In this lesson, our instructor Vincent Selhorst-Jones teaches Systems of Linear Equations. Youll learn how about graphs as location of ‘true and systems as graphs. Vincent also goes over three possibilities for solutionsindependent, inconsistent, and dependent. Youll also learn how to solve these types of problems by substitution and elimination. Vincent ends the lesson by going over graphing with a calculator and systems with more than two variables. Five practice questions and one (non) example conclude the lecture.
where A, B, and C are all constant numbers. Notice that each of the variables has just a power of 1, similar to a linear polynomial only having a degree of 1.
A system of equations is a group of multiple equations that are all true at the same time. The solution to a system is some set of variables that satisfies all the equations in the system at the same time.
When we first introduced graphs in this course, we talked about how a graph can be viewed as the location of all points that make the system true. With this idea in mind, we can graph each equation in a system of equations. Wherever they intersect is
a solution to the system because all of the equations are simultaneously true at an intersection.
There are three possibilities when solving a system of linear equations:
1 Solution-Independent: the equations only agree at one location;
0 Solutions-Inconsistent: it is impossible for the equations to agree;
∞ Solutions-Dependent: the equations agree at infinitely many locations (they are overlapping);
A system of linear equations can be solved by substitution: solving for one variable in terms of the other(s), then "plugging in". This gives an equation we can solve normally, then go back and find the other variable(s).
A system of linear equations can also be solved by elimination: adding a multiple of one equation to the other equation(s) to eliminate variables and give an equation that can be be solved normally.
When working with either of the two above methods, it's possible for the system to be any of the three types mentioned above. Here's how to tell which type you're working with:
Independent (1 solution): The system solves "normally" and you get values for the variables.
Inconsistent (0 solutions): While solving, you get an impossible equation (like 5=8).
Dependent (∞ solutions): While solving, you get an always true equation (like 7=7).
If you have access to a graphing calculator, you can also find the solutions to a system of equations by graphing all the equations, then find any intersection points using the calculator.
If a system of linear equations has more than two variables, the above methods (substitution and elimination) still work great. They become more time-consuming the more variables we have to solve for, but there is no issue in using them for any arbitrary
number of variables.
Systems of Linear Equations
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.