Question

Find the profit function \(\mathrm{p}\) if it yields the values 11 and 7 at \((3,7)\) and \((1,3)\) respectively. (1) \(p=-8 x+5 y\) (2) \(\mathrm{p}=8 \mathrm{x}-5 \mathrm{y}\) (3) \(p=8 x+5 y\) (4) \(p=-(8 x+5 y)\)

Short answer

Answer: The correct profit function is option (1), \(p = -8x + 5y\).

Step-by-step solution

Plug in the point (3,7) for the first function

Evaluate the first function with the point (3,7): \(p = -8x + 5y = -8(3) + 5(7) = -24 + 35 = 11\)

Plug in the point (1,3) for the first function

Evaluate the first function with the point (1,3): \(p = -8x + 5y = -8(1) + 5(3) = -8 + 15 = 7\)

Conclusion for the first function

Since the first function satisfies both given conditions, the profit function is: \(p = -8x + 5y\) There is no need to test the other three options, as we have found the correct profit function. The answer is option (1), \(p = -8x + 5y\).

Key concepts

Coordinate Geometry

Coordinate geometry, also known as "analytic geometry," involves plotting and working with a system of points in a coordinate plane. In this case, we associate each point with coordinates that specify its exact position in the plane. When we talk about a point such as \((3,7)\), it means that the point is located 3 units along the x-axis and 7 units along the y-axis.
This helps give a visual depiction of mathematical problems, especially in functions.

• The x-coordinate (first number in a pair) tells you how far left or right the point is along the x-axis.
• The y-coordinate (second number in a pair) tells you how far up or down the point is from the y-axis.

Having a graph to visually plot the points can be very useful.Visualizing the problem helps to understand where these points lie and how the function affects them.

Linear Equations

A linear equation represents a straight line in a coordinate plane and is usually given in the form of \(Ax + By = C\). In this exercise, the linear equation we explored is \(p = -8x + 5y\). This shows a direct relationship between the variables \(x\) and \(y\). Linear equations are crucial in predicting how a function behaves. If you plug different values of \(x\) or \(y\) into the equation, you can determine the resulting value of the function \(p\).
These equations are called "linear" because the graph of an equation like this forms a straight line. When solving linear equations:

• Identify the slope, which is the coefficient of \(x\). In this case, it is \(-8\).
• Understand how the equation changes with variations in the \(x\) or \(y\) values.
• Linear equations often make it easier to understand and model relationships in various disciplines, like physics and economics.

Function Evaluation

Function evaluation is about substituting values into an equation to find out the output result, like finding the profit in a profit function. In this scenario, evaluating the function involved:

• Taking each point given, such as \((3,7)\) or \((1,3)\), and plugging these values into the equation \(p = -8x + 5y\).

Hence, for \((3,7)\), it becomes\(-8(3) + 5(7)\) and for \((1,3)\), it becomes \(-8(1) + 5(3)\).Through function evaluation:

• We verify if the chosen function correctly predicts or matches the given values.
• We can ascertain which proposed function, if any, returns the expected results. Here, Option 1 fittingly predicted the profit values.
• This effectively tests the validity of our equations under given conditions.

Such evaluations help ensure that the function formulation meets expected performance metrics or criteria.