Problem 9 - A Graphical Approach to Precalculus with Limits

Question

Write the augmented matrix for each system. Do not solve the system. $$\begin{array}{r} x+5 y=6 \\ x=3 \end{array}$$

Step-by-Step Solution

Answer

\(\begin{bmatrix} 1 & 5 & | & 6 \\ 1 & 0 & | & 3 \end{bmatrix}\)

Step 1: Identify the System of Equations

The system of equations is given as: \[\begin{array}{r}x + 5y = 6 \x = 3 \\end{array}\] This involves two equations with two variables, \(x\) and \(y\).

Step 2: Rewrite Each Equation in Standard Form

Ensure both equations are in standard form, \( Ax + By = C \). The equations already fit this form:1. \(x + 5y = 6\) 2. \(x + 0y = 3\) (Note: we explicitly express the absence of \(y\) in the second equation by writing it as \(0y\)).

Step 3: Extract Coefficients for the Augmented Matrix

For the augmented matrix, extract the coefficients of \(x\) and \(y\) and the constants from both equations:1. From \(x + 5y = 6\), coefficients are \(1, 5\) and constant \(6\).2. From \(x + 0y = 3\), coefficients are \(1, 0\) and constant \(3\).

Step 4: Write the Augmented Matrix

Form the augmented matrix using the coefficients and constants extracted:\[\begin{bmatrix}1 & 5 & | & 6 \1 & 0 & | & 3\end{bmatrix}\] This matrix represents the system of equations, where each row corresponds to an equation from the system.

Key Concepts

Systems of Equations

A system of equations is essentially a collection of two or more equations that work together with shared variables. In our everyday context, when two or more equations are considered at the same time, they form a system of equations. In the context of the given exercise, we are working with two equations:

• Equation 1: \( x + 5y = 6 \)
• Equation 2: \( x = 3 \)

These equations involve the same variables, \(x\) and \(y\), and the goal usually is to find values for these variables that make all the equations true at the same time. However, in this exercise, you are only asked to set up the equations in a matrix form and not solve them. This setup provides a clear visual way to represent these equations and explore possible solutions systematically.

Coefficients

Coefficients are the numerical values that are directly placed alongside the variables in an equation. They serve as multipliers of the variables and are crucial in forming the augmented matrix of a system. In our example, each equation has coefficients:

• For the equation \( x + 5y = 6 \), the coefficient of \(x\) is 1 and for \(y\) is 5.
• In the equation \( x = 3 \), the coefficient of \(x\) is 1, while \(y\) is not present, so we can consider its coefficient to be 0 (thus, it's \(x + 0y = 3\)).

These coefficients allow for the equations to be organized within a matrix. In matrix notation, these values are placed in a structured format to help in systematic analysis or solutions, especially in larger systems.

Standard Form

The standard form of a linear equation is expressed as \( Ax + By = C \), where \( A \), \( B \), and \( C \) are constants, and \( x \) and \( y \) are variables. This conventional format makes it easier to organize and solve systems of equations because each equation conforms to the same structure.
In the provided exercise:

• The first equation, \( x + 5y = 6 \), already fits nicely into this form where \( A = 1 \), \( B = 5 \), and \( C = 6 \).
• The second equation, \( x = 3 \), is also in standard form implied by \( x + 0y = 3 \), with \( A = 1 \), \( B = 0 \), and \( C = 3 \).

Ensuring the equations are in standard form is crucial for transforming them into a matrix, leading us to a much more accessible path to their representation as an augmented matrix.

Linear Equations

Linear equations are equations of the first degree, meaning they only include variables raised to the power of one. They form straight lines when graphed on a coordinate plane, hence the name 'linear'. In any linear equation like \( Ax + By = C \), the solution represents all the points \( (x, y) \) that lie on the line defined by the equation.
In context of our exercise, both equations

• \( x + 5y = 6 \)
• \( x = 3 \)

are linear because they do not include any powers or roots of the variables apart from the first power. The clarity and simplicity of linear equations make them foundational in algebra and matrix representations, serving as stepping stones to more complex algebraic concepts.