Problem 59 - A Graphical Approach to College Algebra

Question

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to complete the solution. $$\begin{array}{l}4 x+3 y=-7 \\\2 x+3 y=-11\end{array}$$

Step-by-Step Solution

Answer

The solution is \(x = 2\), \(y = -5\).

Step 1: Set up the system of equations

The given system of equations is:\[ \begin{align*} 4x + 3y &= -7 \ 2x + 3y &= -11 \end{align*} \] We will solve this using Cramer's Rule.

Step 2: Write the coefficient matrix and calculate its determinant

The coefficient matrix \(A\) is \( \begin{bmatrix} 4 & 3 \ 2 & 3 \end{bmatrix} \). The determinant of \(A\) is calculated as follows: \[ D = \begin{vmatrix} 4 & 3 \ 2 & 3 \end{vmatrix} = (4 \times 3) - (3 \times 2) = 12 - 6 = 6 \]Since \(D eq 0\), Cramer's Rule can be applied.

Step 3: Set up matrices to find \(x\) and \(y\)

To find \(x\), we replace the first column of \(A\) with the constants from the right side of the equations: \[A_x = \begin{bmatrix} -7 & 3 \ -11 & 3 \end{bmatrix}\] To find \(y\), we replace the second column of \(A\) with the constants: \[A_y = \begin{bmatrix} 4 & -7 \ 2 & -11 \end{bmatrix}\]

Step 4: Calculate determinants of updated matrices

The determinant of \(A_x\) is: \[D_x = \begin{vmatrix} -7 & 3 \ -11 & 3 \end{vmatrix} = (-7 \times 3) - (3 \times -11) = -21 + 33 = 12\] The determinant of \(A_y\) is: \[D_y = \begin{vmatrix} 4 & -7 \ 2 & -11 \end{vmatrix} = (4 \times -11) - (-7 \times 2) = -44 + 14 = -30\]

Step 5: Apply Cramer's Rule and solve for \(x\) and \(y\)

Cramer's Rule states that \(x = \frac{D_x}{D}\) and \(y = \frac{D_y}{D}\). Using our determinants: \[ x = \frac{12}{6} = 2\]\[ y = \frac{-30}{6} = -5\]Thus, the solution to the system of equations is \(x = 2\), \(y = -5\).

Key Concepts

Systems of Equations

A system of equations is essentially a group of two or more equations that share common variables. In our example, we have two equations:

• The first equation: \(4x + 3y = -7\)
• The second equation: \(2x + 3y = -11\)

The goal is to find values for "x" and "y" that satisfy both equations at the same time. This means that our solution should work in both of these equations and make them true counterparts of one another.
Systems of equations can be solved using various methods, like substitution, elimination, or, as demonstrated here, Cramer's Rule. Each method offers different insights and can be more or less convenient depending on the particular problem. Cramer's Rule is especially useful when the determinant is nonzero, allowing a direct computation of the solutions.

Determinants

Determinants play a crucial role when working with Cramer's Rule. They are numerical values calculated from square matrices and provide insightful properties.

• The determinant of a matrix provides a measure of how "invertible" the matrix is.
• In our example, the determinant of the coefficient matrix \(A\) is 6.
• If this determinant were zero, Cramer's Rule couldn't be applied.

Calculating the determinant involves multiplying across the diagonals of a 2x2 matrix:
\[ D = ad - bc\]where \(a\), \(b\), \(c\), and \(d\) are the elements of the matrix in the form \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \). These calculations are significant as they provide the means to solve for unknown variables.

Coefficient Matrix

The coefficient matrix, denoted by \(A\), is composed of the coefficients in front of the variables in our system of equations. For the given problem:

• The coefficient matrix \(A\) is \( \begin{bmatrix} 4 & 3 \ 2 & 3 \end{bmatrix} \).
• This matrix captures the essential linear relationships articulated by the system.

By organizing these coefficients into a matrix, we can efficiently apply various matrix operations, such as finding the determinant or solving the system using techniques like Cramer’s Rule.
Such a structured approach to organizing and understanding the components of our system makes it easier to visualize and compute solutions systematically. It paves the way for linking algebraic equations to linear algebra's broader aspects.

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, vector spaces, and linear transformations. Among its many operations, it includes systems of linear equations, matrices, determinants, and more.
One of the key features of linear algebra is employing matrices to solve systems of equations, like we did using Cramer's Rule. Here’s why this field is immensely practical:

• It transforms complex systems into more manageable forms.
• Through matrix operations, we can unravel solutions with increased efficiency.
• It provides tools like determinants, essential for solving and validating solutions.

Understanding linear algebra not only helps in solving systems of equations but also enriches mathematical skills in various applications, such as computer graphics, engineering, data science, and physics. It provides a deep and practical insights into multi-variable system challenges.