Systems of linear equations can be used to solve many, but not all, real-world problems. When you chose the problem for your journal entry, how did you know that this problem could be solved using a system of linear equations? What specific characteristics does your problem have that made it a good choice? List at least three reasons why the problem you described in your journal prompt can be solved using a system of linear equations.
Post a detailed response to the questions about your choice of a real-world problem to solve using a system of linear equations. Then respond to two classmates’ posts.
Participate in the discussion to learn more about how to determine if a real-world problem can be solved using systems of linear equations.
When choosing a problem for my journal entry, I knew it could be solved using a system of linear equations based on a few specific characteristics of the problem. The problem I chose was about a farmer who wants to know how many cows and chickens he has on his farm.
One reason why this problem can be solved using a system of linear equations is that it involves multiple variables. In this case, the variables are the number of cows and the number of chickens on the farm. The problem asks for a relationship between these two variables, which is a characteristic of a linear equation.
Another reason is that the problem provides enough information to create a system of equations. The problem states that the farmer has a total of 30 animals on his farm and that the number of cow legs and chicken legs adds up to 92. From this information, we can set up two equations:
- The first equation represents the total number of animals: C + K = 30 (C is the number of cows and K is the number of chickens).
- The second equation represents the total number of legs: 4C + 2K = 92 (since each cow has 4 legs and each chicken has 2 legs).
A third reason why this problem can be solved using a system of linear equations is that it has a unique solution. By solving the system of equations, we can determine the specific number of cows and chickens on the farm. In this case, the solution is C = 18 and K = 12, meaning there are 18 cows and 12 chickens on the farm.
In summary, the problem of determining the number of cows and chickens on a farm can be solved using a system of linear equations because it involves multiple variables, provides enough information to create a system of equations, and has a unique solution. These characteristics make it a good choice for solving using linear equations.
Responses to classmates' posts:
1. Student A:
I agree with your reasoning for choosing a problem that can be solved using a system of linear equations. In addition to the characteristics you mentioned, I would also like to add that another reason for choosing such a problem is that it involves a linear relationship between the variables. In your specific problem, the number of cows and the number of chickens have a linear relationship since changing the number of cows directly affects the number of chickens. This linear relationship can be expressed and solved using a system of linear equations.
2. Student B:
I also chose a problem that can be solved using a system of linear equations. One reason why I knew this problem could be solved using linear equations is that it involved finding the coordinates of a point that lies on two different lines. Since a point that lies on two lines must satisfy both equations, I was able to set up a system of equations to find the coordinates of the point. Another reason is that the problem provided enough information to create a system of equations, which included the slopes and y-intercepts of the lines. Lastly, similar to your problem, the problem I chose had a unique solution, meaning there was only one point that satisfied both equations, allowing for a specific solution to be found using a system of linear equations.