Question

Solve by determinants. $$\begin{aligned} &2 x-3 y=3\\\ &4 x+5 y=39 \end{aligned}$$

Short answer

Using Cramer's Rule, the solution to the system is \(x = 6, y = 3\).

Step-by-step solution

Write the system of equations as a matrix

Formulate the system of linear equations in matrix form as follows: Coefficient matrix A = \(\begin{bmatrix} 2 & -3 \ 4 & 5 \end{bmatrix}\), Constant vector B = \(\begin{bmatrix} 3 \ 39 \end{bmatrix}\), and Variable matrix X = \(\begin{bmatrix} x \ y \end{bmatrix}\), so that AX = B.

Calculate the determinant of coefficient matrix A

Compute the determinant of A, denoted as det(A), using the formula \(\text{det}(A) = a_{11}a_{22} - a_{12}a_{21}\), which gives us \(\text{det}(A) = (2)(5) - (-3)(4) = 10 + 12 = 22\).

Calculate the determinant for x (Dx)

Replace the first column of A with constant vector B to form a new matrix \(A_x\) and compute its determinant, \(\text{det}(A_x)\). Matrix \(A_x = \begin{bmatrix} 3 & -3 \ 39 & 5 \end{bmatrix}\), thus \(\text{det}(A_x) = 3(5) - (-3)(39) = 15 + 117 = 132\).

Calculate the determinant for y (Dy)

Replace the second column of A with constant vector B to form a new matrix \(A_y\) and compute its determinant, \(\text{det}(A_y)\). Matrix \(A_y = \begin{bmatrix} 2 & 3 \ 4 & 39 \end{bmatrix}\), thus \(\text{det}(A_y) = 2(39) - 3(4) = 78 - 12 = 66\).

Solve for x using Cramer's Rule

Apply Cramer's Rule to solve for x. The rule states that \(x = \frac{\text{det}(A_x)}{\text{det}(A)}\). Substituting the calculated determinants, we get \(x = \frac{132}{22} = 6\).

Solve for y using Cramer's Rule

Use Cramer's Rule again to solve for y. It states that \(y = \frac{\text{det}(A_y)}{\text{det}(A)}\). Putting the values in, we get \(y = \frac{66}{22} = 3\).

Key concepts

Cramer's Rule

Cramer's Rule is a straightforward method to solve systems of linear equations using determinants. It provides the solution in terms of the ratio of two determinants. For a system of two equations, like in our exercise, you can find the value of the variables x and y by calculating the determinant of the coefficient matrix and the determinants of matrices obtained by replacing one column of the coefficient matrix with the constant terms.

Use Cramer's Rule as follows:

• For variable x, replace the column corresponding to x in the coefficient matrix with the constants vector to get a new matrix, calculate its determinant (Dx).
• For variable y, do the same but replace the column corresponding to y to get its determinant (Dy).
• Divide Dx and Dy by the determinant of the coefficient matrix to get the values of x and y, respectively.

This rule only applies if the determinant of the coefficient matrix is non-zero, which indicates that the system has a unique solution. Applying this to our exercise resulted in finding that x equals 6 and y equals 3, which is the solution to the system of equations provided.

Determinant of a Matrix

The determinant of a matrix is a special number that can be calculated from its elements and provides important properties about the matrix. For a 2x2 matrix, the determinant is found using the formula \( \text{det}(A) = a_{11}a_{22} - a_{12}a_{21} \).

In our exercise, the determinant of the coefficient matrix is calculated by taking the product of the top left and bottom right elements and subtracting the product of the top right and bottom left elements. Knowing the determinant is crucial because if it's zero, it means that the system of equations does not have a unique solution, possibly having either no solution or infinitely many solutions.

Matrix Representation of Systems

Representing systems of linear equations in matrix form is an organized and powerful approach to finding solutions. This representation involves three main components:

• Coefficient matrix (A): Contains the coefficients of the variables in the system.
• Variables matrix (X): Is a column matrix of the variables we are solving for.
• Constants vector (B): Is a column matrix for the constants on the other side of the equals sign.

The structure AX = B enables us to use various matrix operations and methods, such as Cramer's Rule or other techniques like matrix inversion, to solve the system. The matrix form simplifies complex calculations and highlights relationships between linear equations.

Coefficient Matrix

The coefficient matrix is the cornerstone of the matrix representation of a system of equations. It contains all the coefficients from the system, with each row representing an equation and each column representing the coefficients of a particular variable across those equations. A major aspect of this matrix is that its determinant can reveal whether the system has a unique solution; if the determinant is non-zero, there is indeed one unique solution.

In the context of our exercise, the coefficient matrix is \( A = \begin{bmatrix} 2 & -3 \ 4 & 5 \end{bmatrix} \). Its determinant helps us apply Cramer's Rule. If you are working with systems that have more equations and variables, the coefficient matrix will be larger, but the importance of its determinant in finding solutions remains just as significant.