Nature of Solutions of a System of Linear Equations 06:08 minutes

Video Transcript

Transcript Nature of Solutions of a System of Linear Equations

Little Jimmy is an adventurous eater, and he's always wanted to try sushi. So, his uncle Mike acquiesces, and takes little Jimmy to the city's oldest Japanese restaurant for his birthday. Uncle Mike's relieved when he sees that, even though it's sushi, it's not that expensive. Who's bill will be higher, Jimmy's or Mike's? Let’s help our hungry eaters figure that out by examining the Types of Solutions to Systems of Linear Equations. Looking at the menu, we can see that adults pay $10 for the buffet and kids pay $8, plus either 20 or 30 cents per plate of sushi, depending on what you choose. Uncle Mike isn’t an adventurous eater, so he decides to go for the safest-looking sushi at $0.20 a plate. Jimmy's only interested in the fancy sushi, so he decides to go for the $0.30 plates. After how many plates will the price for Mike be the same as the price for Jimmy? Let’s take a look at a graph to find out. On our graph, we have plates of food on the x-axis and cost on the y-axis. Why did we choose the plates of food for the x-axis? Since we can choose how many plates of sushi to eat, it's the independent variable. And we usually put the independent variable on the x-axis. Let's write an expression for the total cost for Mike, the adult getting the cheaper sushi. We know that the fixed cost is $10 for the buffet and we know that ‘x’ is the number of plates, which we need to multiply by the cost of each plate, $0.20. If Mike ends up not liking sushi and only eats one plate, the total cost will be $10.20, but if he tries it twice and quits, two plates will total $10.40, and so on. We can write a similar expression for the total cost of Jimmy's plate. We know that the fixed cost is $8 for the buffet and $0.30 per plate. If Jimmy eats 5 plates, the total cost will be $9.50. Ten plates will total $11, and so on, and so on. We can see graphically that the two lines intersect at x = 20. This means that the cost of Uncle Mike and little Jimmy's meals will be equal after 20 plates. But is there a way to find this out algebraically? We want to find out when the cost for Mike is the same as the cost for Jimmy, so we write the equation for Mike, 10 + 0.20 times 'x', equals the equation for Jimmy, 8 plus 0.30 times 'x'. First, we want the variable 'x' on only one side, and then we isolate it using opposite operations and bring the 10 over to the other side. Dividing both sides by negative 0.10 gives us 'x' is equal to 20. But what if Mike is more adventurous and also goes for the more expensive sushi? The equation for Jimmy stays the same, 'y' equals 8 plus 0.30x. But the equation for the adventurous Uncle Mike becomes 'y' equals 10 plus 0.30x. After how many plates will Jimmy and Mike's meals cost the same? Let’s adjust our graph for Mike. We start at 10 and add 0.30 for the first plate, and for the second, and so on. Hmmm...these lines don’t look like they’re ever going to meet. Let’s set the equations equal to each other and check just to make sure. Look what happens when we try to isolate the variable. We find that 10 = 8. Whenever the coefficients in front of the variable are the same sign and the same value on opposite sides of the equal sign, it indicates that the lines are parallel and will never meet. Uncle Mike and little Jimmy are having such a good time, they're planning on coming back for the restaurant's International Sushi Day special. All customers pay $9 to get in, and all sushi plates cost $0.25. So the equation for Uncle Mike will be 'y' equals 9 plus 0.25x, and the equation for little Jimmy will also be 'y' equals 9 plus 0.25x. If we plug in numbers for 'x' and graph the results, we see that adults and kids will always pay the same price if they eat the same number of plates. Remember, if the slope and the intercept are both the same, the lines are also the same, which means there are infinite solutions to this system of linear equations. Okay, to review quickly. We start with two lines, each with the equation 'y' equals mx plus 'b'. Some equations give lines that have different slopes and meet at exactly one point. This means there is one solution. Some equations give lines that have the same slope, but have different 'b' values. These lines are parallel and never meet, which means this system of equations has no solution. And some equations give lines that have the same slope and the same 'b' values. This means there are an infinite number of solutions. Let’s get back to our birthday boy Jimmy and his Uncle Mike. That’s a huge stack of plates! The waitress goes to print out the bill and…how much sushi did they eat?!?!