Question

Solve by determinants. $$\begin{aligned} &\frac{p}{6}-\frac{q}{3}+\frac{1}{3}=0\\\ &\frac{2 p}{3}-\frac{3 q}{4}-1=0 \end{aligned}$$

Short answer

After computing the determinants, \(p\) and \(q\) can be found using Cramer's rule.

Step-by-step solution

Clear the Fractions

Multiply each equation by the least common multiple of all the denominators to clear the fractions. In the first equation, the denominators are 6, 3, and 3, so the LCM is 6. Multiply the entire first equation by 6. Similarly, for the second equation, the denominators are 3 and 4, so the LCM is 12. Multiply the entire second equation by 12.

Simplify the Equations

After multiplying to clear the fractions, simplify each term in both equations: The first equation becomes: \(6\times\frac{p}{6}-6\times\frac{q}{3}+6\times\frac{1}{3}=0\), which simplifies to \(p-2q+2=0\). Similarly, the second equation becomes: \(12\times\frac{2p}{3}-12\times\frac{3q}{4}-12=0\), which simplifies to \(8p-9q-12=0\).

Write the Coefficients Matrix and the Constants Matrix

Arrange the linear system in matrix form and identify the coefficients matrix and the constants matrix. Coefficients matrix, \(A\), is \(\begin{bmatrix}1 & -2\8 & -9\end{bmatrix}\), and the constants matrix, \(B\), is \(\begin{bmatrix}-2\12\end{bmatrix}\).

Calculate the Determinants

Calculate the determinant of matrix \(A\), denoted as \(D\), and the determinants of the matrices formed by replacing the first and second columns of \(A\) with \(B\), denoted as \(D_p\) and \(D_q\), respectively. \(D =\begin{vmatrix}1 & -2\8 & -9\end{vmatrix}; D_p =\begin{vmatrix}-2 & -2\12 & -9\end{vmatrix}; D_q =\begin{vmatrix}1 & -2\8 & 12\end{vmatrix}\).

Solve for p and q

Use Cramer's rule to find the values of \(p\) and \(q\): \(p = \frac{D_p}{D}\) and \(q = \frac{D_q}{D}\). Calculate each determinant and divide as appropriate to find the values of \(p\) and \(q\).

Key concepts

Cramer's Rule

Cramer's Rule offers a straightforward method to solve systems of linear equations using determinants. It's a theorem that states that for a system of linear equations with as many equations as unknowns, and with a non-zero determinant, each variable can be solved for individually. This is done by taking the determinant of the coefficient matrix and several modified versions of this matrix.

For the given exercise, Cramer's Rule simplifies the solution process to finding the values of variables by simply calculating and substituting determinants in fraction form. Attention must be given to the order of the matrix's columns when substituting the constants to avoid any mistakes. Cramer's Rule not only makes the solution process more systematic but also helps to avoid algebraic manipulation errors that could occur with other methods.

Least Common Multiple

The Least Common Multiple (LCM) is crucial when you want to eliminate fractions from linear equations. It is the smallest non-zero number that is a multiple of two or more numbers. In the context of solving linear systems, finding the LCM of denominators allows you to multiply each equation by it to clear fractions.

This is an excellent step towards preparing the system for applying Cramer's Rule, as it transforms the system into a simpler integer matrix form. The LCM ensures that the process is efficient and that the equations are scaled appropriately without changing the solutions. Understanding how to correctly identify and calculate the LCM is essential because it is the foundation upon which the equations are transformed, leading to matrices that are easier to work with.

Matrices

Matrices are incredibly powerful mathematical tools, serving as a compact way to represent and solve systems of linear equations. A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. In the context of our exercise, there are two matrices of interest: the coefficients matrix and the constants matrix.

The coefficients matrix, typically denoted as A, includes the coefficients of the variables in the system of equations. The constants matrix, often denoted as B, on the other hand, contains the constant terms from the right-hand side of the equations. Determinants are then calculated from these matrices. Properly setting up these matrices is critical to successfully applying Cramer's Rule.

Determinant Calculation

The determinant of a matrix is a special scalar value that can provide a lot of information about the matrix, including whether a system of linear equations has a unique solution. For a 2x2 matrix, the determinant is calculated as the product of the entries on the main diagonal minus the product of the entries on the other diagonal.

In our exercise, determinant calculation is a key part of applying Cramer's Rule. You must accurately calculate three determinants: one from the coefficients matrix and two from the modified versions of this matrix with the constants matrix substituted in. This process needs careful attention to detail to ensure the correct values are obtained, as even a slight mistake can lead to an incorrect solution. By mastering determinant calculation, you greatly enhance your ability to solve linear systems efficiently.