Hi, this is Anthony Varela. And this tutorial is about an introduction to systems of equations. So we're going to be talking about what a system of equations is. We'll talk about what the solution to a system means. And then we'll take a look at what a solution to a system looks like on a graph.
So what is a system of equations? Well, it's two or more equations with the same variables considered at the same time. So here's an example of a system of equations. y equals 2x minus 3, and y equals negative 5x plus 7.
Now they're grouped together, so you might see these curly braces to indicate that. You might not see that. So here's another example that is separated with a semicolon. Once again, y equals 2x minus 3, and y equals negative 5x plus 7.
The important thing, though, is that they contain the same variables. But more specifically, they have the same definitions. They represent the same quantities. So for example, over here, if we had that x equals cats and y equals dogs for this first equation, and x equals cats and y equals dogs for the second equation, this would represent a system of equations.
However, let's say that we have x equals cats and y equals dogs for this first equation right here. But in our second equation, x represents cats and y represents birds. Because their definitions don't match, this would not be a proper system of equations. So what's the solution then to a system of equations? Well, they have to satisfy all equations in the system, not just one, not just two, not just three, but all of the equations in the system.
So let's consider this system of equations here. We have y equals 2x minus 3, and y equals negative 5x plus 7. So it's easy enough to find a solution to one of the equations. So let's just pull up one of the equations. And I'm going to pick an x value.
So let's plug in 2 for x. So when x equals 2, we can find out what y equals. So 2 times 2 is 4. And then we'll take away 3. So y equals 1. So a solution to my one equation then is the point 2, 1. x equals 2, y equals 1.
Well, let's use x equals 2 and y equals 1 and put that in our second equation, y equals negative 5x plus 7. So plugging in 2 for x and 1 for y. And let's go ahead and see if this is a true statement.
Well, negative 5 times 2 is negative 10. And when I add 7, I get 3 or negative 3. But negative 3 is not equal to 1. So while the point 2, 1 or x equals 2, y equal 1 satisfies one of my equations, it doesn't satisfy the other. So it's not a solution to the system.
So now let's talk about solutions on a graph. So here I have two lines. And graphed together on the same graph, they represent a graph of a system. And taking a look at a point on a line, this is a solution to the line in blue, but not a solution to the system because it doesn't lie on the green line.
Now similarly, the other point that you see here, it's a solution to the green line, but not to the system because that point does not lie on the blue line as well. So what point then lies on both the blue line and the green line? That would be this point of intersection. So on a graph, a solution to a system of equations is where all of the lines that make up that system intersect. So here, there graphically is a solution to this system of two equations.
So I'm going to go ahead and write out the equations to these lines. Here we have y equals x plus 2 is a line you see in blue. And y equals negative x plus 4 is a line you see in green. And that solution, the point of intersection, is the coordinate point 1, 3.
So that means that when x equals 1 and y equals 3, that satisfies both of our equations. Or I should say all of our equations in the system. So plugging in 1 for x and 3 for y, we see then that 3 equals 1 plus 2 and 3 equals 3. That's a true statement.
For equation in green, plugging in 1 for x and 3 for y, we get 3 equals negative 1 plus 4. And 3 equals 3. So this solution point, x equals 1 and y equals 3, works for all of our equations in the system.
So now let's take a look at these two lines. So we notice, based off of this graph, that these are parallel lines. Now parallel lines never intersect. So what does this mean then about a system of equations that's made up of all parallel lines?
Because there's no point of intersection, that means that there's no solution to a system if all of the equations are parallel to each other. They're never going to intersect. So there's never going to be according to point xy that satisfies both equations.
Well, now I'd like to take a look at this line here. We have y equals 2x minus 3. And we're going to consider another equation to make this a system. And this equation is 4x minus 2y equals 6. So let's go ahead then and plot this on our graph.
Well, I'm going to first find some intercepts so I can locate two points on the line in order to graph it. So the x-intercept is when y equals 0. So when y equals 0, I can rewrite this equation then as just 4x equals 6. So in solving for x, I see x equals 2/3.
So x-- or 3/2 I mean, sorry. So when x is 3/2 and y is 0, that would be this point right here. To find the y-intercept, this occurs at x equals 0. And so when x equals 0 here, I can just rewrite this then as negative 2y equals 6.
And solving for y, I see then that y equals negative 3. So when y equals negative 3 and x equals 0, that's this point right here. So now I have two points. So I can connect them and extend the line. And notice that I have the line sharing all points with the line in blue.
So these are then identical lines. The equations might look pretty different, but on a graph, they're the same. They share all points. So what does that mean then with identical lines?
If all of the lines in a salute solution a system are identical lines, they have an infinite number of solutions. That means that there's an infinite number of x and y pairs that would satisfy all of those equations.
So let's review an introduction to a system of equations. System of equations is two or more equations that have the same variables, more specifically their variables represent the same quantity. They're defined the same. A solution to a system must satisfy all equations in the system.
And graphically, it represents the point of intersection on the graph. If a system is made up of entirely parallel lines, there is no solution to the system because there's no point of intersection. If your system is made up of identical lines, you are going to have an infinite number of solutions because they are going to share all points on that line.
So thanks for watching this quick tutorial and an introduction to system of equations. Hope to see you next time.