System of equations word problem: no solution | Mathematics I | High School Math | Khan Academy

TL;DR

The video presents a problem involving a toy factory where machines produce toys and workers pack them, and explains how to solve for the number of machines and workers using two equations and algebraic substitution.

Transcript

• [Voiceover] A factory has machines that produce toys, which are then packed by the factory's workers. One day, each machine produced 14 toys and each worker packed two toys, so that a total of 40 toys remained unpacked. Additionally, the number of workers that day was eight less than seven times the number of machines. How many machines and worke... Read More

Key Insights

• 🧑‍🏭 The problem involves setting up and solving equations to find the number of machines and workers in a toy factory.
• #️⃣ The equations reflect the total number of toys produced and packed, as well as the relationship between the number of workers and machines.
• 🥺 Solving the equations through substitution leads to the discovery that there is no solution that satisfies the given constraints.
• ❓ This problem highlights the importance of careful analysis and consideration of all variables and equations before drawing conclusions.
• 😫 Mathematical problem-solving often requires setting up and manipulating equations to find unknown quantities.
• 💼 In some cases, the given constraints may not have a valid solution, as demonstrated in this problem.
• 🌍 The problem showcases the application of algebraic concepts, such as substitution and simplification, in solving real-world scenarios.

Questions & Answers

Q: What are the unknown variables in the toy factory problem, and how are they denoted in the equations?

The unknown variables in the problem are the number of machines (M) and the number of workers (W). These variables are used to set up the equations that describe the total number of toys produced and packed, as well as the relationship between the number of workers and machines.

Q: How is the linear relationship between the produced and packed toys established in the problem?

The linear relationship is established by subtracting the number of toys packed from the total number of toys produced, which gives us the number of toys that remain unpacked. This allows us to set up an equation relating the variables M and W.

Q: What additional information is provided in the problem to create a second equation?

The problem states that the number of workers on that day was seven times the number of machines, minus eight. This information forms the basis for the second equation relating the variables M and W.

Q: What is the result of solving the two equations?

After performing algebraic operations to solve the equations, it is found that the equations lead to an impossibility, with the equation 16 = 40 being unsolvable. Therefore, there is no specific solution for the number of machines and workers in this scenario.

Summary & Key Takeaways

• The problem states that each machine in the toy factory produced 14 toys and each worker packed two toys, leaving 40 toys unpacked.

• By setting up equations for the total number of toys produced and packed, and the relationship between the number of workers and machines, we can solve for the unknown variables.

• However, after performing algebraic operations, it is discovered that there is no solution to the given constraints, and therefore no specific number of machines and workers.