# 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.

Learning Objectives

At the end of the chapter, the reader should be able to do the following:

- Solve a system of equations of up to three variables.
- Implement solving equations up to five variables in Microsoft Excel.
- Identify topics in the introductory transportation engineering courses that build on the concepts discussed in this chapter.

## 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

Check Your Understanding: How to Set Up a System of Equations

### How to solve systems of equations graphically

Check Your Understanding: How to Solve Systems of Equations Graphically

### Testing a solution to a system of equations

Check Your Understanding: Testing a Solution to a System of Equations

### Solving System of Equations with Substitution

Check Your Understanding: Solving System of Equations with Substitution

### Solving Systems of Equations with Elimination

Check Your Understanding: 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

Check Your Understanding: Solving Linear Systems with Three Variables

### Solving Linear Systems with Three Variables: No Solution

Check Your Understanding: 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

Check Your Understanding: Solving Systems of Equations Using Matrices

### Solving System of Equations Using Excel

Check Your Understanding: Solving System of Equations Using Excel

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

Check Your Understanding: 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.

Key Takeaways

- There are two main approaches to traffic assignment in the travel demand modeling process: user equilibrium (UE) and system optimal (SO). Both of these approaches involve solving a system of equations to determine the number of trips that will be made on each route.
- In highway design equations that help ensure that the final design meets all necessary requirements and constraints need to be solved using the methods described in the chapter.
- In traffic operations, different segments of roadways with uninterrupted flow may have different relationships between speed, flow, and density, and these relationships can be used to set up equations for important parameters such as capacity flow rate, jam density, and free flow speed.

## Glossary: Key Terms

**Augmented Matrix[1]** – a matrix whose elements are the coefficients of a set of simultaneous linear equations with the constant terms of the equations entered in an added column

**Matrix[1]** – a rectangular array of mathematical elements of simultaneous linear equations that can be combined to form sums and products with similar arrays having an appropriate number of rows and columns

## Media Attributions

Note: All Khan Academy content is available for free at (www.khanacademy.org).

### Videos

- Video 1: “How to set up a system of equations” by Khan Academy is licensed by Creative Commons NonCommercial-ShareAlike 3.0 United States (CC BY-NC-SA 3.0 US)
- Video 2: “How to solve systems of equations graphically” by Khan Academy is licensed by Creative Commons NonCommercial-ShareAlike 3.0 United States (CC BY-NC-SA 3.0 US)
- Video 3: “Testing a solution to a system of equations” by Khan Academy is licensed by Creative Commons NonCommercial-ShareAlike 3.0 United States (CC BY-NC-SA 3.0 US)
- Video 4: Solving System of Equations with Substitution by Khan Academy is licensed by Creative Commons NonCommercial-ShareAlike 3.0 United States (CC BY-NC-SA 3.0 US)
- Video 5: “Solving Systems of Equations with Elimination” by Khan Academy is licensed by Creative Commons NonCommercial-ShareAlike 3.0 United States (CC BY-NC-SA 3.0 US)
- Video 6: “Introduction to Linear Systems with Three Variables” by Khan Academy is licensed by Creative Commons NonCommercial-ShareAlike 3.0 United States (CC BY-NC-SA 3.0 US)
- Video 7: “Solving Linear Systems with Three Variables” by Khan Academy is licensed by Creative Commons NonCommercial-ShareAlike 3.0 United States (CC BY-NC-SA 3.0 US)
- Video 8: “Solving Linear Systems with Three Variables: No Solution” by Khan Academy is licensed by Creative Commons NonCommercial-ShareAlike 3.0 United States (CC BY-NC-SA 3.0 US)
- Video 9: “Using Matrices to Solve Systems of Linear Equations” by Linda Green is licensed by Creative Commons Attribution 3.0 Unported (CC BY 3.0)
- Video 10: “Solving System of Equations Using Excel” by Roel Van de Paar is licensed by Creative Commons Attribution 3.0 Unported (CC BY 3.0)
- Video 11: How to Use the Solver Tool in Excel to Solve Systems of Linear Equations in Algebra by Rebecca Ong is licensed by Creative Commons Attribution 3.0 Unported (CC BY 3.0)

## References

- Farid, A. (2022). Transportation Planning 2. Personal Collection of Ahmed Farid, California Polytechnic State University, San Luis Obispo, CA.
- Farid, A. (2022). Highway Design. Personal Collection of Ahmed Farid, California Polytechnic State University, San Luis Obispo, CA.