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17Calculus Precalculus - Systems of Equations

Single Variable Calculus
Multi-Variable Calculus

Types of Systems of Equations

Linear with the same number of equations as unknowns - covered on this page

Linear with fewer equations than unknowns - covered on the dependent system page

Non-Linear - covered on the non-linear system page

Systems of equations are made up of a set of two or more equations that contain multiple variables. Linear systems have all linear terms, i.e. all the variables have a power of one and none of the variables are multiplied together. Nonlinear systems have at least one variable with a power other than one, like a square root or a squared term, or have variables multiplied together.

Linear Systems of Equations - 3 Cases

For example, here is a linear system of two equations and two unknowns, x and y.
\( \begin{array}{rcrcl} 2x & + & y & = & 3 \\ x & - & 3y & = & 0 \end{array} \)
Notice that we have two variables and two equations. In this case, we have three possible types of solutions.
1. We have one, unique solution. If we think of these equations as lines in the plane, there is one point where they intersect.
2. We have no solution. In this case, the lines would be parallel and never intersect.
3. We have an infinite number of solutions, i.e. the lines completely overlap and the two equations represent the same line. When we have an infinite number of solutions or we have fewer equations than variables, then we have what is called a dependent system of equations. We can get some equations that 'solve' the system, i.e. we can get a set of equations that represent the infinite set of solutions. We cover that idea on a separate page. On this page, we will stick with linear systems where we have the same number of equations as variables.

Before we go on, let's watch a quick video clip about some terminology related to these three cases of linear systems.

Thinkwell - Three Cases for Linear Systems [1min-10secs]

video by Thinkwell

Graphically, the three possible (2-dimensional) cases are shown here. The idea is to find where the lines intersect.

1. One, Unique Solution

2. Infinite Solutions

3. No Solution

Case 1: One, Unique Solution
In this case, the two lines intersect at exactly one point. This is the easiest, most common and the best case possible. The other two cases are usually considered special situations.

Case 2: Infinite Solutions
In this case, the two lines are exactly the same. In the plot above, it looks like we have only one line. However, when we have two equations and we plot them, the lines are the same and so in graphs it looks like there is only one line. Since the lines are the same, the lines intersect at every point and, therefore, we say there are an infinite number of solutions.

Case 3: No Solution
When the lines are parallel, they will never intersect. So we say there is no solution since there is no point where they intersect.

Note: Some books and instructors switch the two cases above and call case 2 no solution and case 3 infinite solutions. The order does not matter, so it is best to follow your instructor and textbook.

Here is another good video showing all three solutions, side-by-side including graphs.

Khan Academy - Solving systems of equations (2x2), the special solution cases [9min-11secs]

video by Khan Academy

Overview of Techniques

Solving systems of equations can be intuitively thought of as finding where the equations intersect. This is a great way to get a feel for what is happening. There are several ways to solve systems of equations.

Using Basic Algebra and Graphing




Using Matrices

Gaussian Elimination/Row Reduction

inverse matrices

Cramer's Rule

This introduction video goes into a little more detail on the first 3 techniques. The presenter actually crosses out the use of matrices option but on this site we cover those techniques.

Brightstorm - Introduction to Systems of Equations [3min-14secs]

video by Brightstorm

Important Last Step: Checking Your Answers

Before we discuss specific techniques, let's talk about how to make sure you have the correct answer. I know what you are thinking, you are busy and it takes time to check your answer. Besides, you were careful when you worked the problem. That's true. However, think about it this way. If you made a mistake, you will not get full credit for your work and some of your time was wasted. And you have a way to check your answer (not always possible in higher math) and if you don't check it, you will regret it later. If only I had checked my answer! I might have gotten an A! It takes just a few seconds and if you do find an error, you may learn how to not repeat it and improve your grade on your next exam. So it is well worth the time to check.

To check your answer, just plug the point into all of the original equations. Using the original equations is extremely important, since there could be a mistake at any point. And make sure and check all of the equations, since your answer may work in one but not all of them.

Okay, let's start with solving these problems by discussing 3 basic techniques, graphing, substitution and elimination, starting with graphing.

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3 Cases

Overview of Techniques

Important Last Step


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