Problem 19 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use Cramer's Rule to solve (if possible) the system of equations. \(\left\\{\begin{array}{r}-7 x+11 y=-1 \\ 3 x-9 y=9\end{array}\right.\)

Step-by-Step Solution

Answer

Based on Cramer's Rule, the solutions to the system of equations are x = -2.1 and y = 2.

Step 1: Formulate the Coefficient Matrix and the Constant Vector

The first step is to construct the coefficient matrix, \[ A = \[ \[ -7, 11 \], \[ 3, -9 \] \] \] and the constant vector, \[ B = \[ -1, 9 \] \]. From these two elements we will be able to solve the system.

Step 2: Compute the Determinant of Coefficient Matrix

The next step is to calculate the determinant of the coefficient matrix A: \[ det(A) = -7*(-9) - 11*3 = 63 - 33 = 30 \]. The determinant is non-zero, so Cramer's rule can be applied to the given system of equations.

Step 3: Apply Cramer's Rule to find x

Cramer's rule for x involves replacing column 1 (x's coefficients) in matrix A with vector B. Find det(Ax) = det(\[ \[ -1, 11 \], \[ 9, -9 \] \]), and then x = det(Ax) / det(A) = -63 / 30 = -2.1.

Step 4: Apply Cramer's Rule to find y

Apply Cramer's rule for y means replace column 2 (y's coefficients) in matrix A with vector B. Compute det(Ay) = det(\[ \[ -7, -1 \], \[ 3, 9 \] \]), and derive y = det(Ay) / det(A) = 60 / 30 = 2.

Key Concepts

System of Equations

A system of equations is a collection of two or more equations with a common set of variables. In this instance, the system of equations is \[\begin{cases}-7x + 11y = -1 \3x - 9y = 9 \end{cases}\]This system consists of two equations with two variables, namely, \(x\) and \(y\). The solution to such systems is any pair of \(x\) and \(y\) that satisfies both equations simultaneously.
There are multiple methods for solving systems of equations, including substitution, elimination, and matrix methods like Cramer's Rule. Here, we use Cramer's Rule, which is applicable provided the determinant of the coefficient matrix is non-zero.

Determinant Calculation

The determinant of a matrix is a special number that can be calculated from its elements. It is a crucial component in solving systems of linear equations using Cramer's Rule. For a 2x2 matrix, the determinant can be calculated using the formula: \[ \text{det}(A) = a_{11} \cdot a_{22} - a_{12} \cdot a_{21} \]For our coefficient matrix \[ A = \begin{bmatrix} -7 & 11 \3 & -9 \end{bmatrix},\]the determinant is computed as: \[ -7 \times (-9) - 11 \times 3 = 63 - 33 = 30. \]A non-zero determinant (det(A) = 30) indicates that the system of equations can be solved using Cramer's Rule.

Coefficient Matrix

The coefficient matrix is a matrix that contains only the coefficients of the variables in a system of linear equations. In this system of equations:\[\begin{cases}-7x + 11y = -1 \3x - 9y = 9 \end{cases}\]the coefficient matrix \( A \) is formed by the coefficients of \( x \) and \( y \) from both equations:\[ A = \begin{bmatrix} -7 & 11 \3 & -9 \end{bmatrix}. \]This matrix acts as the foundation for computing the determinant, and ultimately for applying Cramer's Rule to solve the system.

Constant Vector

A constant vector in system equations refers to the column of numbers on the right side of the equations. These numbers are constant values derived from the linear equations. In our case:\[\begin{cases}-7x + 11y = -1 \3x - 9y = 9 \end{cases}\]the constant vector \( B \) would be:\[ B = \begin{bmatrix} -1 \9 \end{bmatrix}. \]This vector is juxtaposed with the coefficient matrix inside Cramer's Rule calculations, determining how the system's structure impacts the solution.