Problem 5 - Advanced Engineering Mathematics

Question

In Problems \(1-10\), solve the given system of equations by Cramer's rule. $$ \begin{aligned} &2 x+y=1 \\ &3 x+2 y=-2 \end{aligned} $$

Step-by-Step Solution

Answer

The solution is \(x = 4\) and \(y = -7\).

Step 1: Write the System in Matrix Form

First, represent the given system of equations as a matrix equation. The system \(\begin{aligned}&2 x+y=1 \&3 x+2 y=-2\end{aligned}\) can be written as \(AX = B\), where \(A\) is the coefficient matrix \(\begin{bmatrix} 2 & 1 \ 3 & 2 \end{bmatrix}\), \(X\) is the column matrix of variables \(\begin{bmatrix} x \ y \end{bmatrix}\), and \(B\) is the column matrix of constants \(\begin{bmatrix} 1 \ -2 \end{bmatrix}\).

Step 2: Calculate the Determinant of Matrix A

Compute the determinant of the coefficient matrix \(A = \begin{bmatrix} 2 & 1 \ 3 & 2 \end{bmatrix}\). The determinant, \(\det(A)\), is calculated as \(2 \cdot 2 - 1 \cdot 3 = 4 - 3 = 1\).

Step 3: Calculate the Determinant of Matrix A_x

To find \(x\) using Cramer's Rule, replace the first column of \(A\) with the matrix \(B\): \(A_x = \begin{bmatrix} 1 & 1 \ -2 & 2 \end{bmatrix}\). The determinant \(\det(A_x)\) is calculated as \(1 \cdot 2 - 1 \cdot (-2) = 2 + 2 = 4\).

Step 4: Calculate the Determinant of Matrix A_y

To find \(y\), replace the second column of \(A\) with the matrix \(B\): \(A_y = \begin{bmatrix} 2 & 1 \ 3 & -2 \end{bmatrix}\). The determinant \(\det(A_y)\) is calculated as \(2 \cdot (-2) - 1 \cdot 3 = -4 - 3 = -7\).

Step 5: Solve for x and y using Cramer's Rule

Cramer's rule states that \(x = \frac{\det(A_x)}{\det(A)}\) and \(y = \frac{\det(A_y)}{\det(A)}\). Substitute the obtained values: \[x = \frac{4}{1} = 4,\]\[y = \frac{-7}{1} = -7.\]

Step 6: Conclusion

The solution to the system is \(x = 4\) and \(y = -7\).

Key Concepts

Determinant Calculation

Calculating the determinant is a key step in Cramer's Rule, especially when solving systems of linear equations. The determinant is a scalar value that helps us understand the properties of a matrix, such as whether it is invertible. To compute the determinant of a 2x2 matrix, you apply a simple formula. Given a matrix \(A\):\[A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\]the determinant, denoted as \(\det(A)\), is calculated as:\[\det(A) = a \cdot d - b \cdot c\]For example, for the matrix\[\begin{bmatrix} 2 & 1 \ 3 & 2 \end{bmatrix}\]the determinant would be: \[2 \cdot 2 - 1 \cdot 3 = 4 - 3 = 1\]Understanding how to compute the determinant is vital since it tells us:

• If the determinant is zero, the matrix is singular and the system of equations may have no unique solutions.
• For nonzero determinants, the system can be solved uniquely with Cramer's Rule, as demonstrated in the example.

Matrix Equations

Matrix equations are a compact way of expressing systems of linear equations. Instead of writing out each equation individually, we can express the entire system as a single matrix equation. Consider a standard linear system:\[\begin{aligned}&2x + y = 1 \&3x + 2y = -2\end{aligned}\]This can be rewritten in the form \(AX = B\), where:

• \(A\) is the coefficient matrix containing the coefficients of the variables: \[\begin{bmatrix} 2 & 1 \ 3 & 2 \end{bmatrix}\]
• \(X\) is the column matrix of variables: \[\begin{bmatrix} x \ y \end{bmatrix}\]
• \(B\) is the column matrix of constants: \[\begin{bmatrix} 1 \ -2 \end{bmatrix}\]

Writing equations in matrix form simplifies the manipulation of equations, allowing for more streamlined solution methods like those using matrices.

System of Linear Equations

A system of linear equations consists of two or more linear equations with the same set of variables. The goal is to find a set of values for these variables that satisfy all equations simultaneously. Cramer's Rule is particularly useful for solving such systems when the number of equations equals the number of variables.For a system like:\[\begin{aligned}&2x + y = 1 \&3x + 2y = -2\end{aligned}\]We have two linear equations with two variables \(x\) and \(y\). To solve this system:

• First, we represent it in a matrix form with a coefficient matrix \(A\), as explained earlier.
• Then, we apply Cramer's Rule, which states that for each variable in \(X\), we calculate the determinant of matrix \(A\) and two modified matrices where one column of \(A\) is replaced with \(B\), as seen in the problem solution.

Cramer's Rule is particularly handy since it provides a direct formula to calculate the variables, provided that the determinant of \(A\) is not zero, ensuring a unique solution.