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Chapter 2 Systems of Linear Equations and Matrices
• Systems of Linear Equations:
• An Introduction
• Unique Solutions
• Underdetermined and
• Overdetermined Systems
• Multiplication of Matrices
• The Inverse of a Square Matrix
• Leontief Input-Output Model
2.1 Systems of Linear Equations: An Introduction
•
Recall that a system of two linear equations in two variables
may be written in the general form
ax  by  h
cx  dy  k
where a, b, c, d, h, and k are real numbers and neither a and b nor
c and d are both zero.
• Recall that the graph of each equation in the system is a
straight line in the plane, so that geometrically, the solution to
the system is the point(s) of intersection of the two straight
lines L1 and L2, represented by the first and second equations
of the system.
Systems of Equations
•
Given the two straight lines L1 and L2, one and only one of the
following may occur:
1. L1 and L2 intersect at exactly one point.
y
L1
y1
Unique
solution
(x1, y1)
(x1, y1)
x1
x
L2
Systems of Equations
•
Given the two straight lines L1 and L2, one and only one of the
following may occur:
2. L1 and L2 are coincident.
y
L1, L2
Infinitely
many
solutions
x
Systems of Equations
•
Given the two straight lines L1 and L2, one and only one of the
following may occur:
3. L1 and L2 are parallel.
y
L1
L2
No
solution
x
Example: a system of equations with exactly one solution
•
Consider the system
3x  y  2
x  y  10
•
Solving the first equation for y in terms of x, we obtain
y  3x  2
•
Substituting this expression for y into the second equation
yields
x  (3x  2)  10
4 x  2  10
4 x  12
x3
Example: a system of equations with exactly one solution
•
Finally, substituting this value of x into the expression for y
obtained earlier gives
y  3x  2
 3(3)  2
7
•
Therefore, the unique solution of the system is given by
x = 3 and y = 7.
Example: a system of equations with exactly one solution
•
Geometrically, the two lines represented by the two equations
that make up the system intersect at the point (3, 7):
y
12
3x  y  2
10
8
(3, 7)
6
4
x  y  10
2
–2
2
4
6
8
10
12
x
Example: a system of equations with infinitely many solutions
•
•
Consider the system
2 x  y   4
4x  2 y  8
Solving the first equation for y in terms of x, we obtain
y  2x  4
•
Substituting this expression for y into the second equation
yields
4 x  2(2 x  4)  8
4x  4x  8  8
00
•
•
which is a true statement.
This result follows from the fact that the second equation is
equivalent to the first.
Example: a system of equations with infinitely many solutions
• Thus, any order pair of numbers (x, y) satisfying the equation
y = 2x – 4 constitutes a solution to the system.
• By assigning the value t to x, where t is any real number, we
find that y = 2t – 4 and so the ordered pair (t, 2t –4) is a
solution to the system.
• The variable t is called a parameter.
• For example:
•
Setting t = 0, gives the point (0, –4) as a solution of the
system.
•
Setting t = 1, gives the point (1, -2) as another solution of
the system.
Example: a system of equations with infinitely many solutions
•
•
Since t represents any real number, there are infinitely many
solutions of the system.
Geometrically, the two equations in the system represent the
same line, and all solutions of the system are points lying on
the line:
y
8
6
2 x  y  4
4x  2 y  8
4
2
-6 -4
-2
-2
-4
–6
2
4
6
x
Example: a system of equations that has no solution
•
•
Consider the system
2 x  y   4
4 x  2y  3
Solving the first equation for y in terms of x, we obtain
y  2x  4
•
Substituting this expression for y into the second equation
yields
4 x  2(2 x  4)  3
4x  4x  8  3
0  5
•
•
which is clearly impossible.
Thus, there is no solution to the system of equations.
Example: a system of equations that has no solution
•
To interpret the situation geometrically, cast both equations in
the slope-intercept form, obtaining
• y = 2x – 4 and y = 2x – 3/2
which shows that the lines are parallel.
• Graphically:
y
3
4x  2 y  3
2 x  y   4
2
1
-2
-1
-1
-2
-3
–4
1
2
3
4
x
A System of 3 Linear Equations
a1 x  b1 y  c1 z  d1
a2 x  b2 y  c2 z  d 2
a3 x  b3 y  c3 z  d3
can have
Infinitely many
solutions (line)
No solution
(parallel)
Linear Equations with n Variables
•
A linear equation in n variables x1 , x2 ,..., xn is one of the
form a1 x1  a2 x2  ...  an xn  c where a1 , a2 ,..., an  not all zero 
and c are constant.
Ex. 2x1  3x2  4 x3  5 x4  10 x5  12 is a linear equation in
the five variables, x1 , x2 , x3 , x4 , and x5 .
Example
The total number of passengers riding a certain city bus during
the morning shift is 1000. If the child’s fare is $0.50, the adult’s
fare is $1.25, and the total revenue from the fares in the morning
shift is $987.5, how many children and how many adults rode
the bus during the morning shift? (Formulate but do not solve
the problem.)
Solution
•
•
Let x = the number of children who rode the bus during the
morning shift
Let y = the number of adults who rode the bus during the
morning shift
We have the system of linear equations:
x +
y = 1000 (total passengers)
0.50x + 1.25y = 987.5 (total revenue)