Systems of Linear Equations. Methods for Solving Them. - Revision Notes

In these revision notes for Systems of Linear Equations. Methods for Solving Them., we cover the following key points:

• What are linear equations?
• How do we plot the graph of a linear equation?
• What are the systems of linear equations?
• What is the solution set of a system of linear equations?
• How do we solve systems of linear equations?
• How to verify whether a system of linear equations is solved correctly or not?

Systems of Linear Equations. Methods for Solving Them. Revision Notes

Linear equations are first - order equations with two variables. They are called "linear" because their graph is a straight line.

All points in the graph of first - order equations with two variables are collinear. We can connect them with each other so that we obtain a straight line known as a linear graph. We need only two points (pairs of coordinates) to plot a linear graph. It is preferred that these two points be the intercepts with the X- and Y-axes, i.e. the points A(0, y) (the Y - intercept) and B(x, 0) (the X - intercept), but it is up to you to choose the points you see as the most appropriate.

It is impossible to solve a single linear equation, as the number of possible solutions is infinite. The only thing we can do to calculate exact values for variables in such conditions is to simultaneously solve a pair of linear equations. In this way, we (perhaps) obtain a common pair of numbers as a solution, shown graphically by the intercept point of the two lines that represent each line involved.

Often, you will see linear equations involved in a system written in the form ax + by = -c instead of the standard form ax + by + c = 0.

There are three methods for solving a system of linear equations, where two of which are numerical methods and the third one is graph - based.

1. Elimination method, which consists of the elimination of one of the variables through the addition of the two equations to each other. This is done by multiplying one or both equations by suitable coefficients, to make it therefore possible to cancel the variable we choose to eliminate from the resulting equation. Then, we continue with the solution because the new equation obtained, now has only one variable. After calculating the value of this variable, we substitute it in any of the original equations to find the value of the other variable (the one that was eliminated before).
2. Substitution method, which consists of choosing an equation from the system for expressing one of the variables in terms of the other variable and then, substituting the corresponding expression in the other equation. In this way, we obtain a first - order equation with one variable.
3. Graphing method, which consists of plotting the graphs of each linear equation in the system and checking the coordinates of the meeting point, as they represent the solution set of the system. This is because at the meeting point both equations have the same values for the variable x and also for the other variable y.