Solving+Systems+of+Equations

=Introduction=

In order to excel in a physics class, it is vital to master the algebraic concept of solving **systems of equations**. In order to achieve this, it is first important to know what the term "system of equations" means. If you have taken an algebra class, you will know that an **equation** is a mathematical representation of two expressions being equal to one another. Equations are often used to represent lines or shapes on a graph. To "solve" an equation means to calculate any **unknown variables** in the equation. Therefore, in order to solve a "system" of equations, you are essentially calculating any unknown variables in multiple equations.

When solving physics problems, there are often occurences in which you will end up with two equations, each with two variables. To find the value of these variables, you must know how to solve a system of equations.

toc =﻿ Review: Single Equations with Two Variables=

Before we discuss how to solve a system of equations, let's review how we solve a two variable equation to determine the value of a single variable.

Lets work with the following equation: 1**2x + 3y = 6.**

Now, though there is multiple variables in this equations, lets first only worry about finding the value of variable "y".

(1) Subtract 12x from both sides of the equal sign. (2) Divide 3 from both sides of the equal sign.

12x + 3y = 6 3y = -12x + 6 y = -4x +2

Now lets find the value of variable "x".

(1) Subtract 3y from both sides of the equal sign. (2) Divide 12 from both sides of the equal sign.

12x + 3y = 6 12x = -3y + 6 x = -1/4y +1/2

In both of these solutions, one undefined variable remains. How do we find the number value of both variables? With a single equation, it is impossible to calculate the number value of two variables. However, by using a system of equations, we can accomplish this easily.

=How to Solve Systems of Equations=

Now that we've reveiwed a simple two variable equation, lets take a look at a system of equations:


 * -x + y = 1**
 * 2x + 3y = 8**

As you can see, this system of equations has two variables, "x" and "y". We will be able to calculate the numerical values of both variables.

Systems of equations can be solved using a variety of different methods. However, the most basic yet effective ways to solve a system of equations problem is by using the the concepts of **substitution** and **elimination**.

Substitution
Substitution is when you insert the value of a variable into an equation. Substitution is a major algebraic concept and is used in a variety of different algebraic equations and problems.

Simple Substitution Application
Before we apply this concept to systems of equations, let's look at a basic application of substitution. (If you are familiar with the concept of substitution, then skip down below to the section headed "Solving a System of Equations Using Substitution".)


 * Ex.) When x = 3, Find the value for the equation y = 5x + 5.**

Because we know what the value of variable "x" is, we are able to find the value of "y" by essentially inserting, or "substituting", the value of "x" into the equation.

y = 5(3) + 5 y = 15 + 5 y = 20

Systems of equations uses this same concept, except instead of inserting a single number into other equations, we insert a combination of numbers of other variables.

General Approach:
(1) Solve one of the equations for a single variable, as demonstrated above in the section labeled "Reveiw: Single Equations with Two Variables". When choosing which equation and variable you want to work with, consider which would be the simplest to solve for. This will save you time. (2) Substitute this calculated value of the variable into the second equation. (3) You may notice the last step cancels out a variable. Therefore, with only one variable to worry about, solve the equation to find the numerical value of the remaining variable. (4) Substitute this numerical value into one of the original equations to find the value of the other variable. (5) The solution of a system of equations is written as (x, y).

Example:

 * -x + y = 1**
 * 2x + 3y = 8**

(1) -x + y = 1 y = x + 1

(2) 2x + 3y = 8 2x + 3(x + 1) = 8

(3) 2x + 3x + 3 = 8 2x + 3x = 5 5x = 5 x = 1

(4) -x + y = 1 -(1) + y = 1 -1 + y = 1 y = 2

(5) (1,2)

Elimination
Elimination is a method to solving systems of equations in which one variable is "eliminated" in order to find the value of the other variables. When this value is found, it is then substituted into one of the original equations in order to find the value of another variable. Elimination can be used to find more more than two variables.

General Approach:
(1) In this method, a variable is "eliminated" by essentially adding the equations together. However, in order to do this, you must alter the equations so that when one of the two variables are added to one another, one variable sum will equal "0", therefore, "eliminating the variable. This can be done by multiplying equations by a number that will give products that equal zero when added together. (2) Now that the equation(s) have been altered, add the equations together and eliminate a variable. (3) Now solve for the remaining variable. (4) Substitute the value you calculated in Step 3 into one of the two orignal equations and solve for the other variable.. (5) The solution of the system can be written as (x, y).

Example:
Lets use the same system of equations as we did previously in the "substitution section".


 * -x + y = 1**
 * 2x + 3y = 8**

(1) 2(-x + y = 1) -2x + 2y = 2

(2) -2x +2y = 2 + 2x + 3y = 8 5y = 10

(3) 5y = 10 y = 2

(4) -x + y = 1 -x + 2 = 1 -x = -1 x = 1

(5) (1, 2)

(This matches the answer we received for the substitution problem above.)

Solving Systems of Equations with Elimination: Three Variables
In some cases, you may encounter systems of equations that contain three variables.

General Approach:
(1) Pick two of the three equations. Alter them (as you did in Step 1 of the previous section) so that when added, one variable will be eliminated. Add the equations. (In the example below, the equations do not need to be altered.) (2) Pick a different two equations and repeat step 1. (3) Take the two new equatons that you have created and alter them so that when added, another variable willl be eliminated. Add the two new equations. (4) Solve to find the numerical value of the remaining variable. (5) Take the value you found in the last step and substitute it into one of the two new equations you worked with in step 4. Solve to find the numerical value of the second variable. (6) Plug in both the numerical values you have found so far into one of the original equations. Solve to find the numerical value of the last variable. (7) The solution of the system can be written as (x, y, z).

Example:
(1) x - 3y + 3z = -4 + 2x + 3y - z = 15
 * x - 3y + 3z = -4**
 * 2x + 3y - z = 15**
 * 4x - 3y - z = 19**

3x + 2z = 11

(2) 2x + 3y - z = 15 + 4x - 3y - z = 19

6x - 2z = 34

(3) 3x + 2z = 11 + 6x - 2z = 34

9x = 45

(4) 9x = 45 x = 5

(5) 3x + 2z = 11 3(5) + 2z = 11 15 + 2z = 11 2z = -4 z = -2

(6) 2x + 3y -z = 15 2(5) + 3y - (-2) = 15 10 + 3y + 2 = 15 12 + 3y = 15 3y = 3 y = 1

(7) (5, 1, -2)

= =