# Solving Systems of Multivariate Equations

## Objectives

After reviewing this unit you will be able to:

• Use substitution techniques to solve systems of equations.
• Use the addition/subtraction techniques to solve systems of equations.

## Introduction

In the last unit we reviewed how to deal with solving for one variable when given a single equation that contains more than one variable. When we have only one equation that contains more than one variable, we cannot find a numeric answer for any one of those variables. However, often you will have situation where you will have not only more than one variable, but also more than one equation. In this unit we will be dealing with multiple equations, which are what we call a system of equations. The following example about supply and demand in economics demonstrates a system of two equations.

 Suppose that the demand for coal in the state of Maine is represented by the equation: –1/3x + 22 = D where x represents the quantity of coal mined per week. Let the supply be represented as: 2x + 8 = S If we would like to find out when the supply and demand are in equilibrium, meaning when D = S, we can set the equations equal and solve for our variable, x.

In this unit we will review how to solve for the variables given in systems of equations like the one above. We will first review some of the basic tools you will need, then two methods used for solving systems of equations.

## Solving Systems of Equations

When we have a system of equations, we can find a numeric value for more than one equation. As mentioned above, there are two methods used for solving systems of equations:

• substitution method

Note that either method may be used to solve any problem, but depending on the given information, one method may be advantageous. In this unit we will give guidelines for helping you determine which method you wish to use. When solving any system of equations, there must be the same number of equations as variables. If there are two variables, there must be two equations; three variables, three equations, etc. NOTE: In this unit we will only refer to systems of equations with two equations. You can use the same process for systems with more equations, but this requires using the steps in an iterative process. Once you have done this technique with two equations, you can apply it to systems with more.

### When to Use the Substitution Method

The substitution method is most useful when one of the equations can easily be solved for one variable. When solving for a variable, just take one of the equations and isolate a variable. For example, if you are given the following equations:

(a) y – 3x = 5, and
(b) y + x = 3

We could choose to solve for either variable in either of the equations. For example:

Given (a) y – 3x = 5 we can solve for y = 3x + 5

As we mentioned, either method could be used to solve any system of equations. But you should learn to do both, since there may be times when one is easier to use than the other.

### When to Use the Addition/Subtraction Method

The addition/subtraction method is most useful when one variable from both equations has the same coefficient in both equations, or the coefficients are multiplies of one another. For example, if we are given the following system of two equations:

(c) 2y – 6x = 4, and
(d) 4y + 5x = 3

If we look at the term with the y variable in each equation, the numerical coefficients of both are multiples of 2, this means the addition/subtraction method could be used to solve this system of equations.

 When to Use the Substitution Method There must be the same number of equations as variables. If there are two variables, there must be two equations; three variables, three equations, etc. One of the equations can easily be solved for one variable. When to Use the Addition/Subtraction Method There must be the same number of equations as variables. One variable from both equations has the same coefficient in both equations, or the coefficients are multiplies of one another.

If you'd like to review these methods in detail click on the appropriate link below:

Now try the practice problems. In each case you can choose which method to solve each problem.