Solve the following system of equations with the help of matrices: 3x + y + 2z = 3, 2x - 3y - z = -3, and x + 2y + z = 4. Given that marginal revenue (M.R.) = 12 - 8x + x^2, determ... Solve the following system of equations with the help of matrices: 3x + y + 2z = 3, 2x - 3y - z = -3, and x + 2y + z = 4. Given that marginal revenue (M.R.) = 12 - 8x + x^2, determine revenue and demand functions where x denotes the number of units of output sold.
Understand the Problem
The question is asking us to solve a system of equations using matrices and to determine revenue and demand functions based on given marginal revenue. This requires applying matrix methods for the first part and understanding revenue concepts for the second part.
Answer
The values are \( x = 1, y = 2, z = 0 \). The revenue function is \( R(x) = 12x - 4x^2 + \frac{x^3}{3} + C \) and the demand function is \( P(x) = 12 - 8x + x^2 \).
Answer for screen readers
The values of ( x, y, z ) calculated are:
$$ x = 1, \quad y = 2, \quad z = 0. $$
For the revenue function:
$$ R(x) = 12x - 4x^2 + \frac{x^3}{3} + C \quad \text{(where C is the constant of integration)}. $$
And the demand function is:
$$ P(x) = 12 - 8x + x^2. $$
Steps to Solve
- Write the system of equations in matrix form
The given equations are:
[ \begin{align*} 3x + y + 2z &= 3 \ 2x - 3y - z &= -3 \ x + 2y + z &= 4 \end{align*} ]
This can be represented in matrix form as
$$ A \mathbf{x} = \mathbf{b} $$
where
$$ A = \begin{bmatrix} 3 & 1 & 2 \ 2 & -3 & -1 \ 1 & 2 & 1 \end{bmatrix}, \quad \mathbf{x} = \begin{bmatrix} x \ y \ z \end{bmatrix}, \quad \text{and } \mathbf{b} = \begin{bmatrix} 3 \ -3 \ 4 \end{bmatrix}. $$
- Find the inverse of matrix A
To solve for $\mathbf{x}$, we need to calculate the inverse of matrix $A$. The formula for the inverse of a 3x3 matrix is complex, so we can simply compute it using a calculator or a method such as Gauss-Jordan elimination.
- Calculate the values of x, y, and z
Once we have the inverse matrix, we apply it:
$$ \mathbf{x} = A^{-1} \mathbf{b}. $$
- Substituting values into the marginal revenue function
Next, we substitute the value of $x$ obtained from our solution into the marginal revenue function:
$$ \text{M.R.} = 12 - 8x + x^2. $$
- Determine the revenue function
The revenue function $R(x)$ is the integral of marginal revenue:
$$ R(x) = \int \text{M.R.} , dx. $$
- Calculate the demand function
Differentiate the revenue function:
$$ \text{Demand function } P(x) = \frac{dR}{dx}. $$
The values of ( x, y, z ) calculated are:
$$ x = 1, \quad y = 2, \quad z = 0. $$
For the revenue function:
$$ R(x) = 12x - 4x^2 + \frac{x^3}{3} + C \quad \text{(where C is the constant of integration)}. $$
And the demand function is:
$$ P(x) = 12 - 8x + x^2. $$
More Information
The solution shows how to apply matrix methods to solve a system of equations and derive revenue and demand functions from a marginal revenue equation. Understanding these concepts is crucial in economics for managing production and profit optimization.
Tips
- Forgetting to check if the matrix is invertible before attempting to find its inverse.
- Mistakes in arithmetic while calculating the matrix inverse.
- Confusing the revenue function with marginal revenue and demand functions.