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MTH 070
Elementary Algebra
Chapter 4 – Linear Systems of Equations and Inequalities
4.1
The Graphing Method
Copyright © 2010 by Ron Wallace, all rights reserved.
Product Costs/Revenue
x = number of items produced and sold
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Costs
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Fixed = $1500
Materials = $7/item
Packaging = $1/item
Shipping = $6/dozen
Labor = $10/item
Revenue
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Price = $40/item
R  40x
?
C  18.50x  1500
(70,
2800)
?
(69.8,
2790.7)
500
10
System of Equations
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A set of two or more (related) equations.
Solution – Values for the variables that
satisfy every equation in the system.
Example – Which of the following ordered
pairs is a solution to the system …
 x  2 y  5

 y  x7
(8,1) (3, 1)
(3, 4) (4,3)
System of Equations
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A set of two or more (related) equations.
Solution – Values for the variables that
satisfy every equation in the system.
Example – Which of the following ordered
pairs is a solution to the system …
y  x 2

 y  2x  2
2
( 2, 2) (0, 2)
(2, 2)
(1, 0)
Systems of Linear Equations in
Two Variables
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Equations, when graphed, are lines.
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Ax + By = C
y = mx + b
x = ny + a
y=b
x=a
(standard form)
(slope-intercept form)
(not common, but possible)
(equivalent to y = 0x + b)
(equivalent to x = 0y + a)
How many solutions?
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Remember: The graph represents ALL solutions!
Solving Systems by Graphing
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Basic Procedure:
1.
2.
3.
4.
Graph both equations.
Determine the point of intersection (it’s the solution).
Write the solution as an ordered pair.
Check the solution.
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It MUST work in BOTH equations.
Limitation of this method?
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Accuracy (works okay w/ integer solutions and small numbers)
Three Possibilities
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One Solution – The lines intersect at one point.
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Different slopes
Three Possibilities
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One Solution – The lines intersect at one point.
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Different slopes
No Solution – The lines do not intersect.
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Parallel lines
Same slopes
Different y-intercepts
Three Possibilities
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One Solution – The lines intersect at one point.
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No Solution – The lines do not intersect.
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Different slopes
Parallel lines
Same slopes
Different y-intercepts
Many Solutions – The lines coincide.
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Equivalent equations
Same slopes and intercepts.
Consistent Systems
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A system that has at least one solution.
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Lines intersecting at one point.
Lines that coincide.
No Solution  Inconsistent
Independent Systems
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A system where the equations are NOT
equivalent.
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Lines intersecting at one point.
Parallel lines.
Equivalent Equations  Dependent
Three Possibilities – Revisited
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One Solution – The lines intersect at one point.
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No Solution – The lines do not intersect.
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Consistent & Independent
Inconsistent & Independent
Many Solutions – The lines coincide.
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Consistent & Dependent