Question

The cost of each table or each chair cannot exceed Rs 13. If the cost of 5 tables and 7 chairs cannot exceed Rs 250 , then the inequations which best represents the above information are (1) \(x>13, y>13,5 x+7 y>250\) (2) \(x>0, y>0,5 x+7 y<250\) (3) \(x<13, y<13,5 x+7 y \leq 25\) (4) \(0

Short answer

(1) \(x>13, y>13, 5x +7y >250\) (2) \(x>0, y>0, 5x+7y<250\) (3) \(x<13, y<13,5 x+7 y \leq 25\) (4) \(0<x \leq 13,0<y \leq 13,5 x+7 y \leq 250\) Answer: (4) \(0<x \leq 13,0<y \leq 13,5 x+7 y \leq 250\)

Step-by-step solution

Identify the constraints for the cost of tables

We know that the cost of each table (x) cannot exceed Rs 13. This means that the value of x must be less than or equal to 13. In mathematical terms, this can be represented as: \(x \leq 13\)

Identify the constraints for the cost of chairs

Similarly, we know that the cost of each chair (y) cannot exceed Rs 13 as well. This means that the value of y must be less than or equal to 13. In mathematical terms, this can be represented as: \(y \leq 13\)

Identify the constraint for the combined cost of 5 tables and 7 chairs

We also know that the combined cost of 5 tables and 7 chairs (5x + 7y) cannot exceed Rs 250. This means that the value of (5x + 7y) must be less than or equal to 250. In mathematical terms, this can be represented as: \(5x + 7y \leq 250\)

Combine the constraints to find the best representations

Now, we'll take the constraints we found in Steps 1, 2, and 3 and see which of the options best represent this information: (1) \(x>13, y>13, 5x +7y >250\) - This is incorrect because the values of x and y should be less than or equal to 13, and the combined cost should be less than or equal to 250. (2) \(x>0, y>0, 5x+7y<250\) - This option is partially correct since the cost of tables and chairs must be positive, but it doesn't match the constraint \(x \leq 13\) and \(y \leq 13\) as well as, it should be \(5x+7y \leq 250\). (3) \(x<13, y<13,5 x+7 y \leq 25\) - This option is incorrect because it does not account for the equal sign in the constraint \(x \leq 13\) and \(y \leq 13\) and also it's incorrect as 5x+7y should be less or equal to 250 and not 25. (4) \(0<x \leq 13,0<y \leq 13,5 x+7 y \leq 250\) - This option correctly represents the constraints we found in Steps 1, 2, and 3. Therefore, the correct answer is option (4): \(0<x \leq 13,0<y \leq 13,5 x+7 y \leq 250\)

Key concepts

Linear Inequalities

Linear inequalities are mathematical expressions involving variables where the expressions are not equal but are instead related by 'greater than', 'less than', 'greater than or equal to', or 'less than or equal to' signs. In the context of our exercise about tables and chairs, the inequalities describe limitations on the costs.
For example, the inequality \(x \leq 13\) means that the cost of one table (x) should not be more than 13 rupees. Similarly, \(y \leq 13\) represents the maximum cost of one chair. These are constraints that confine the values that the variables can take.
Another key inequality we used is \(5x + 7y \leq 250\), which indicates that the total cost of 5 tables and 7 chairs cannot exceed 250 rupees. This is achieved by understanding that linear inequalities bind the variable values within a feasible region on a coordinate plane, which is fundamental in several applications, especially economics and business analyses.

Constraints in Problem Solving

Constraints are pivotal in problem-solving as they provide a set of rules or limitations that a solution must adhere to. In optimization and real-world problem scenarios, constraints help in narrowing down vast possibilities to workable solutions.
In our exercise, the constraints are the inequalities: \(x \leq 13\), \(y \leq 13\), and \(5x + 7y \leq 250\). Each inequality delineates a boundary that the variables cannot exceed.

• \(x \leq 13\): restricts the price of each table.
• \(y \leq 13\): limits the price of each chair.
• \(5x + 7y \leq 250\): controls the overall expenditure on tables and chairs.

By understanding and aligning decisions within these constraints, one can effectively find an optimal solution that satisfies all conditions, ensuring that resources such as money, time, or materials are not overused or wasted.

Optimization Problems

Optimization problems focus on finding the best possible solution from a set of feasible options given certain constraints. The core of such problems is to maximize or minimize a function, usually dependent on some cost, efficiency, or return condition.
In the context of our table and chair example, optimization would mean determining how to allocate Rs 250 most efficiently across purchasing tables and chairs without surpassing the individual costs. The inequalities help frame the boundaries within which optimal solutions can be sought.
Typically, optimization problems use linear programming techniques to analyze the inequalities. The feasible region formed by these inequalities represents all the possible combinations of tables and chairs that can be purchased within the given budget constraints. The goal is to find the point(s) in this region that maximize or minimize a desired outcome, whether it's minimizing cost or maximizing the number of items purchased.