Determining The Cost Of A Television Set A Mathematical Problem

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In this article, we will delve into a fascinating mathematical problem that involves determining the cost of a television set based on its relationship with the cost of a video recorder. This problem presents an excellent opportunity to apply algebraic concepts and problem-solving skills. By carefully analyzing the given information and formulating equations, we can arrive at the solution and gain a deeper understanding of how mathematical principles can be used to solve real-world scenarios. So, let's embark on this mathematical journey together and uncover the cost of the television set.

The problem states that a video recorder costs $250 less than a television set. Additionally, the television set costs 3 times as much as the video recorder. Our objective is to determine the cost of the television set. This problem involves two key pieces of information that need to be carefully considered: the price difference between the video recorder and the television set, and the multiplicative relationship between their costs. By translating these pieces of information into mathematical equations, we can then solve for the unknown variable, which represents the cost of the television set. The challenge lies in accurately representing the given information in mathematical form and then applying the appropriate algebraic techniques to solve for the unknown.

To solve this problem, let's use variables to represent the unknowns. Let 'x' be the cost of the video recorder and 'y' be the cost of the television set. Based on the given information, we can formulate two equations:

  1. The video recorder costs $250 less than the television set: This can be written as x = y - 250. This equation directly translates the first piece of information into a mathematical expression. It states that the cost of the video recorder (x) is equal to the cost of the television set (y) minus $250. This equation establishes a relationship between the two variables, allowing us to relate their values.
  2. The television set costs 3 times as much as the video recorder: This can be written as y = 3x. This equation represents the second piece of information, expressing the cost of the television set as a multiple of the cost of the video recorder. It indicates that the cost of the television set (y) is three times the cost of the video recorder (x). This equation provides another crucial link between the two variables.

These two equations form a system of equations that can be solved to determine the values of 'x' and 'y'. By using techniques such as substitution or elimination, we can find the values that satisfy both equations, thus revealing the costs of the video recorder and the television set.

Now that we have our equations, we can solve them using the substitution method. Since we know that x = y - 250, we can substitute this expression for 'x' in the second equation:

y = 3(y - 250)

Now, we can simplify and solve for 'y':

y = 3y - 750

Subtracting 3y from both sides, we get:

-2y = -750

Dividing both sides by -2, we find:

y = 375

Therefore, the cost of the television set is $375. To find the cost of the video recorder, we can substitute the value of 'y' back into the equation x = y - 250:

x = 375 - 250

x = 125

So, the cost of the video recorder is $125. By using the substitution method, we have successfully solved the system of equations and determined the costs of both the television set and the video recorder.

To ensure that our solution is correct, we can substitute the values of x and y back into the original equations and check if they hold true:

  1. x = y - 250: Substituting x = 125 and y = 375, we get 125 = 375 - 250, which simplifies to 125 = 125. This equation holds true.
  2. y = 3x: Substituting x = 125 and y = 375, we get 375 = 3 * 125, which simplifies to 375 = 375. This equation also holds true.

Since both equations are satisfied by our solution, we can confidently conclude that the cost of the television set is indeed $375 and the cost of the video recorder is $125. This verification step is crucial in ensuring the accuracy of our solution and confirming that we have correctly interpreted and solved the problem.

In conclusion, by carefully analyzing the given information and setting up a system of equations, we have successfully determined that the cost of the television set is $375. This problem demonstrates the power of algebraic techniques in solving real-world problems. By translating word problems into mathematical expressions, we can utilize the tools of algebra to find solutions and gain a deeper understanding of the relationships between different variables. This problem serves as a valuable example of how mathematical concepts can be applied to solve practical scenarios and enhance our problem-solving abilities. The process of setting up equations, solving them, and verifying the solution is a fundamental aspect of mathematical problem-solving and is applicable in various fields and disciplines.

Keywords and SEO Optimization

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