Now that you've learned what a system of linear equations is and when they are singular or non-singular, it is time for some visualizations. It turns out that linear equations can easily be visualized as lines in the coordinate plane. This is because you have two variables. If you have three variables, they are planes in space. If you have more variables, they look like high-dimensional space. But let's not worry about that yet. Since linear equations can be represented as lines, then systems of linear equations can be represented as arrangements of lines in the plane. This way, you can visualize their solutions and their singularity or non-singularity in a much clear way. How can you visualize, for example, the equation a plus b equals 10 as a line? First, let us get a grid in which the horizontal axis represents a, which is the price of an apple, and the vertical axis represents b, which is the price of a banana. Now let's look at solutions to this equation a plus b equals 10. In other words, pairs of numbers that add to 10. What you'll do is put them in this plot. Two obvious solutions are the point 10, 0. The a coordinate, the price of an apple, is 10 and the b coordinate, the price of a banana, is zero because 10 plus 0 is 10. Another obvious solution is the point 0, 10 where a is zero and b equals 10. Other solutions are the point 4, 6 because 4 plus 6 equals 10. This is a equals 4 and b equals 6 or the point 8, 2 where a equals and b equals 2. Notice that you can also have negative solutions, for example minus 4,14. Now, this makes no sense in the world problem because an apple cannot cost minus 4. But these are two numbers that add to 10, minus 4 plus 14 equals 10. This is a legitimate solution to the equation. You can also have negative solutions like 12, minus 2. Now notice that all these points form a line. In fact, every single point in this line is solution to the equation. You can then associate the equation a plus b equals 10 with this line. Now let's do another equation. Say the equation a plus 2b equals 12. That means points for which the horizontal coordinate plus two times the vertical coordinate add to 12. Some solutions for this equation are the point 0,6 since 0 plus 2 times 6 equals 12. The point 12,0 because 12 plus 2 times 0 is 12. The point 8,2 because 8 plus 2 times 2 is 12. Again, negative solutions like minus 4,8 for example, because minus 4 plus 2 times 8 is 12. Again, these points form a line and every point in the line is a solution to this equation. The line is associated with the equation a plus 2b equals 12. One small assignment, be familiarized with the notions of slope and y-intercept in a line. The slope is the ratio of rise over run which in the line on the left is minus 1. As for every unit you move to the right, the line moves one unit down. The down is the minus the negative. For the line on the right, the slope is minus a half because for every unit you move to the right, the line moves half a unit down. For the y-intercept for a line on the left, it is 10 as this is the height of the intersection between the line and the vertical y-axis, and for the line on the right it is six. Now here's what's interesting. Each equation is associated to a line. What happens with the system of two equations? Well, the system of two equations is simply associated with two lines in the same plane. Notice that these two lines cross at a unique point. The point 8,2 for a equals 8 and b equals 2. The point is precisely unique solution to that system of equations. This is exactly what we got before algebraically but now we can see it geometrically. Now that we know how to plot the equation of line a plus b equals 10, let's try another one. How about 2a plus 2b equals 20? Well, notice that that line still goes through the point 0,10 and 10,0. As the line is defined by only two points then the line is exactly the same as the one with equation a plus b equals 10. We called that a few lessons ago, you learned that equation a plus b equals 10 and 2a plus 2b was 20 carry the same information. This is a visual confirmation for that. Now, when we want to find the solution to this set of equations, there is no single intersection point. Instead, the two lines overlap each other, they are the same line. What happens now is that every point that belongs to both lines is a solution to the set of equations a plus b equals 10 and 2a plus 2b equals 20. That means we have infinitely many solutions because every point in that line is a solution. Finally, let's look in our system of equations, the one with equations a plus b equals 10 and 2a plus 2b equals 24. Let's plot the one on the right. Notice that the lines of equation 2a plus 2b equals 24 goes through the point 0,12 and 12,0 because 2 times 0 plus 2 times 12 is 24 and 2 times 12 plus 2 times 0 is 24 and therefore it has to be this line over here. Is very similar to the original line except it's translated up by two units. When we try to get the solutions to this set of equations, take a look. The system of two equations is simply associated to these two lines in the same plane that are parallel. Parallel lines never meet so there are no solutions to this system. There's no point that belongs to both the lines. The system has no solutions. Let's now summarize what you've seen in this video. There are three systems of equations. The first one has equations a plus b equals 10 and a plus 2b equals 12. The second one has equations a plus b equals 10 and 2a plus 2b equals 20 and the third one has equations a plus b equals 10 and 2a plus 2b equals 24. Here are the plots for the three. The first one corresponds to two lines that intersect at a unique 8,2. That's the unique solution to the system. The second one corresponds to two lines that are exactly the same line corresponding to a system that has infinitely many solutions. The third one corresponds to two parallel lines that never meet which means the system has no solutions. We can use the exact same nomenclature as we used with equations and with systems of sentences. Since the first system has a unique solution, it is complete and it is non-singular because every line brings something new to the table. The second system has infinitely many solutions because the second line is exactly the same as the first one so the system is redundant and singular. The second line brings nothing new to the table because it's exactly the same as the first line. Finally, since the third system corresponds to two lines and never meet, it means the second equation contradicts the first one. We have no solutions. Therefore, the system is contradictory and singular. Now you're ready for another quiz. Problem 1 says, which of the following plots correspond to the systems of equations 3a plus 2b equals 8 and 2a minus b equals 3? Problem 2 says, by looking at the plot for Problem 1, do you conclude that the system is singular or non-singular? The answer is this. In order to plot these lines, you can notice that the line with equation 3a plus 2b goes through the point 0,4 and 8/3,0 and the line with equation 2a minus b equals 3 goes to the point 0 minus 3 and 3/2,0. Notice that the two lines cross at the point 2,1 which is precisely the unique solution to the system of equations. That's a equals 2 and b equals 1. Since the two lines intersect at a unique point, then the system is non-singular.
