# Systems of Linear Equations

### What Is a System of Linear Equations?

A system of linear equations is a set of two or more linear equations. The graph of a system of two equations is a pair of lines in the plane. Now you know what a system of linear equations is, let’s talk about how to solve systems of linear equations. For example, let’s say we have this problem. Problem Find the difference between two numbers using a system of linear equations. Solutions How to Solve a Problem using a System of Linear Equations Lets start by starting with a really simple system of equations. Problem Find the difference between x2 and y2 Solution x²+y² = x-y²+x-y = x-3y²-x+y²-2y²-3 = x3x3-2y2x+x3y3 What’s happening here? x2+y²+x²=x2+y2x²+x2-y2x²+x2y3. The answer is 9, so 8 + 2 = 9. What we just did was to solve the problem.

### The Basics of Systems

Mathematically, a system is a mathematical structure used to evaluate different sets of two or more linear equations. The formal definition of a system of two linear equations is given in, and may be expanded upon using some of the previously given examples. Practical Examples The following are examples of systems of two linear equations. These are examples of how they are used and constructed, and may not be given the formal definition that was given above. Example 1 : A system of two linear equations A system of two linear equations is written as a pair of equations. The first equation is called the “topic” and the second equation is called the “expectation”. The expression for the system is written as the sum of the topics and expectation.

### Algebraic Methods for Solving Systems

In each line, the first two variables are plotted as X and Y. Substituting the coordinates of the point you wish to solve a system of linear equations into the equation of the line, you can solve for a variable. Let’s say you want to find the equation of the line of (x, y) – 1, which is (2π/3). 2π/3 times 5 gives you 3, which is the equation of (5 + 2π/3). In terms of the variables x and y, you can solve for their xy values. So xy = (x + 2π/3) and yy = (y + 2π/3). You now have two equations that tell you how to turn x and y into a graph, and then use that graph to find y. Analogous with linear algebra, if you have the equation for y, you can work backwards to the equation for x, which tells you how to turn x into a graph.

### Graphing Systems

Graphing systems as above We can also express linear equations as matrices. So we can see that two linear equations x = 0.1 and y = 0.5 are matrices of size 1x and 2y. But matrices of a very simple shape can be displayed as a small black and white matrix in the workspace. This example shows that matrices can be used to display pretty complex shapes in a realistic way. Matrix Multiplication and Submatrices One of the cool things about matrix multiplication is that it can be expressed as a matrix commute: Since we can also represent the diagonal as an isomorphism, we can do a bit more. Given two matrices, we can list their corresponding submatrices. For example, a linear system x = 0.3 and y = 0.

### Other Methods to Solve Systems

To solve system of linear equations numerically, the system of equations is transformed into the matrix equations. The sum and difference of two matrices gives the sum and difference of a single matrix. There are other methods as well, to solve a system of linear equations, but they are not used as frequently in classroom. Here is an example of matrix transformation. Matrix Transformation Example Given the system of equations, the matrices A and B are transformed into matrix A ^ 1 and A ^ 2 . This results in the following matrix equation: 0 ^ 1 = 1 ^ 2 = 2 ^ 2 ^ 1 ^ 2 = 0 ^ 1 ^ 2 . Now solve the matrix equation using linear algebra and answer the question.

### Conclusion

The next time you are stuck for an answer, skip the explanations and focus on the flowchart below. You can quickly understand a system of linear equations and get a clearer understanding of what you should do next. The flowchart shows the connection between the variables x and y and the equations shown in the example above. When you focus on the diagram rather than the equations and the terminology, you are more likely to notice the errors in your own reasoning. When you do, you can immediately correct them. The diagram is also easier to understand and can be used by those who may have difficulty learning math. The diagram allows the diagram-to-flick learning method to work for anyone, no matter their learning ability.

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