Problem 2 - College Algebra and Calculus: An Applied Approach

Question

Find \(x\) and \(y\). $$ \left[\begin{array}{rr} x & -7 \\ 9 & y \end{array}\right]=\left[\begin{array}{ll} 5 & -7 \\ 9 & -8 \end{array}\right] $$

Step-by-Step Solution

Answer

The solution to this exercise is \(x = 5\) and \(y = -8\).

Step 1: Identify the corresponding elements

The elements in the same positions in two equal matrices must be the same. Therefore, for these two 2x2 matrices, we have \(x\) from the first matrix equal to 5 in the same position in the second matrix, and \(y\) from the first matrix equal to -8 in the same position in the second matrix.

Step 2: Solve for x

From the analysis in Step 1, we have \(x = 5\) because these are the corresponding elements in the top left position of both matrices.

Step 3: Solve for y

From the analysis in Step 1, we have \(y = -8\) because these are the corresponding elements in the bottom right position of both matrices.

Key Concepts

Matrix Equality

When two matrices are said to be equal, it means that they have the exact same dimensions and their corresponding elements are also identical. This is a fundamental aspect in linear algebra that simplifies comparisons and calculations with matrices. In the case of the matrices given in the exercise, both are 2x2 matrices. This means they have two rows and two columns.
If we were to check for equality of two 2x2 matrices, each element within one matrix must be equal to the corresponding element in the other:

• The element in row 1, column 1 of the first matrix equals the element in row 1, column 1 of the second matrix.
• The element in row 1, column 2 of the first matrix equals the element in row 1, column 2 of the second matrix.
• The element in row 2, column 1 of the first matrix equals the element in row 2, column 1 of the second matrix.
• The element in row 2, column 2 of the first matrix equals the element in row 2, column 2 of the second matrix.

This means that for any matrix equality comparison, checking the individual element equality is mandatory. This way, you can ensure the matrices are truly the same.

Solving Systems of Equations

Matrices are not only tools for organizing data but can also be used to solve systems of equations. In this exercise, identifying equal matrices transforms into straightforward equations that can be solved easily.
For instance, when matrices are stated to be equal, corresponding elements can be equated, effectively setting up a system of equations.

• For the given matrices, we derive the equations such as \( x = 5 \) by matching the top left elements, and \( y = -8 \) by equating the bottom right elements.

What this really means is that understanding what each matrix's elements represent allows you to solve for variables efficiently. Rather than attempting to solve equations independently outside of matrix structures, using the concept of matrix equality streamlines the process.

Linear Algebra Concepts

Linear algebra is a branch of mathematics that deals with vectors, vector spaces (sometimes called linear spaces), and linear transformations. One of the core ideas in linear algebra is the concept of matrices, which are powerful tools used to express linear equations and transformations compactly and efficiently.

• Linear algebra helps in modeling and solving real-world problems where systems of linear equations arise.
• By representing systems of equations as matrices, we can use a range of algebraic techniques to find solutions, such as Gaussian elimination or using matrix inverses.

In our given example, solving for the variable elements within matrices is a part of linear algebra's wider applications. It helps in performing operations that would otherwise be time-consuming and complex. Understanding the structure and mathematical footing of matrices allows students to appreciate and apply linear algebra concepts effectively in various fields, from engineering to computer science.