Problem 21 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use Cramer's Rule to solve (if possible) the system of equations. \(\left\\{\begin{array}{l}4 x-3 y=-10 \\ 6 x+9 y=12\end{array}\right.\)

Step-by-Step Solution

Answer

The solution to the system of equations using Cramer's Rule is \(x = -1\) and \(y = 2\).

Step 1: Formulate the Coefficient Matrix and the Constant Vector

Firstly, formulate the coefficient matrix \(A\) and the constant vector \(B\) from the system: \(A= \[\[4,-3\], \[6,9\]\]\) and \(B= \[\[-10\],\[12\]\]\)

Step 2: Calculate Determinant of the Coefficients Matrix (det A)

The second step is to calculate the determinant of the coefficient matrix \(A\), denoted as \(det(A)\). Using the formula for the determinant we get: \(det(A) = 4*9 - (-3)*6 = 36 + 18 = 54. The determinant of the coefficient matrix is not equal to zero, so the system of equations has a unique solution.

Step 3: Formulate Two New Matrices by Replacing Columns

Next, formulate two new matrices by replacing each column of the coefficient matrix \(A\) with the constant vector \(B\), one at a time. The first matrix \(A_1\) is obtained by replacing the first column with the constant vector \(B\): \(A_1 = \[\[-10,-3\], \[12,9\]\]\). The second matrix \(A_2\) is obtained by replacing the second column with the constant vector \(B\): \(A_2 = \[\[4,-10\], \[6,12\]\]\).

Step 4: Calculate the Determinants of the New Matrices

The fourth step is to calculate the determinants of these new matrices. These determinants are found with the same formula for calculating the determinant: \(det(A_1) = -10*9 - (-3)*12 = -90 + 36 = -54\) and \(det(A_2) = 4*12 - (-10)*6 = 48 + 60 = 108.

Step 5: Compute the Values of Variables x and y

Finally, apply Cramer's rule, which states that each variable is equal to the determinant of the matrix obtained by replacing the column of coefficients of that variable with the constant vector, divided by the determinant of the coefficients matrix (det A). Hence, the solutions are \(x = det(A_1)/det(A) = -54/54 = -1\) and \(y = det(A_2)/det(A) = 108/54 = 2\).

Key Concepts

Understanding System of Linear Equations

A system of linear equations is a collection of one or more linear equations involving the same set of variables. For instance, a system with two equations and two unknowns looks like: \[\begin{equation}\left\{\begin{array}{l}ax + by = e \ cx + dy = f\end{array}\right.\end{equation}\]In our exercise, we have the system: \[\begin{equation}\left\{\begin{array}{l}4x - 3y = -10 \ 6x + 9y = 12\end{array}\right.\end{equation}\]The goal is to find the values of the variables, in this case, x and y, that satisfy both equations simultaneously. Techniques to solve such systems include graphing, substitution, elimination, and matrix methods, like Cramer's Rule.
Cramer's Rule is applicable when there are as many equations as variables and when the matrix composed of the coefficients of the variables is nonsingular—that is, it has a nonzero determinant. This method provides a systematic and quick way of finding the solution to a system of linear equations.

Deciphering Determinants

The determinant of a matrix is a unique number assigned to a square matrix that provides important information about the matrix, including whether it has an inverse and the volume of the geometric shape it represents. In Cramer's Rule, determinants play a pivotal role:\[\begin{equation}det(A) = ad - bc\end{equation}\]

Why Are Determinants Important?

• Determinants can indicate if a system has a unique solution, no solution, or infinitely many solutions.
• If the determinant of the coefficient matrix is zero (det(A) = 0), the system is either inconsistent or has infinite solutions.
• A non-zero determinant, as in our exercise (det(A) = 54), suggests a unique solution.

In the given solution, by determining the matrix's determinant, we could assert that a unique solution exists. This is crucial before proceeding with Cramer's Rule.

Coefficient Matrix and Its Significance

A coefficient matrix is the matrix formed from the coefficients of variables in a system of linear equations. It is central to solving systems with matrix operations. For our problem, the coefficient matrix is: \[\begin{equation}A = \begin{pmatrix}4 & -3 \6 & 9\end{pmatrix}\end{equation}\]

Purpose of the Coefficient Matrix:

• It simplifies the system, condensing all the variable coefficients into a clear, tabulated form.
• It helps in performing matrix operations to find solutions to the system.
• Its determinant (det(A)) plays a significant role in applying Cramer's Rule.

When applying Cramer's Rule, each variable's value is found by replacing the relevant column in the coefficient matrix with the constants from the equation, forming new matrices. Then, their determinants are used to solve for the variables, as shown in the steps of the solution.