Use the graph to identify the solution to the system of equations or identify that the system has 0 0 or infinitely many solutions. If there is a solution, check that the point satisfies both equations.

Everybody. So now that we have understand how to solve system of equations by graphing, let's take a look at our first practice problem here. So we're told to use the graph to identify the solution to the system of equations or identify that the system has zero or infinitely many solutions. If you remember, the solution to a system of equations, the way you find that graphically, is just the point where the two lines intersect. Alright? If they don't intersect, it's zero. And if they're the same line, it's infinitely many solutions. Now if there is a solution, what this question asks us to do, let's go one step further and check that the point actually satisfies both of those equations. Okay? So let's go ahead and get started here. We can see that these two lines are sort of going to crisscross like this, and they are going to intersect. And remember that the solution is basically that point where they intersect, which is going to be right over here. Alright? Now the coordinate for that is going to be zero comma 3, and that's really the answer. The answer is that the solution to the system is zero comma three. Now what ends up happening here is typically sometimes if you're drawing these things on a hand drawn graph, sometimes you can draw the lines incorrectly. And so maybe what you've drawn is maybe not actually the point zero comma three. It just looks like it. So we're going to go ahead and check the point satisfies both of those equations so we can really, really be sure that that's the correct answer. Okay? So I'm going to start off with this equation or this line that goes sort of up towards the top right, that positively sloped line. And I'm going to go ahead and write out the equation for this in slope intercept form. So that's y equals mx plus b. So I'm just going to write y equals something times x and then plus something else. So the second number is always just the point where it crosses the y axis. That's the y intercept. That's what goes at the end. And clearly, we can see here that the y coordinate where it crosses the y axis is three. Now, I want to be really clear here. It's not three because that's the point where the two lines cross. It's three because that's where this line crosses the y axis. Alright. That's the reason it's three. Okay? So now in order to get the slope, we just pick any two points across this line, then we just figure out the rise over the run. Okay? So I'm gonna go ahead and pick this point and this one, but you could use really any two points. You're just trying to figure out what's the rise over the run. So I have to go one up and then one over, one up and one over, and so on and so forth. So in other words, the slope is basically just one over one. In other words, the slope of this line is just one. So we have y equals one x plus three. You don't have to write that one, but I'll just leave it there. Okay? Now we're gonna do the same exact thing for this line over here, the one that goes down towards the bottom right. So we're going to do the same exact thing. What's the y intercept, the place where it crosses the y axis? It's actually the same. It's still plus three. Again, not because that's where these two lines cross each other, but that's where this line crosses the y axis over here. All right? So now it's the slope. We just pick any two points across that line, and we just compute the rise over the run. Alright? So I have to go one down over, one down and then one over, one down and then one over. And then you know when you when you go down, it's gonna be a negative one over one, negative one over one. So in other words, the slope of this line is negative x plus three. Okay? Now we've written both of these equations. We can go ahead and check that point actually satisfies those those equations. So we're gonna go ahead and check, again, the point zero comma three. You just plug the x coordinate in for the x, the y coordinate in for the y, and then so on and so forth. Alright? So we can see here is three, does that actually equal one times zero plus three? It's sort of like a question mark there for that equal sign. And you can see that one times zero is just going to be zero, leaving you with just positive three on both sides of the equations. So that's going to be true. That's going to be a true statement. If you do the same exact thing over here, you're going to just check the points zero comma three, and you're going to end up with three. Does that equal negative one times zero plus three? Do the same exact thing, right? So your negative one times zero doesn't matter, it's still just zero. So that just goes away. And you have positive three on both sides. So it definitely ends up being a true statement. So because this point satisfied both of those equations, that's the only point that's going to satisfy this system of equations here. And that's your answer. So again, let me know if you have any questions, and if not, let's move on.