Problem 8 - Precalculus with Limits

Question

In Exercises 7-16, use Cramer's Rule to solve (if possible) the system of equations. \(\begin{cases} 4x - 3y = -10 \\ 6x + 9y = 12 \end{cases}\)

Step-by-Step Solution

Answer

The solution to the system of equations is \(x = -1, y = 2\)

Step 1: Formation of the Coefficient Matrix and its Determinant

Construct the system's coefficient matrix \(A\) and compute its determinant. Coefficients matrix \(A\) is given by \[\begin{matrix} 4 & -3 \ 6 & 9 \end{matrix}\]. The determinant \(D\) can be calculated as \(D = 4*9 - -3*6 = 36 + 18 = 54\)

Step 2: Calculate Determinant Dx

Formulate a matrix \(Dx\) by replacing the first column in matrix \(A\) with the constants in the system of equations. The determinant of \(Dx\) can then be calculated. Matrix \(Dx\) is \[\begin{matrix} -10 & -3 \ 12 & 9 \end{matrix}\]. The determinant \(Dx = -10 * 9 - -3 * 12 = -90 + 36 = -54\)

Step 3: Calculate Determinant Dy

Formulate a matrix \(Dy\) by replacing the second column in matrix \(A\) with the constants in the system of equations. The determinant of \(Dy\) can then be calculated. Matrix \(Dy\) is \[\begin{matrix} 4 & -10 \ 6 & 12 \end{matrix}\]. The determinant \(Dy = 4 * 12 - -10 * 6 = 48 + 60 = 108\)

Step 4: Use Cramer’s Rule to Find the Solution

According to Cramer's Rule, the solutions can be found by calculating \(x = Dx/D\) and \(y = Dy/D\). So, \(x = -54/54 = -1\) and \(y = 108/54 = 2\). Thus, the solution to the system of equations is \(x = -1, y = 2\)

Key Concepts

Determinant

The determinant is a special number calculated from a square matrix. In the context of solving a system of equations using Cramer's Rule, understanding how to calculate a determinant is crucial. The matrix determinant provides important information about the system of linear equations, particularly whether the system has a unique solution or not. When dealing with a 2x2 matrix, you can calculate the determinant by using the simple formula:\[D = ad - bc\] where \(a, b, c,\) and \(d\) are elements of the 2x2 matrix:\[\begin{matrix} a & b \ c & d \ \end{matrix}\]
In the exercise provided, the determinant of the coefficient matrix \(A\) is 54, which is non-zero. This non-zero determinant implies that the system of equations has a unique solution. Calculating the determinant correctly ensures the application of Cramer's Rule for finding the solution to the system.

Coefficient Matrix

A coefficient matrix is key in representing a system of linear equations. It consists solely of the coefficients of the variables in the system, making it an essential element for methods like Cramer's Rule. Consider the system of equations: \[ \begin{cases} 4x - 3y = -10 \ 6x + 9y = 12 \end{cases} \]
Here, the coefficient matrix \(A\) is:\[\begin{matrix} 4 & -3 \ 6 & 9 \end{matrix}\]
This matrix is essential in determining the determinants necessary for Cramer's Rule. The consistency and layout of coefficients in matrix form allow for straightforward computations such as determinants, which are necessary to solve the system of equations.

System of Equations

A system of equations is a set of equations with multiple variables that you aim to solve simultaneously. Each equation provides constraints on the variables, and together they can define a unique set of solutions. The example in the exercise presents the system:\[\begin{cases} 4x - 3y = -10 \6x + 9y = 12\end{cases}\]
To solve this, we use tools like Cramer's Rule, which leverages the determinants of specially modified matrices to find the values of the variables. In our exercise, the matrix determinants of \(Dx\) and \(Dy\), derived from the original coefficient matrix, allow us to calculate the values of \(x\) and \(y\). Solving the system reveals that \(x = -1\) and \(y = 2\), illustrating how such systems can often be systematically solved using linear algebra techniques.