Introduction to Simultaneous Linear Equations 05:08 minutes

Video Transcript

Transcript Introduction to Simultaneous Linear Equations

Fans of the beloved 80's pop duo, Vintage Silence, were shocked and saddened when the band broke up. Rumor was, they were experiencing some "creative differences". Thomas wanted to take his music in a more digital direction and Dieter wanted to unplug and focus on songwriting. But they were so great together! How will they ever make it on their own? Maybe we can find out by investigating Simultaneous Linear Equations. After breaking up, Dieter and Thomas both released solo records in the same month. Let's see how they're doing. This table shows Dieter's monthly record sales, in millions, for the first five months after his album is released. In the first month, he sold half a million records and in the second month, he sold one million! In fact, every month he sells half a million records more than he did the previous month. That means we can write this as a linear equation using variables! Since the amount of records sold depends on the month we can call the number of the month 'x' and the number of records sold that month, 'y'. Because Dieter's monthly sales are increasing by one half million each month, we can write 'y' is equal to zero point five times the month, 'x'. Notice both the variables are raised to the first power. So this must be a linear equation. Let's put this on a coordinate graph by starting at the origin then going up by half a million each month. Notice that all of the points in our table are also points on this graph. But what about Thomas? We don't have a table for his sales, we just have this equation. Since both of the variables are raised to the first power, we know this is a linear equation. So we can graph it with substitution, then by connecting the points with a line. If we plug in 0 for x we see that the number of record sales will be 3 million. Interesting, I guess he must have had some pre-sales. To find out how many records he sold in the first month, we just plug in 1 for 'x' then we get...2.9 for y? Oh no! It looks like his monthly sales decreased by 0.1 million! That's not a good sign for his solo career. By looking at the graph, we can notice a couple things. Because they are selling records at the same time, we are able to graph these on the same coordinate plane. This means they are Simultaneous Equations. Also notice that in month five, both Dieter and Thomas sell two and a half million records. When two graphs intersect like this, it means they have a simultaneous solution. The solution in this graph is the point where the month and the number of records sales are the same for both equations. Simultaneous equations are often written like this: Notice that a bracket is used to show that the two equations are happening simultaneously. We can use substitution to check the solution we found on our graph. Just plug in the coordinates for the solution into both equations. So if we set 'x' equal to 5, and 'y' equal to 2.5 in Dieter's equation... We end up with 2.5 equals 2.5.. This is a true statement, so this point works as a solution for Dieter. Now, let's check and verify that this is also a solution for Thomas. Using the same point, we set 'x' equal to 5 and 'y' equal to 2.5 in Thomas's equation and we end up with the true statement 2.5 equals 2.5. One point works for both equations, so this means we definitely have a simultaneous solution. To review: Simultaneous linear equations are equations that happen at the same time. So, you graph them on the same coordinate plane and see their solution at the point of intersection. You can check the solution algebraically with substitution. But what does all this mean for Vintage Silence? Not only are Thomas's album sales tanking but nobody turned up for his concert except for one person. No way... Is that...? I can't believe it! They're finally back together! Aww, who needs fans and fame when you have each other?