Chapter 3: Systems of Linear Equations

The ability to solve systems of multiple equations is a critical component of an introductory transportation course. Systems of equations are helpful in several key transportation analyses, including travel demand estimation (e.g., traffic assignment), design (e.g., roadway alignment), and operations (e.g., speed-density relationships). This chapter highlights how to solve systems of equations with two variables, three variables, and when no solution occurs. It will also explain how to use Microsoft Excel to solve system equations.

Solving Systems of Equations with Two Variables

In this section, you will learn how to setup a system of equations and solve linear equations by reading each description along with watching the videos. Also, short problems to check your understanding are included.

How to set up a system of equations

How to solve systems of equations graphically

Testing a solution to a system of equations

Solving System of Equations with Substitution

Solving Systems of Equations with Elimination

Solving Systems of Equations with Three Variables

In this section, you will learn how to set up a system of equations with three variables and solve a system of equations when no solution occurs by watching the videos. Short problems to check your understanding are included.

Introduction to Linear Systems with Three Variables

Solving Linear Systems with Three Variables

Solving Linear Systems with Three Variables: No Solution

Systems of Equations in Microsoft Excel

In this section, you will learn how to set up a system of equations using Microsoft Excel and matrices and how to use the Microsoft Excel solver tool by watching the videos. Short problems to check your understanding are included.

Solving Systems of Equations Using Matrices

Solving System of Equations Using Excel

How to Use the Solver Tool in Excel to Solve Systems of Linear Equations in Algebra

Relevance to Transportation Engineering Coursework

This section explains the relevance of traffic assignment and operation along with highway design to transportation engineering coursework.

Traffic Assignment

In travel demand modeling process, the last step is the traffic assignment for routes, which involves estimating the number of trips made on each route on the roadway network. Two commonly applied traffic assignment approaches, which involve solving a system of equations, are the user equilibrium (UE) and system optimal (SO) assignments. For simple UE problems helpful in conceptual understanding of these approaches, the system of equations is typically linear. The number of variables to solve for depends on the number of routes being considered for traffic assignment. The reader is referred to the above section titled “Solving Systems of Equations with Two Variables” for solving a system of linear equations with two variables and the section titled “Systems of Equations in Microsoft Excel” for solving a system of linear equations with three variables. For solving a system with more than three variables one requires use of computer-based tools which are discussed in “Systems of Equations in Microsoft Excel”.

Highway Design

Analysis and design of roadway (or railway) elements, particularly horizontal alignment and vertical profiles, often require solving a system of equations. These equations are set up based on the geometric constraints of the design. For example, the roadway elevation at a certain location may be fixed due to an intersecting element. These constraints are used to set up system of simultaneous equations that require the use of methods discussed in the three previous sections in this chapter for solving them.

Traffic Operations

The relationships between highway speed, lane density (a measure of congestion), and traffic flow (number of vehicles past a point) are governed by a universal relationship of uninterrupted flow, i.e., flow rate is a multiple of speed and density. In addition, different segments are characterized by certain relationships between any of the variable pairs (i.e., speed-density; speed-flow; or flow-density). These relationships help set up equations for key parameters for roadway segments such as capacity flow rate, jam density, and free flow speed.