Problem 8 - A Graphical Approach to College Algebra

Question

Write the augmented matrix for each system. Do not solve the system. $$\begin{aligned} &3 x+5 y=-13\\\ &2 x+3 y=-9 \end{aligned}$$

Step-by-Step Solution

Answer

The augmented matrix is \( \begin{bmatrix} 3 & 5 & | & -13 \\ 2 & 3 & | & -9 \end{bmatrix} \).

Step 1: Write Coefficient Matrix

Identify the coefficients of the variables in the given system of equations. The coefficients for the first equation, \(3x + 5y = -13\), are 3 and 5. For the second equation, \(2x + 3y = -9\), the coefficients are 2 and 3. These coefficients form the coefficient matrix: \( \begin{bmatrix} 3 & 5 \ 2 & 3 \end{bmatrix} \).

Step 2: Write Constant Matrix

Identify the constants from the right-hand side of each equation. For \(3x + 5y = -13\), the constant is -13 and for \(2x + 3y = -9\), it is -9. These form the constant matrix: \( \begin{bmatrix} -13 \ -9 \end{bmatrix} \).

Step 3: Form the Augmented Matrix

Combine the coefficient matrix and the constant matrix to form the augmented matrix. Place the constant matrix to the right of the coefficient matrix, separated by a vertical line. The augmented matrix is: \( \begin{bmatrix} 3 & 5 & | & -13 \ 2 & 3 & | & -9 \end{bmatrix} \).

Key Concepts

Coefficient Matrix

When working with systems of equations, one of the key components is the coefficient matrix. This matrix is constructed using the numerical coefficients of the variables in each equation.
For example, consider a system of equations given as:

• \( 3x + 5y = -13 \)
• \( 2x + 3y = -9 \)

To form the coefficient matrix, simply extract the coefficients of the variables for each equation. Here, the coefficients are:

• For the first equation: 3 and 5 (from \(3x + 5y\))
• For the second equation: 2 and 3 (from \(2x + 3y\))

Place these values into a matrix, aligning them by their respective variables:\[\begin{bmatrix}3 & 5 \2 & 3\end{bmatrix}\]The coefficient matrix is essential as it uniquely represents the structure of the linear equations' left side, excluding any constants.

Constant Matrix

While the coefficient matrix focuses exclusively on the variables' coefficients within a system of equations, the constant matrix brings our attention to the right-hand side values. These are the constants you're trying to equate the linear expressions to.

• In the equation \(3x + 5y = -13\), the constant is -13.
• In the equation \(2x + 3y = -9\), the constant is -9.

With these constants, form the constant matrix by placing them into a single column:\[\begin{bmatrix}-13 \-9\end{bmatrix}\]The constant matrix is crucial when you wish to represent a system of equations in matrix form, serving as part of the augmented matrix, which provides a comprehensive view of both sides of the equations.

Systems of Equations

A system of equations is a collection of two or more equations with the same set of variables. Solving these systems is all about finding values for the variables that satisfy all equations simultaneously. Here's the system we're dealing with:

• \(3x + 5y = -13\)
• \(2x + 3y = -9\)

Different methods can be applied to solve these systems, including:

• Substitution: Solve one equation for a variable and substitute this expression into the other equation.
• Elimination: Add or subtract equations to eliminate one of the variables, simplifying to a single variable equation.
• Augmented Matrices: Convert the system to an augmented matrix form and apply row operations to find the solution.

In this case, although you do not need to solve it, forming the augmented matrix is a useful step, as it combines the coefficient and constant matrices into one:\[\begin{bmatrix}3 & 5 & | & -13 \2 & 3 & | & -9\end{bmatrix}\]An augmented matrix serves as a compact and straightforward representation, setting the stage for matrix techniques to solve the system.