Now that we’ve laid the groundwork for multi-variable equations by looking at linear equations, we’re going to look at systems of equations.
A system of equations is a set or a group of equations where the variables in however many equations you have represent the same quantities in each equation
The following transcript is provided for your convenience.
For instance, if we have the system of equations 3x+4y = 10, and 4x+8y = 12, we have x and y in both of these equations, and this x, and this x are the same, and this y and this y are the same.
And so, essentially you have two pieces of information about these variables instead of one, and the general rule is that with systems of equations, if you have the same number of equations as you have variables, you can solve for values for the variables. You’ll recall that the single two-variable equation, we couldn’t solve for values for x and y, we can only solve for those variables in terms of the other variable.
So, with a system of equations, if we have the same number of equations as variables, generally, we can solve for values for those variables. Now, there are two exceptions to that rule. One is if the two equations are contradictory.
The other situation we can’t solve for values is if the two equations are redundant.
So, for instance, if we have x+y = 2, and 2x+2y = 4. These two equations are redundant because they tell us the exact same thing. If we divide each term in the second equation by 2, what we’re left with is equation 1. And so, it’s really no better than having just one equation. We can’t solve it for the values for these two variables. One way to think about solving systems of equations, at least linear equations like this, is to think about it graphically.
The definition of what we’re looking for here can be shown very easily on a graph. Let’s say that this first line, the graph looks like this, and this second line, the graph looks like this, and we have some x- and y-axis here, let’s say this is x, and this is y, and this point right here is where these two lines intersect. And so, this point has an x coordinate, and a y coordinate, and the values of this x and y coordinate are the values for x and y that satisfy both of these equations, that’s how you can kind of think about systems of equations graphically.
And, in this case here, where we have the contradictory equations, this would essentially be parallel lines. The equations x+y = 5 and x+y = 1 would look something like this. You have these two parallel lines that never intersect, and so that’s why you can’t solve these for either variable. And, in this case, with the redundant equations, this is as though you have one line drawn on top of another line. The solution for this is the entire line.
So, in both of these cases, there’s no way to solve for x and y, but if you have the same number of equations as variables, and they’re not contradictory and they’re not redundant, you should be able to find what is graphically the intersection point, and solve for values for the variables. In the next couple of videos, we’re going to demonstrate a of different methods for solving systems of equations.