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A system of equations involves two or more equations that contain two or more variables. The goal is to find the value of these variables that satisfy all the equations simultaneously. In this exercise, we have two linear equations that describe the motions of two joggers. Let's break those down:

1. When jogging toward each other, the combined speed of the joggers equals the distance divided by time, which gives us the equation \( v_A + v_B = 10 \).
2. When jogging in the same direction, the relative speed between them allows one to catch up, providing the equation \( v_A - v_B = 1 \).

These equations together form what's known as a system of linear equations, and solving them gives us the speeds of each jogger in miles per hour.

Cramer's Rule is a mathematical theorem used to solve systems of linear equations with as many equations as unknowns, provided that the determinant of the coefficient matrix is non-zero. It's a straightforward method to find solutions to linear systems using determinants.

Here's how Cramer's Rule functions in our exercise:

• For each variable, replace the respective column in the coefficient matrix with the constants' column and calculate the determinant of this new matrix.
• Divide this determinant by the determinant of the original coefficient matrix to obtain the value of the variable.

Cramer's Rule is particularly useful because it gives us a neat, formulaic way to find our variables without needing additional algebraic juggling.

Determinants are a special number calculated from a square matrix. In the context of solving linear equations, determinants describe the scaling factor applied to a space spanned by the matrix, such as its area or volume.

Key points about determinants:

• They can tell us if a matrix can be inverted; a determinant of zero means no unique solution exists.
• In our case, the determinant of the coefficient matrix ensures the system has a unique solution, allowing further calculations with Cramer's Rule.

With the given equations, the determinant calculated as \(-2\) confirmed that the set up was correct and that further progress towards a solution was possible.

Matrix algebra is a powerful tool for solving systems of linear equations. It utilizes matrices, which are rectangular arrays of numbers, to transform and solve complex mathematical systems.

For this exercise:

• The coefficient matrix is created from the coefficients of our variables \( v_A \) and \( v_B \).
• The solution process becomes systematic by transforming and multiplying matrices.

By expressing our system of equations in matrix form, we can leverage matrix operations such as calculating determinants and implementing Cramer's Rule to find the solution efficiently. Matrix algebra simplifies handling even larger or more complex systems of equations with ease.