Problem 21 - Precalculus with Limits

Question

In Exercises 21-26, write the system of linear equations represented by the augmented matrix. (Use variables \(x\), \(y\), \(z\), and \(w\), if applicable.) \( \left[\begin{array}{rrrr} 1 & & 2 & & \vdots & & 7 \\ 2 & & -3 & & \vdots & & 4 \end{array}\right] \)

Step-by-Step Solution

Answer

The system of linear equations is: \(x + 2y = 7\) and \(2x - 3y = 4\).

Step 1: Form the first equation

Examine the first row. It represents the first equation. The first number (1) is the coefficient of the first variable (x), the second number (2) is the coefficient of the second variable (y) and the number on the right side of the dots (7) is the constant term. Therefore, the equation is \(x + 2y = 7\).

Step 2: Form the second equation

Examine the second row of the matrix. It forms the second equation. The first number (2) is the coefficient of the first variable (x), the second number (-3) is the coefficient of the second variable (y) and the number on the right side of the dots (4) is the constant term. Therefore, the equation is \(2x - 3y = 4\).

Key Concepts

Augmented Matrix

An augmented matrix is a compact and organized way to represent a system of linear equations. This special type of matrix combines both the coefficients of the variables and the constants from the equations into one neat table. For instance, when delving into the matrix given in the exercise, it appears as follows: \[\left[\begin{array}{cccc|c} 1 & 2 & 0 & 0 & 7 \2 & -3 & 0 & 0 & 4 \end{array}\right]\] In this matrix, the left side contains the coefficients of the variables, while the right side (after the vertical bar) displays the constants. Each row corresponds to an equation, and each column corresponds to a variable or the constant term. Utilizing augmented matrices lets us elegantly solve systems of linear equations, especially when employing tactics like row reduction or Gaussian elimination.

System of Equations

A system of equations is a collection of equations with similar variables that must be solved together. Each equation in the system provides a restriction that applies to all the unknowns, thereby narrowing down the possible solutions. In this exercise, converting the augmented matrix into its representative equations, we see: - Equation 1: \(x + 2y = 7\) - Equation 2: \(2x - 3y = 4\) These linear equations denote lines in a coordinate plane, where the solution represents the point at which these lines intersect. Solving systems gives precise values for the variables whereby both equations remain true. This process can involve various methods such as substitution, elimination, or using matrices. Understanding this also helps bridge gaps to more advanced topics in algebra and calculus.

Variables x y z w

Variables like \(x\), \(y\), \(z\), and \(w\) are often used in mathematics to symbolize unknown quantities. In linear algebra, these variables act as placeholders that allow us to express mathematical models or equations unclutteredly. They help in forming equations from observations about the world or hypothetical problems. In context with the exercise, the variables \(x\) and \(y\) appear in the equations derived from the augmented matrix: - Equation 1: \(x + 2y = 7\) - Equation 2: \(2x - 3y = 4\) The goal is to determine the specific values of these variables that satisfy all equations in the system. Although \(z\) and \(w\) were not present in this exercise, they act as placeholders for cases with more complex or larger systems. Mastery of using such variables is foundational for engaging with mathematical theories and applications.