Question

Solve by determinants. $$\begin{aligned} &5 p+4 q-14=0\\\ &17 p=31+3 q \end{aligned}$$

Short answer

Using determinants, the solution to the system of equations is \( p = 2 \) and \( q = 1 \).

Step-by-step solution

Write the given system of equations in the matrix form

Represent the system of equations as a matrix equation of the form AX = B, where A is the coefficient matrix, X is the vector of variables, and B is the constant matrix. For the given equations, this yields \( A = \begin{pmatrix} 5 & 4 \ 17 & -3 \end{pmatrix} \), \( X = \begin{pmatrix} p \ q \end{pmatrix} \), and \( B = \begin{pmatrix} 14 \ 31 \end{pmatrix} \).

Find the determinant of matrix A

Calculate the determinant of matrix A. The determinant is \( \text{det}(A) = (5)(-3) - (4)(17) = -15 - 68 = -83 \).

Solve for p using Cramer's Rule

Replace the first column of A with B to form a new matrix \( A_p \), then calculate its determinant. \( A_p = \begin{pmatrix} 14 & 4 \ 31 & -3 \end{pmatrix} \), and \( \text{det}(A_p) = (14)(-3) - (4)(31) = -42 - 124 = -166 \). Use Cramer's Rule to find the value of p: \( p = \frac{\text{det}(A_p)}{\text{det}(A)} = \frac{-166}{-83} = 2 \).

Solve for q using Cramer's Rule

Replace the second column of A with B to form a new matrix \( A_q \), then calculate its determinant. \( A_q = \begin{pmatrix} 5 & 14 \ 17 & 31 \end{pmatrix} \), and \( \text{det}(A_q) = (5)(31) - (14)(17) = 155 - 238 = -83 \). Use Cramer's Rule to find the value of q: \( q = \frac{\text{det}(A_q)}{\text{det}(A)} = \frac{-83}{-83} = 1 \).

Key concepts

Determinants

Determinants are a mathematical property assigned to square matrices. They play a crucial role in linear algebra, particularly when it comes to solving systems of linear equations, like the ones presented in our exercise. Technically, the determinant is a scaling factor for the transformation that a matrix represents. For example, a 2x2 matrix \( A = \begin{pmatrix} a & b \ c & d \end{pmatrix} \) has a determinant calculated by \( \text{det}(A) = ad - bc \).

In the exercise, we use determinants to apply Cramer's Rule. The determinant tells us whether a unique solution exists, and if so, how to find it. If the determinant of the coefficient matrix is zero, the system may not have a unique solution. Fortunately, in our example, the determinant of matrix \( A \) is not zero, allowing us to proceed with Cramer's Rule to find the solutions for \( p \) and \( q \).

System of Linear Equations

A system of linear equations consists of two or more linear equations with the same set of variables. We're dealing with such a system in the given exercise, where we have two equations with two variables, \( p \) and \( q \). The goal is to find the values of these variables that satisfy both equations at the same time.

Linear systems like these can be visualized graphically as lines on a coordinate plane. The solution to the system is the point or points where the lines intersect. Solving these systems can be done in various ways, including graphically, algebraically through substitution or elimination, or with more advanced methods such as matrix operations or Cramer's Rule, which is used in this exercise. The chosen method typically depends on the complexity and specific requirements of the problem at hand.

Matrix Equations

Matrix equations are a compact and efficient way to represent and solve systems of linear equations. They consist of matrices, which are rectangular arrays of numbers, organized in rows and columns. In our exercise, the system of linear equations is transformed into the matrix equation \( AX = B \) where:

• \( A\) is the coefficient matrix,
• \(X\) is a column vector of the variables we want to solve for,
• \(B\) is a column vector representing the constants from the right-hand side of the equations.

Matrix equations are powerful because they can be manipulated and solved using various matrix operations. One of the most common is matrix inversion, but when the matrix is not invertible or to avoid computational complexity, methods like Cramer's Rule come into play. This rule takes advantage of determinants to solve for each variable directly, provided the determinant of the coefficient matrix is non-zero. In the given exercise, after converting the system into a matrix equation, we leverage the determinants of altered versions of the coefficient matrix to find the values for \( p \) and \( q \), effectively solving the system.