is 1 or -1
• To solve a system of equations by substitution:
– Solve one equation for one of the variables
– Substitute (replace) into the OTHER equation
• DO NOT substitute into the equation that was just used
to solve for a variable!
– What results is an equation in 1 variable
– After solving this equation, we now have a partial
coordinate
– Evaluate the partial coordinate into either equation to
get the full coordinate
• Often easier to plug into the substitution
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Solving Linear Systems of Equations by
Substitution (Example)
Ex 1: Solve the system by substitution:
2 x  y  2  0
a) 
4 x  y  5  0
1
1
x y 8

b)  5 2

x  y  20
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Solving Systems of Non-Linear
Equations by Substitution (Example)
Ex 2: Solve the system by substitution:
 y  2x
a) 
2
y

x
1

xy  1  0

b) 
2 x  4 y  7  0
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Inconsistent & Dependent Systems
• Inconsistent systems: a system of equations which
has NO solution
– Graphically, the equations will never intersect
• Dependent systems: a system of equations which
has an INFINITE number of solutions
– Graphically, the equations lie on top of each other
– “All real numbers” does NOT apply
– Use a parameter variable to express the solution
• Will get a better idea after an example
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Inconsistent & Dependent Systems
(Example)
Ex 3: Solve the system by substitution:
 6 x  5 y  3
a) 
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 x  6 y  7
 x  3 y  12
b) 
2 x  6 y  24
x  2 y  4
c)  2
x  y  0
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Summary
• After studying these slides, you should be able to:
– Solve linear and non-linear systems of equations by
substitution
• Additional Practice
– See the list of suggested problems for 7.1
• Next lesson
– Two-Variable Linear Systems (Section 7.2)
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