Hello. In this video, we're going to learn about solving simultaneous equations using multiple methods. To start, as we define simultaneous equations, so simultaneous equations are a set of equations with multiple variables that are being solved together. So for example, on the left here, we have two equations of two parameters X and Y, and we'd like to solve for the value of X and Y that satisfy these two equations. On the right, we have three equations in three unknowns X, Y, and Z, and we'd like to find the values of X, Y, and Z that simultaneously satisfy those three equations. In our circuit class, those variables X, Y, and Z can be the current and voltage at different branches and nodes respectively, and we'd like to find the value of the current and voltage that satisfy some certain constraints using KCLs, KVLs, and node and loop analysis. Okay, so to reiterate, our goal is to find the solution where all equations in the system are simultaneously satisfied.

In order to do so, we have four popular methods to solve simultaneous equations. One is the substitution method. The second one is the elimination method. The third one is the graphical method. We are going to cover these three methods in this video. And the fourth one is the matrix method, and we're going to cover it over the next two videos. So let's start with the method of substitution.

With the method of substitution, let's assume that we have two equations like these. What we're going to do is we're going to use one of the two equations to solve for one variable as a function of the other variable. So here, I can solve for X as a function of Y, so I can write that X = 4 - Y / 2. Once I have this expression for X, I'm going to substitute it into the second equation. Okay, so in the second equation, I have X, so 4 - Y over 2 (so this is X) and then minus Y. So what we have here is X equal to 1. So now the second equation after substituting what we have for X is now one equation in one variable, which is Y. We can solve for Y. Once I have a solution for Y, I can substitute Y back here into that expression for X, and this completes my solution.

So let's see the steps here. So I have the two equations. I'm going to solve or express one variable in terms of the other using one of the two equations. So here, we use equation number one to solve for X as a function of Y. Once I have this expression for X, I'm going to substitute that X expression into equation number two. So in equation number two, I have X here. I'm going to substitute that X via the expression that we have from equation number one. Now this is one equation in one unknown Y. I can algebraically modify it and solve for Y. Once I have this final solution for Y, I can use this expression or either of the two equations to solve for X, since the value of Y now is known. So use any equation, substitute Y here is equal to two. This is the solution that we obtain for Y. We can immediately solve for X. So this method is called the substitution method.

The second method that we have is the method of elimination. With the method of elimination, we look at our equations and we see how we can algebraically add them or subtract them so that we can get rid of some of the variables. So for example, in this example here, if I add equation one to equation two, you can see that I'm going to get rid of Y, and then I'm going to end up having one equation and one unknown, which is X. I can solve for X, and once I have a value for X, I can use either of the two equations to solve for Y. So let's see this example in more detail.

So here, I have equation one and equation two. Algebraically multiply by any constant and then add or subtract so that you can get rid of one of the two variables. So here, I don't need to multiply by anything because addition directly will get rid of Y. So the equation that I have would be 3X = 3. This would give me a solution for X. Once I have this solution of X, I can substitute in any of the two equations to solve for Y. So here, we choose equation number one, substitute X by its solution, which is one. Now we have only one equation in Y, which can give us Y = 2. And note that this is the same solution that we obtained in the method of substitution. These methods are different ways to solve the system of equations, but the solution would be the same no matter which approach you use.

The third approach that we have is the graphical method. But this graphical method would be convenient if and only if we have two variables, like for example here X and Y. But if we have X, Y, and Z, we will need to plot a three-dimensional figure, which might not be convenient. And the previous two methods are more convenient than the graphical method. So if you have two equations in two unknowns, what I'm going to do is these two equations are linear equations, so they can be plotted as a line equation in the XY space or variable one and variable two. Okay, and again, this XY can be the current in element number one, or this is maybe the voltage at node number one in our circuit analogy. Okay, but here for generality, let's just keep them as two variables X and Y. So all what I need to do is plot a line, and as you know, to plot a line, I need two points on that line. Each equation corresponds to a line, and we are looking for the solution XY that satisfies the two equations. So the solution would be the point of intersection of the two lines.

So for example, the two equations that we have throughout this video here, I can represent this line as Y = 4 - 2X, and I can plot this line using two values for X. I can get the corresponding value of Y and generate a line. Likewise here, we can write that Y is equal to X + 1, so this is another line equation. The solution, so here I have the two lines, the solution is the point of intersection because this point of intersection is the point that satisfies the two lines, i.e., the solution that satisfies the two equations. So this would be our solution. You can see here the value of X is equal to one, and the value of Y is equal to two. So this is our solution: X is equal to 1 and Y is equal to 2.

So these are the three methods that we can use for solving simultaneous equations. The fourth method is by constructing a matrix form, and this is what we're going to learn in the next video. Thank you.