Hi and welcome. My name is Anthony Varela, and today I'm going to be solving a system of equations by graphing. So what that involves is graphing our equations on a single grid, and we're going to be looking for points of intersection. That's going to represent our solution to the system. And so we're also going to take a look then of a system that has no solution and a system that has an infinite number of solutions.
So taking a look at a system of equations on a graph, so we're going to have several lines or curves, and that point of intersection represents a specific x value and a specific y of value that satisfies all of the equations in your system. So we're looking for a point of intersection. Now, be aware that solving by graphing may not be the most accurate or precise way to solve for a system.
So let's take a look at this example. We can clearly see that there is a point of an intersection, but depending on the scale or where that point of intersection is, you might only get approximate answers and not exact answers, which could be a good enough for you depending on your situation. So as we've said before, the solution to a system on a graph is that point of intersection. So here we see two lines. We have 1 line, 5x plus 3y equals 15, and we have 2x minus 3y equals 6.
Now on this graph, we can clearly see where that point of intersection is. This occurs at the points 3, 0. So what this means then is that when x equals 3 and y equals 0, this satisfies all of the equations in my system. So let's go ahead and show that. So plugging in three for x and zero for y, we have that 5 times 3 plus 3 times 0 is 15, and we can see that that's true. For our second equation, 2 times 3 minus 3 times 0 equals 6, and we could see that that is true.
All right, so now let's go ahead and solve a system by graphing. So we're given a system of equations, and we need to first plot this on our grid then look for points of intersection. So I have these 2 equations written in standard form, and I know that standard form makes it pretty easy to find x and y-intercepts. And as long as I can plot 2 points on a line, I can just connect those lines and graph it. So to find the x-intercept, this is when y equals 0. And to find the y-intercept, this is when x equals 0. So I'm going to call our two equations equations a and b, and now let's find the x and y-intercepts of line a.
So this is when y equals 0. So when y equals 0, x equals negative 6. This is just 0, right? The whole y term is 0. So this tells me then that the x-intercept is going to be located at negative 6, 0. X equals negative 6, y equals 0. To find the y-intercept. This is when x equals 0. So when x equals 0, I then just have negative 2y equals negative 6. So I can divide both sides by negative 2, and I get y equals 3. So this tells me that the y-intercept is located at the point 0, 3. So here is my line of my first equation.
Let's do the same thing for equation b. So when y equals 0, I have that 2x equals 8. So this tells me then that x equals 4, so one of my intercepts is at 4, 0. To find the y-intercept, x equals 0, so I have y equals 8. So I know then that's the y-intercept is at the point 0, 8, so there's my other line. So now we're looking for where the two lines intersect, and this occurs at the points 2, 4. So because my point of intersection is 2, 4, this means that x equals 2 and y equals 4 satisfies both of these equations. So this would be 2 minus 8 equals negative 6. That's true, and we can work this one out as well. 4 plus 4 equals 8, because 2 times 2 plus 4 equals 8. So we've confirmed that this is indeed a solution to our system.
So now let's solve a system that has three equations. So what I'm going to do then is plot all three of these lines and look for points of intersection. So let's find our x and y-intercepts so we can get this on our graph. So looking at the equation a first, when y equals 0, I just have x equals 3, so my x-intercept is at 3, 0. So that's on our graph. The y-intercept would then be when x equals 0, so I have negative y equals 3, which means that y equals negative 3. So my y-intercept is at 0, negative 3. There it is on the graph. So there's our line of x minus y equals 3.
Now, let's go ahead then and graph line b. So when y equals 0, I have 2x equals 9, so x equals 4.5. So now I have to then locate on my graph the point 4.5, 0, so we see that in between 4 and 5 on our x-axis. And for the y-intercept when x equals 0, y equals 9. So my y-intercept then occurs at 0, 9. So I have that on my graph. So there is the line of 2x plus y equals 9.
And finally for equation c, our x-intercept is when y equals 0, so I have x equals negative 3. So I can plot the point negative 3, 0. The y-intercept x equals 0, so I have 2y equals negative 3. So y equals negative 3 over 2, or negative 1.5, so I'm going to plot that on my graph, and there is the equation then on the graph for x plus 2y equals negative 3.
So now let's look for points of intersection. Now this is tricky, because I see some points where 2 of our lines intersect, but this does not mean that we have three solutions. We actually have no solutions, because we're looking for points of intersection in all of the lines, not just one. So there is no point where all three of our lines meet, so there is no solution here.
We're going to go through one final example here where we're going to graph the line y equals 2x minus 2, and we're going to graph 4x minus 2y equals 4. So graphing y equals 2x minus 2, I can see by this negative 2 that I have a y-intercept at y equals negative 2. So let's go ahead and put that down on our graph. And interpreting the slope, the slope is 2, so I have a rise of 2 and a run of 1, so I can go up 2 over 1, and that's another point on our line, so there is the line y equals 2x minus 2. To graph 4x minus 2y equals 4, let's pull out our chart again to find x and y-intercepts.
For the x-intercept, y is 0, so I have 4x equals 4 or x equals 1. So I can plot the 1, 0 on our graph. And notice we already have it on there. Hmm, that's interesting. To find the y-intercept, this is when x equals 0, so I'm going to have negative 2y equals 4. Divide both sides by negative 2, I get y equals negative 2. So my y-intercept is at 0, negative 2.
Once again, I already have that on my graph. So the line for equation b is the same as the line for equation a. Now, what does this mean then about their points of intersection? They actually share all points because they're the same line. They're identical equations. So here we have an infinite number of solutions.
So let's review solving a system of equations by graphing. A solution on a graph is a point of intersection between all lines in the system, not just some. So here we saw a graph simple two equations system, that one point of intersection is that single solution. Here we saw that there might be some points of intersection between two lines, but if it's not between all lines in the system, there's no solution. And we also saw an example where if we have identical lines that make up a system, there is an infinite number of solutions because they share all points. So thanks for watching this tutorial on solving a systems of equations by graphing. Hope to see you next time.