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3.1 Systems of Equations in Two
Variables

Translating

Identifying Solutions

Solving Systems Graphically
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
System of Equations
A system of equations is a set of two or
more equations, in two or more variables, for
which a common solution is sought.
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Slide 3- 2
Example
T-shirt Villa sold 52 shirts, one kind at $8.25 and
another at $11.50 each. In all, $464.75 was taken in
for the shirts. How many of each kind were sold?
Solution
1. Familiarize. To familiarize ourselves with this
problem, guess that 26 of each kind of shirt was
sold. The total money taken in would be
26  $8.25  26  $11.50  $513.50
The guess is incorrect, now turn to algebra.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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2. Translate. Let x = the number of $8.25 shirts
and y = the number of $11.50 shirts.
Kind of
Shirt
Number
sold
Price
$8.25
shirt
x
$11.50
shirt
y
$8.25
$11.50
Amount
$8.25x
$11.50y
Total
52
$464.75
We have the following system of
equations: x  y  52,
x + y = 52
8.25x + 11.50y
= 464.75
8.25 x  11.50 y  464.75.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Identifying Solutions
A solution of a system of two equations in
two variables is an ordered pair of numbers
that makes both equations true.
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Slide 3- 5
Example
Determine whether (1, 5) is a solution of the system
x  y  4,
2 x  y  7.
Solution
x–y=–4
1–5 –4
–4=–4
TRUE
2x + y = 7
2(1) + 5 7
7=7
TRUE
The pair (1, 5) makes both equations true, so it is a solution
of the system.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Solving Systems Graphically
One way to solve a system of two equations
is to graph both equations and identify any
points of intersection. The coordinates of
each point of intersection represent a
solution of that system.
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Slide 3- 7
Example
Solve the system graphically.
x  y  1,
x y 5
Solution
y
x+y=5
7
x–y=1
6
5
4
It appears that (3, 2) is the
solution. A check by
substituting into both
equations shows that (3, 2)
is indeed the solution.
3
2
(3, 2)
1
-5 -4 -3 -2 -1
-1
1
2 3 4 5
x
-2
-3
-4
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 8
Example
Solve the system graphically.
y  2 x  3,
y  2x 1
y
7
y = 2x – 1
y = 2x + 3 6
Solution
5
The lines have the same
slope and different
y-intercepts, so they are
parallel. The system has
no solution.
4
3
2
1
-5 -4 -3 -2 -1
-1
1
2 3 4 5
x
-2
-3
-4
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 9
Example
Solve the system graphically.
3x  y  6,
2 y  6 x  12
Solution
The same line is drawn
twice. Any solution of one
equation is a solution of the
other. There is an infinite
number of solutions. The
solution set is
( x, y) | 3x  y  6.
y
3x – y = –6
7
6
5
2y – 6x = 12
4
3
2
1
-5 -4 -3 -2 -1
-1
1
2 3 4 5
x
-2
-3
-4
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 10
When we graph a system of two linear equations in
two variables, one of the following three outcomes
will occur.
1. The lines have one point in common, and that
point is the only solution of the system. Any
system that has at least one solution is said to be
consistent.
2. The lines are parallel, with no point in common,
and the system has no solution. This type of
system is called inconsistent.
3. The lines coincide, sharing the same graph. This
type of system has an infinite number of solutions
and is also said to be consistent.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 11
When one equation in a system can be
obtained by multiplying both sides of
another equation by a constant, the two
equations are said to be dependent. If two
equations are not dependent, they are said
to be independent.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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