Solve each problem. A group of students bought 3 soft drinks...

Solve each problem. A group of students bought 3 soft drinks and 2 boxes of popcorn at a movie for $\$ 18.50 .$ A second group bought 4 soft drinks and 3 boxes of popcorn for $\$ 26$. (a) Find a matrix equation $A X=B$ whose solution gives the individual prices of a soft drink and a box of popcorn. Solve this matrix equation by using $A^{-1}$ (b) Could these prices be determined if both groups had bought 3 soft drinks and 2 boxes of popcorn for $\$ 18.50 ?$ Try to calculate $A^{-1}$ and explain your results.

Solve each problem.
A group of students bought 3 soft drinks and 2 boxes of popcorn at a movie for $\$ 18.50 .$ A second group bought 4 soft drinks and 3 boxes of popcorn for $\$ 26$.
(a) Find a matrix equation $A X=B$ whose solution gives the individual prices of a soft drink and a box of popcorn. Solve this matrix equation by using $A^{-1}$
(b) Could these prices be determined if both groups had bought 3 soft drinks and 2 boxes of popcorn for $\$ 18.50 ?$ Try to calculate $A^{-1}$ and explain your results.

Key Concepts

System of Linear Equations

This concept involves setting up equations that represent relationships between unknown quantities—in this case, the price of a soft drink and a box of popcorn. Each purchase scenario provides a linear equation, and the solution to the system gives the individual prices. This overall approach is key to translating real-world problems into a mathematical framework that can be systematically solved.

Matrix Equation Representation

The system of linear equations can be succinctly expressed using matrices, where the coefficients of the variables are arranged in a matrix (A), the unknowns form a vector (X), and the totals form another vector (B). This representation, written as AX = B, facilitates the use of linear algebra techniques such as matrix inversion to solve the equations efficiently.

Matrix Inversion

Finding the inverse of a matrix (A?¹) is a method used to solve matrix equations when A is invertible. The solution vector X is computed as A?¹B. This method is only valid if the matrix A has an inverse, which requires that the determinant of A is non-zero. Matrix inversion is a fundamental tool in linear algebra for solving systems of equations.

Invertibility and Uniqueness of Solutions

A key aspect of solving systems using matrix inversion is ensuring that the coefficient matrix is invertible. Invertibility implies that the system of equations has a unique solution. If the matrix is singular, or non-invertible, it means that the system either has no solution or infinitely many solutions. This concept is critical when analyzing if the system can be uniquely solved.

Degeneracy and Linearly Dependent Equations

This concept deals with situations where the equations in the system are not independent, meaning one equation may be a multiple or combination of another. In such cases, the coefficient matrix becomes singular and does not have an inverse, leading to indeterminacy. This is particularly important when analyzing hypothetical scenarios where the same purchase pattern leads to redundant equations, thus preventing the determination of unique prices.

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