Transcript Solution

Chapter 8
Systems of Linear
Equations and
Problem Solving
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8.1
Systems of Equations in Two
Variables
• Translating
• Identifying Solutions
• Solving Systems Graphically
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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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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? Set up the equations but do not solve.
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.
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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.
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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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Example
Determine whether (1, 5) is a solution of the system
x  y  4,
2 x  y  7.
Solution
x – y = –4
1–5
2x + y = 7
–4
–4 = –4
TRUE
2(1) + 5 7
7=7
TRUE
The pair (1, 5) makes both equations true, so it is a
solution of the system.
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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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Example
Solve the system graphically.
x  y  1,
x y 5
x–y=1
Solution
(3, 2)
Graph both equations.
x+y=5
It appears that (3, 2) is the
solution. A check by
substituting into both
equations shows that (3, 2)
is indeed the solution.
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Example
Solve the system graphically.
y  2 x  3,
y  2x 1
Solution
Graph both equations.
The lines have the same
slope and different
y-intercepts, so they are
parallel. The system has no
solution.
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Example
Solve the system graphically.
3x  y  6,
2 y  6 x  12
Solution
Graph both equations.
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.
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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.
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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.
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