Problem 28 - Precalculus

Question

Use Cramer's rule to solve each system of equations, if possible. $$\begin{array}{l} 0.5 x-0.4 y=-3.6 \\ 10 x+3.6 y=-14 \end{array}$$

Step-by-Step Solution

Answer

The solution is \(x = -3.2\) and \(y = 5\).

Step 1: Write the system of equations in standard form

The given system of equations is already in standard form:\[\begin{aligned}0.5x - 0.4y &= -3.6 \10x + 3.6y &= -14\end{aligned}\]

Step 2: Define the coefficient matrix and constants

Identify the coefficient matrix \(A\) and the constant matrix \(B\):\[A = \begin{bmatrix} 0.5 & -0.4 \ 10 & 3.6 \end{bmatrix}, \quad B = \begin{bmatrix} -3.6 \ -14 \end{bmatrix}\]

Step 3: Calculate the determinant of the coefficient matrix

The determinant of matrix \(A\), denoted as \(\det(A)\), is calculated as:\[\det(A) = (0.5)(3.6) - (-0.4)(10) = 1.8 + 4 = 5.8\]Since \(\det(A) eq 0\), Cramer's rule can be used.

Step 4: Form the first augmented matrix for \(x\)

Replace the first column of \(A\) with \(B\) to form matrix \(A_x\):\[A_x = \begin{bmatrix} -3.6 & -0.4 \ -14 & 3.6 \end{bmatrix}\]

Step 5: Calculate the determinant of the first augmented matrix \(A_x\)

Compute \(\det(A_x)\):\[\det(A_x) = (-3.6)(3.6) - (-0.4)(-14) = -12.96 - 5.6 = -18.56\]

Step 6: Calculate the value of \(x\)

Use Cramer's rule: \(x = \frac{\det(A_x)}{\det(A)}\):\[x = \frac{-18.56}{5.8} \approx -3.2\]

Step 7: Form the second augmented matrix for \(y\)

Replace the second column of \(A\) with \(B\) to form matrix \(A_y\):\[A_y = \begin{bmatrix} 0.5 & -3.6 \ 10 & -14 \end{bmatrix}\]

Step 8: Calculate the determinant of the second augmented matrix \(A_y\)

Compute \(\det(A_y)\):\[\det(A_y) = (0.5)(-14) - (-3.6)(10) = -7 + 36 = 29\]

Step 9: Calculate the value of \(y\)

Use Cramer's rule: \(y = \frac{\det(A_y)}{\det(A)}\):\[y = \frac{29}{5.8} \approx 5\]

Step 10: Final Result

The solution to the system of equations is \(x = -3.2\) and \(y = 5\).

Key Concepts

Determinant in Cramer's Rule

When solving a system of equations using Cramer's Rule, the determinant plays a crucial role. The determinant is a special number calculated from a square matrix. For a 2x2 matrix, it is found using the formula \[(ad - bc)\], where the matrix is given by \[{\begin{bmatrix} a & b \ c & d \end{bmatrix}}\]. In the provided system of equations, the coefficient matrix is \\[{A = \begin{bmatrix} 0.5 & -0.4 \ 10 & 3.6 \end{bmatrix}}\], and its determinant, \({\det(A)}\), is calculated as follows: \(0.5 \times 3.6 - (-0.4 \times 10) = 1.8 + 4 = 5.8\). Since the determinant is non-zero, it indicates that the system is consistent and has a unique solution. If the determinant were zero, Cramer's Rule could not be applied because the system would either have no solutions or infinitely many solutions.
Calculating the determinant gives us more than just a check for Cramer's Rule. It can also tell us important properties about the matrix, like its invertibility. In this case, because \(\det(A) eq 0\), the matrix is invertible. This means that it is possible to find its inverse, further empowering various methods in linear algebra, including those related to solving systems of equations.

Understanding Systems of Equations

A system of equations is essentially a collection of two or more equations with a set of unknowns. The goal is to find values for these unknowns that satisfy each equation within the system simultaneously. In other words, you're looking for a common solution. Take our example system: \(0.5x - 0.4y = -3.6\) and \(10x + 3.6y = -14\). Each equation represents a line on a graph, and the solution corresponds to the intersection of these lines.
Solving systems of equations is fundamental in numerous fields, providing answers to questions about supply and demand, physics problems, and engineering challenges. Systems can be solved through various methods: graphically, by substitution, elimination, or using matrix algebra techniques like Cramer's Rule. Each of these methods has its own advantages, depending on the system's complexity and context.

• Consistent and Independent: The system has a single solution, as is the case here with the determinant of the coefficient matrix being non-zero.
• Consistent and Dependent: If a determinant is zero, solutions may exist along an infinite line.
• Inconsistent: No solutions exist when the determinant is zero, and there is no intersection.

Cramer's Rule is especially useful for small systems where you can quickly and efficiently calculate the determinant and use it to find solutions.

Matrix Algebra Basics

Matrix algebra forms the backbone of many procedures in linear algebra, providing a structured way to handle linear equations. Matrices are rectangular arrays of numbers—like charts or tables. In solving systems of equations using matrix methods, a typical approach involves forming one or more matrices to represent the system.
For our exercise, the coefficient matrix \({A}\) is created using the coefficients of the variables from the equations. The constant matrix \[B = \begin{bmatrix} -3.6 \ -14 \end{bmatrix}\], represents the results of the equations. This structured setup allows us to apply algebraic operations, such as calculating determinants, to solve for unknowns.
Matrix operations like addition, multiplication, and derivation of determinants and inverses are foundational skills in matrix algebra. When you replace columns of the coefficient matrix with the constant matrix, as done in Cramer's Rule, you produce matrices that help isolate solutions for each variable. This transformation translates abstract algebraic manipulation into concrete numerical results. Whether in computational scenarios or theoretical explorations, understanding matrix algebra equips you with the tools to navigate complex systems and derive meaningful conclusions effectively.