College Algebra 9th txtbk

496 CHAPTER 7 SYSTEMS OF EQUATIONS AND MATRICES
70. If
A c a
c
and
show that det (AB) (det A)(det B).
It is clear that x 0, y 0, z 0 is a solution to each of the
systems given in Problem 71. Use Cramer’s rule to determine
whether this solution is unique. [Hint: If D 0, what can you
conclude? If D 0, what can you conclude?]
71. (a). x 4y 9z 0 (b). 3x y 3z 0
4x y 6z 0 5x 5y 9z 0
x y 3z 0 2x y 3z 0
72. Prove Theorem 2 for y.
APPLICATIONS
b
d d
73. REVENUE ANALYSIS A supermarket sells two brands of coffee:
brand A at $p per pound and brand B at $q per pound. The daily demand
equations for brands A and B are, respectively,
x 200 6p 4q
y 300 2p 3q
(both in pounds). The daily revenue R is given by
R xp yq
B c w
y
x
z d
(1)
(A) To analyze the effect of price changes on the daily revenue, an
economist wants to express the daily revenue R in terms of p and q
only. Use system (1) to eliminate x and y in the equation for R, expressing
the daily revenue in terms of p and q.
(B) To analyze the effect of changes in demand on the daily revenue,
the economist now wants to express the daily revenue in
terms of x and y only. Use Cramer’s rule to solve system (1) for p
and q in terms of x and y and then express the daily revenue R in
terms of x and y.
74. REVENUE ANALYSIS A company manufactures ten-speed and
three-speed bicycles. The weekly demand equations are
p 230 10x 5y
q 130 4x 4y
where $p is the price of a ten-speed bicycle, $q is the price of a
three-speed bicycle, x is the weekly demand for ten-speed bicycles,
and y is the weekly demand for three-speed bicycles. The weekly
revenue R is given by
R xp yq
(A) Use system (2) to express the daily revenue in terms of x and y
only.
(B) Use Cramer’s rule to solve system (2) for x and y in terms of
p and q, and then express the daily revenue R in terms of p and q
only.
(2)
CHAPTER
7 Review
7-1 Systems of Linear Equations
A system of two linear equations in two variables is a system of the
form
ax by h
cx dy k
where x and y are variables; a, b, c, and d are real numbers called
the coefficients of x and y, and h and k are real numbers called the
constant terms in the equations. The ordered pair of numbers
(x 0 , y 0 ) is a solution to system (1) if each equation is satisfied by the
pair. The set of all such ordered pairs of numbers is called the solution
set for the system. To solve a system is to find its solution set.
In general, a system of linear equations has exactly one
solution, no solution, or infinitely many solutions. A system of linear
equations is consistent if it has one or more solutions and
inconsistent if no solutions exist. A consistent system is said to be
independent if it has exactly one solution and dependent if it has
more than one solution.
To solve a system by substitution, solve either equation for
either variable, substitute in the other equation, solve the resulting
linear equation in one variable, and then substitute this value into the
expression obtained in the first step to find the other variable.
(1)
Two systems of equations are equivalent if both have the same
solution set. To solve a system of equations using elimination by
addition, use Theorem 2 to find a simpler equivalent system whose
solution is obvious.
As stated in Theorem 2, a system of linear equations is transformed
into an equivalent system if:
1. Two equations are interchanged.
2. An equation is multiplied by a nonzero constant.
3. A constant multiple of another equation is added to a given
equation.
The solution set S of a dependent system is often expressed
in terms of a parameter. Any element in S is called a particular
solution.
Any equation that can be written in the form
ax by cz k
where a, b, c, and k are constants (not all a, b, and c zero) is called
a linear equation in three variables. The method of elimination
by addition can be used for systems of linear equations in three
variables.