Mathematics for Economists

Chapter 3 : Equations and sums

By Lund University

Most of chapter 3 is devoted to equations but we have also included a section on the summation sign. We will introduce all the terminology that we need for a fundamental understanding of what it means to solve an equation. We will focus most of our attention on quadratic equations and systems of linear equations.

Equations: Rules and terminology

In this section, we will look at the basic terminology for equations and the rules that you can use to solve equations. We will look at the distinction between a variable and a constant in an equation and we will learn how to classify an equation allowing us to apply the appropriate methods to solve the equation.

Equivalent equations and inequalities

Equations, terminology

Equations, constants and variables

Classification of equations

Solving equations

Quadratic equations

Quadratic equations is an important class of equations. We will review the formula for solving a general quadratic equation and a reduced quadratic equation (which is a quadratic equation where the constant in front of the x-square term is one). In this section we will also look at how to factor a quadratic expression and learn how to completing the square of a quadratic expression.

Reduced quadratic equations

Quadratic equations

Factoring a reduced quadratic expression and completing the square

Factoring a quadratic expression and completing the square

Single equations: Problems

Exercises on quadratic equations and other equations

Problem: Solve linear and quadratic equations

Problem: Understanding the quadratic formula

Problem: Solve linear and nonlinear equations

Problem: Simplify by factoring

Problem: Solve equations

Problem: Factor expressions

Problem: Solve inequalities

Problem: Solve text-equations

Problem: Solve equations containing constants

Problem: Solve equations containing constants

Problem: Solve quadratic equations

Problem: Solve quadratic equations

Problem Solve quadratic text-equations

Problem: Solve equations containing square roots

Systems of linear equations

This section is an introduction to systems of linear equations. We will restrict the study to two linear equations and look at two methods for solving such a system. The first method is called the substitution method and the second is the elimination method.

Systems of equations

Problem: Solve systems of equations

Problem: Solve systems of equations

Problem: Solve systems of equations

Problem: Solve non-linear equations

Sums and the summation sign

The final section of this chapter is not really about equations but it could not find a better home. Here, we will look at the summation sign and the rules that you can use to simplify expressions using the summation sign. These rules are not new, they are the same rules that we met in chapter 2, such as the distributive law. However, in chapter 2 we only considered the sum of two elements. We need to extend this to sums of an arbitrary number of elements.

Summation sign

Summation rules

Problem: Evaluate sums

Problem: Calculate sum

Problem: True or false for sums

Problem: Sums involving the mean