Problem 15 - Precalculus with Limits

Question

In Exercises 15-20, write the augmented matrix for the system of linear equations. \( \left\\{ \begin{array}{l} 4x - 3y = -5 \\ -x + 3y = 12 \end{array} \right. \)

Step-by-Step Solution

Answer

The augmented matrix for the system of linear equations is \[ \begin{bmatrix} 4 & -3 & -5 \ -1 & 3 & 12 \end{bmatrix} \]

Step 1: Identify the coefficients and constants

For the given system of linear equations, \n 4x - 3y = -5 and \n -x + 3y = 12, \n the coefficients and constants for each equation are: \n For the first equation, the coefficient of x is 4, the coefficient of y is -3, and the constant is -5; \n For the second equation, the coefficient of x is -1, the coefficient of y is 3, and the constant is 12.

Step 2: Write the augmented matrix - 1

Write the coefficients and constants of the equations as rows of a matrix. For the first row, write the coefficients of x and y, and the constant from the first equation as follows: [4, -3, -5].

Step 3: Write the augmented matrix - 2

For the second row, write the coefficients of x and y, and the constant from the second equation as follows: [-1, 3, 12].

Step 4: The augmented matrix

By placing the numbers train in the matrix, the augmented matrix for the given system of linear equations becomes \n \[ \begin{bmatrix} 4 & -3 & -5 \ -1 & 3 & 12 \end{bmatrix} \]

Key Concepts

Understanding Systems of Linear Equations

A system of linear equations is a set of two or more linear equations involving the same set of variables. They are named 'linear' because they each represent a line in a coordinate plane when plotted. The main idea is to find the point or points where these lines intersect, which corresponds to the solution of the system. Systems can have:

• a single unique solution, meaning the lines intersect at one point,
• infinitely many solutions, which occurs when the lines are on top of each other, or
• no solution, which is when the lines are parallel and never meet.

In this exercise, we are looking at a system with two equations and two unknowns, which means we're seeking points on a 2D plane where both equations are simultaneously satisfied.

Exploring Coefficients

Coefficients are the numbers that multiply the variables in an equation. In the context of systems of linear equations, coefficients accompany variables such as \( x \) and \( y \) and show the influence each variable has on the equation. For example, in the equation \(4x - 3y = -5\), 4 is the coefficient of \( x \) and -3 is the coefficient of \( y \). Similarly, in the equation \(-x + 3y = 12\), the coefficient of \( x \) is -1 and 3 is the coefficient for \( y \). These coefficients are essential because they help determine the slope and position of the line represented by the equation when it is graphed. When combined with their constants, these coefficients give us a complete equation that contributes to forming the augmented matrix used to solve the system.

Performing Row Operations

Row operations are techniques used to manipulate matrices to simplify the solving process of systems of linear equations. These operations are crucial for transforming the matrix into simpler forms like row-echelon form or reduced row-echelon form, from which solutions can be easily deduced. There are three primary types of row operations:

• Row swapping: Exchanging two rows in the matrix.
• Row scaling: Multiplying all entries of a row by a non-zero scalar.
• Row addition: Adding a multiple of one row to another row.

In this exercise, although we primarily focus on converting the system of equations into an augmented matrix, understanding row operations is essential. These operations can then be applied to the augmented matrix for further simplification and to find the solutions effectively.