Math 141: Business Mathematics I
Fall 2015
§2.1 Systems of Linear Equations: An Introduction
Instructor: Yeong-Chyuan Chung
Outline
• What are systems of linear equations?
• Geometric interpretation of solutions of such systems
• Setting up a system of linear equations from a word problem
Systems of Two Linear Equations in Two Variables
A system of linear equations is simply a list of linear equations. In this context, we often
write the terms involving the variables on the same side of the equation. For now, we will
consider systems where the number of equations is the same as the number of variables.
Later on in the chapter, we will consider systems where these two numbers are different.
We have already encountered examples of systems of linear equations when we were discussing break-even points and market equilibrium. In those situations, we had systems of
two linear equations in two variables, and we discussed how to find the point of intersection
of the corresponding lines. The coordinates of the point of intersection can also be called
the solution of the system of linear equations.
In general, given two linear equations in two variables, there are three possibilities with
regard to the solution of the system, depending on the positions of the lines.
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§2.1 Systems of Linear Equations: An Introduction
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Example (Exercises 2,4,8 in the text). Solve the following systems of linear equations.
(a)
2x − 4y = −10
3x + 2y = 1
(b)
3x − 4y = 7
9x − 12y = 14
(c)
5x − 6y = 8
10x − 12y = 16
§2.1 Systems of Linear Equations: An Introduction
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Systems with More Equations and More Variables
A linear equation may contain more than two variables. For instance, a linear equation in
three variables may be written as ax + by + cz = d where a, b, c, d are constants. Such an
equation represents a plane in three-dimensional space. If we have a system of three linear
equations in three variables, then we have three planes, and the point(s) of intersection of
the three planes is/are the solution(s) of the system.
There are again three possibilities with regard to the solution of the system, depending on
the positions of the planes.
Figure 1.1: Image from Finite Mathematics for the Managerial, Life, and Social Sciences
We can also consider systems with more than three variables. In this case, we no longer
have the geometric interpretation we had in the cases of two or three variables because we
cannot visualize higher dimensional spaces. However, the algebraic techniques that we will
introduce later will work just as well regardless of the number of variables.
Setting Up Systems of Linear Equations from Word
Problems
For now, we will get some practice in setting up systems of linear equations from word
problems. Later on, we will develop techniques to solve such systems.
A word problem may contain lots of information. It is crucial that we extract the relevant
ones and organize them. The first step is to identify what the variables should be and to
define them. The question at the end of the problem usually provides a hint. The next step
is to translate the sentences into equations.
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§2.1 Systems of Linear Equations: An Introduction
In the following examples, formulate but do not solve the problem.
Example (Exercise 36 in the text). A theater has a seating capacity of 900 and charges $4
for children, $6 for students, and $8 for adults. At a certain screening with full attendance,
there were half as many adults as children and students combined. The receipts totaled $5600.
How many children attended the show?
Example (Exercise 44 in the text). Joan and Dick spent 2 weeks (14 nights) touring four
cities on the East Coast - Boston, New York, Philadelphia, and Washington. They paid
$240, $400, $160, and $200 per night for lodging in each city, respectively, and their total
hotel bill came to $4040. The number of days they spent in New York was the same as the
total number of days they spent in Boston and Washington, and the couple spent 3 times as
many days in New York as they did in Philadelphia. How many days did Joan and Dick stay
in each city?