Math 010 - Cooley
Elementary Algebra
OCC
Section 7.1 – Systems of Equations and Graphing
A system of linear equations consists of two or more equations (in x and y) where a common solution is
sought. The solution of this system will be an ordered pair (x, y) that satisfies all the equations in the system.
y  x  5
A system of equations is usually denoted by: 
, where the solution is written as an ordered pair. In
 y  2x  4
this example, the ordered pair, (1, 6) satisfies both equations. Each equation corresponds to a line, when
graphed, and the ordered pair, (1, 6), is the point of intersection of the two lines.
When a system of two linear equations is graphed, three physical situations (solutions) are possible:
Name of System:
Number of
Solutions:
Consistent , Independent
1
(Exactly 1 solution)
What to look for:
Different slopes.
What going on:
Non-parallel lines.
Inconsistent , Independent
0
(No solution)
1. Same slope.
2. Different y-intercepts.
Parallel lines.
Consistent , Dependent
Infinitely many
(Infinite solutions)
1. Same slope.
2. Same y-intercepts.
Same line.
There are three techniques for solving a system of equations that are discussed in our text:
1. Graphically or Graphical Method. (see Section 7.1)
2. Substitution Method. (see Section 7.2)
3. Addition or Elimination Method. (see Section 7.3)
Method:
Procedure:
Efficiency Rating:
Explanation:
Graphical
Pictorial
Worst
This method produces an
approximate graphical
solution. It is hard to get
accurate results, but it does
show students how the type
of solution relates to the
physical situation.
Substitution
Computational
Okay
This method produces an
exact solution. It is a
medium difficulty
computational technique.
Generally, the Substitution
Method involves fractions
and is a little more time
consuming than the
Addition/Elimination
Method.
Addition/Elimination
Computational
Best
This method produces an
exact solution. It is also the
quickest and most efficient
method, and is preferred by
the vast majority of students.
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Math 010 - Cooley
Elementary Algebra
OCC
Section 7.1 – Systems of Equations and Graphing
Consistent – A system of equations that has at least one solution.
Inconsistent – A system of equations that has no solution.
Independent – A system of equations with no more than one solution.
Dependent – A consistent system of equations that has infinitely many solutions..
 Examples:
Determine the solution to each system of equations graphically. If the system is dependent or inconsistent, state
so.
x  2 y  6
 y  2x  1
2 x  y  3
a)
b)
c)



2 y  2 x  6
2 y  4 x  6
2 y  4 x  6
 Solution:
After putting each of the equations in slope–intercept form, we get…
 y   12 x  3

y  x  3
 y  2x  1

 y  2x  3
 y  2x  3

 y  2x  3
Before we graph, examine the slopes and the y-intercepts. See if you can describe the nature of the lines (i.e.,
parallel, non-parallel, coincident). Then based off that information, how many solutions do you expect for each
system? Then, what type of solution do you think we have? (i.e., consistent, inconsistent, dependent, independent).
Lines:
___________________
___________________
___________________
# of Solutions: ___________________
___________________
___________________
Type:
___________________
___________________
___________________
Graph of solution (physical situation):
 y   12 x  3

y  x  3
 y  2x  1

 y  2x  3
 y  2x  3

 y  2x  3
Algebraic Solution:
(4, 1)  Exactly one solution.
Consistent & Independent
No Solution
Inconsistent & Independent
Infinite Solutions
Consistent & Dependent
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Math 010 - Cooley
Elementary Algebra
OCC
Section 7.1 – Systems of Equations and Graphing
 Exercises:
Determine the solution to each system of equations graphically. State whether the system is consistent or
inconsistent as well as dependent or independent.
1)
8
x  y  6

x  y  4
y
6
4
2
x
–8
–6
–4
–2
2
4
6
8
–2
–4
–6
–8
2)
 y   13 x  1

4 x  3 y  18
8
y
6
4
2
x
–8
–6
–4
–2
2
4
6
8
–2
–4
–6
–8
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