Problem 5 - Contemporary Precalculus

Question

In Exercises \(5-8,\) the augmented matrix of a system of equations is given. Express the system in equation notation. \left(\begin{array}{rrr} 3 & -5 & 4 \\ 9 & 7 & 2 \end{array}\right)

Step-by-Step Solution

Answer

Question: Convert the given augmented matrix to a system of linear equations in equation notation. Augmented Matrix: \[ \left[ \begin{array}{cc|c} 3 & -5 & 4 \\ 9 & 7 & 2 \end{array} \right] \] Answer: The given augmented matrix can be expressed in equation notation as: \begin{cases} 3x - 5y = 4\\ 9x + 7y = 2 \end{cases}

Step 1: Identify the coefficients and constant terms

From the matrix, we can pick the columns as representing variable coefficients and constants. First row: coefficients (3, -5) and constant term (4). Second row: coefficients (9, 7) and constant term (2).

Step 2: Write down the linear equations using coefficients and constants

Now we can write down the linear equations by using the coefficients and constants identified in step 1. First Equation: \(3x - 5y = 4\) Second Equation: \(9x + 7y = 2\) Thus, the given augmented matrix can be expressed in equation notation as follows: \begin{cases} 3x - 5y = 4\\ 9x + 7y = 2 \end{cases}

Key Concepts

Augmented Matrix

An augmented matrix is a tool that helps us translate systems of linear equations into a more compact form, where everything is neatly organized. This matrix includes both the coefficients of the variables in each equation as well as the constants from the other side of each equation. It typically looks like a rectangular array of numbers. For example, in our exercise, the augmented matrix is:

• \( \begin{pmatrix} 3 & -5 & 4 \ 9 & 7 & 2 \end{pmatrix} \)

Here, the first two columns represent the coefficients of the variables (like \(x\) and \(y\)), while the third column represents the constants on the right side of the equations.
Understanding the structure of an augmented matrix simply requires recognizing these parts: the left section for coefficients and the right section for constants. This allows us to more easily handle and manipulate systems of equations, especially when dealing with more equations and variables.

Systems of Equations

Systems of equations consist of multiple equations that share common variables. The goal of solving a system of equations is to find the values of these variables that satisfy all the equations simultaneously. In the given exercise, we are working with the following system:

• First Equation: \(3x - 5y = 4\)
• Second Equation: \(9x + 7y = 2\)

These equations are connected because they both involve the variables \(x\) and \(y\). Solving systems like this can reveal whether there's a single solution, no solution, or infinitely many solutions.
Systems of equations can be expressed in multiple ways. For beginners, expressing them from an augmented matrix back into their equation form can be a helpful practice. Different methods, such as substitution, elimination, or matrix operations, can be used to find solutions.

Coefficients and Constants

Coefficients and constants are essential enough in the world of algebra that their roles deserve some attention. Each coefficient in a system of equations corresponds to the number multiplying a variable. In contrast, constants are the numbers on the right-hand side that do not change based on the variable value. For instance, in our exercise, for the equation \(3x - 5y = 4\):

• Coefficients: \(3\) for \(x\) and \(-5\) for \(y\)
• Constant: \(4\)

Understanding these distinctions is critical because they shape the behaviour of linear equations.
When solving or interpreting systems of equations, it's important to correctly identify which are coefficients and which are constants to accurately formulate the equations from a matrix or any other representation. This fundamental understanding aids in simplifying and solving complex mathematical problems efficiently.