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Elementary
Mathematical
Modeling
Da Zheng
3.5 Systems of
Linear
Equations
Elementary Mathematical Modeling
Chapter 3. Straight Lines and Linear Functions
3.5. Systems of Linear Equations
Da Zheng
University of Houston
March 5, 2014
Introduction
Elementary
Mathematical
Modeling
Da Zheng
3.5 Systems of
Linear
Equations
Remember in chapter 2, we learned how to solve the
intersection of two functions, that is, by applying the
crossing graph method. When it comes to the case of
solving a system of linear equations, indeed, we will see
that we can also use this method.
Before we learn the methods to solve systems of linear
equations, let’s look at the definition of a system of
linear equations.
Introduction
Elementary
Mathematical
Modeling
Da Zheng
3.5 Systems of
Linear
Equations
A system of linear equations is pair of linear equations of
the following form:
a1 x + b 1 y = c 1
a2 x + b 2 y = c 2 .
We put the two equations together to mean that we are
solving for x and y which satisfy the two equations
simultaneously. So, to obtain the solutions, we have to
make use of both equations.
Algebraic Way of Solving Linear Systems
of Equations
Elementary
Mathematical
Modeling
Da Zheng
3.5 Systems of
Linear
Equations
First, we introduce a way to solve by hand. To explain it,
let’s see the following example:
Example
You have $36 to spend on refreshments for a party. Large
bags of chips cost $2.00 and drinks cost $0.50. You need
to buy five times as many drinks as bags of chips. How
many bags of chips and how many drinks can you buy?
Algebraic Way of Solving Linear Systems
of Equations
Elementary
Mathematical
Modeling
Da Zheng
3.5 Systems of
Linear
Equations
Let’s say that we buy x bags of chips and y drinks. So
the total cost will be 2x + 0.5y.
Since we have $36 to spend, 2x + 0.5y = 36.
However, we need to buy five times as many drinks as
bags of chips. That is, y = 5x. Hence, we obtain the
following system of linear equations:
2x + 0.5y = 36 (1)
y = 5x (2)
Algebraic Way of Solving Linear Systems
of Equations
Elementary
Mathematical
Modeling
Da Zheng
3.5 Systems of
Linear
Equations
To solve it, we first consider equation (2). It tells us
y = 4x. So, replacing the variable y in equation (2) by
y = 4x, we have
2x + 0.5 · 5x = 36.
This is merely a usual linear equation. Solve it, we have
4.5x = 36 =⇒ x = 8.
Now, use equation (2), i.e. y = 5x, we obtain y = 40.
In-class Problems
Elementary
Mathematical
Modeling
Da Zheng
3.5 Systems of
Linear
Equations
Problem
Solve the following system of linear equations
3x + 2y = 14
2x − y = 0
Graphical Way of Solving Linear Systems
of Equations
Elementary
Mathematical
Modeling
Da Zheng
3.5 Systems of
Linear
Equations
We try solve the following system of linear equations by
using another method.
2x + 0.5y = 36 (1)
y = 5x (2)
Recall the technique of reversing the roles of variables,
apply this technique to equation (1), we have
y = 72 − 4x.
Graphical Way of Solving Linear Systems
of Equations
Elementary
Mathematical
Modeling
Da Zheng
3.5 Systems of
Linear
Equations
But equation (2) tells us we must also have y = 5x. So
we are indeed solving the intersection of the two
functions: y = 5x and y = 72 − 4x. Now, apply the
crossing-graphs method in chapter 2, we have x = 8,
y = 40.
In-class Problems
Elementary
Mathematical
Modeling
Da Zheng
3.5 Systems of
Linear
Equations
Problem
Solve the following system of linear equations
3x + 2y = 6
4x − 3y = 8