Problem 31 - Algebra and Trigonometry

Question

Use Cramer's Rule to solve each system. $$\left\\{\begin{aligned}4 x-5 y-6 z &=-1 \\\x-2 y-5 z &=-12 \\\2 x-y &=7\end{aligned}\right.$$

Step-by-Step Solution

Answer

The solution to the given system of equations using Cramer's Rule is \(x = -6.125\), \(y = -6.583\), and \(z = -2.125\)

Step 1: Write down the system of equations

The system of equations is given by:\n \(4x - 5y - 6z = -1\)\n\(x - 2y - 5z = -12\)\n\(2x - y = 7\) Note: The third equation does not have the variable \(z\), hence for this equation \(z = 0\)

Step 2: Formulate the coefficient matrix and calculate its determinant

Form the coefficient matrix, \(A\), of the system and calculate its determinant, \(\Delta\), as follows: \[A = \begin{bmatrix} 4 & -5 & -6 \\ 1 & -2 & -5 \\ 2 & -1 & 0 \end{bmatrix}\]Then calculate the determinant of A:\[\Delta = \begin{vmatrix} 4 & -5 & -6 \\ 1 & -2 & -5 \\ 2 & -1 & 0 \end{vmatrix} = 4(-2(0)-(-1)(-5)) - (-5)(1(0)-2 * -5) - (-6)(1*-1 - 2*-2) = 4*5 - 5 * 10 - 6 * 3 = 20 - 50 - 18 = -48

Step 3: Calculate the determinants for \(x\), \(y\) and \(z\)

Replace each column in turn by the column vector on the right of the equals sign (-1, -12, 7), and solve for the determinant in each case:For \(x\), \(\Delta_x\) replace the first column of the coefficient matrix with the vector on the right side:\[\Delta_x = \begin{vmatrix} -1 & -5 & -6 \\ -12 & -2 & -5 \\ 7 & -1 & 0 \end{vmatrix} = -1(-2*0-(-1*-5)) - (-5)(-12*0 - 7*-5) - (-6)(-12*-1 - 7*1) = -1*-5 - 5*-35 + 6*19 = 5 +175 + 114 = 294\]For \(y\), \(\Delta_y\) replace the second column of the coefficient matrix with the vector on the right side:\[\Delta_y = \begin{vmatrix} 4 & -1 & -6 \\ 1 & -12 & -5 \\ 2 & 7 & 0 \end{vmatrix} = 4*(-12*0 - 7*-5) - (-1)(1*0 - 2*-5) - (-6)(1*7 - 2*-12) = 4*35 - 1*-10 + 6*31 =140 - 10 + 186 =316\]For \(z\), \(\Delta_z\) replace the third column of the coefficient matrix with the vector on the right side:\[\Delta_z = \begin{vmatrix} 4 & -5 & -1 \\ 1 & -2 & -12 \\ 2 & -1 & 7 \end{vmatrix} = 4*(-2*7 - -1*-12) - (-5)(1*7 - 2*-12) - (-1)(1*-1 - 2*-2) = 4*-14 + 5*31 + 1*3 =-56 + 155 +3 = 102\]

Step 4: Solve for x, y and z using Cramer's Rule

Cramer's Rule states that the solution of the system is given by \(x = \Delta_x/\Delta\), \(y = \Delta_y/\Delta\) and \(z = \Delta_z/\Delta\)Substituting for \(\Delta_x, \Delta_y, \Delta_z\) and \(\Delta\) yields:\[ x = \Delta_x/\Delta = 294/-48 = -6.125 \]\[ y = \Delta_y/\Delta = 316/-48 = -6.583 \]\[ z = \Delta_z/\Delta = 102/-48 = -2.125 \]

Key Concepts

System of Equations

A system of equations consists of multiple equations that share common variables. In the exercise given, we deal with three equations that involve the variables \(x\), \(y\), and \(z\). Solving such a system means finding the values of these variables that satisfy all the equations simultaneously.
The purpose of solving a system of equations is to find these unknowns, which might represent various practical situations like shadows in geometry or currents in an electrical circuit.

• To express a system of linear equations, we often write it in a matrix form using coefficients of the variables.
• In linear algebra, systems can be linear (straight-line relationships) or non-linear (involving curves).

In this particular problem, note that the third equation doesn't include the variable \(z\), but the system is classified as a linear system since the highest power of each variable is 1.

Determinants

The determinant is a special number that can be calculated from a square matrix. It provides important properties about the matrix and the system it represents.
In the context of systems of equations, determining whether the determinant of the coefficient matrix is zero or non-zero tells us a lot:

• If the determinant is zero, the system may have no solutions or an infinite number of solutions.
• If the determinant is non-zero, the system has a unique solution.

For the system of equations in our problem, the coefficient matrix has a determinant of \(\Delta = -48\), which is non-zero. This implies that the system has a unique solution for the variables \(x\), \(y\), and \(z\). Understanding how to compute the determinant is key in methods like Cramer's Rule used here.

Linear Algebra

Linear algebra is a field of mathematics focusing on vector spaces and linear mappings. It's core to solving systems of linear equations.
By representing equations as matrices and vectors, linear algebra provides powerful tools and methods:

• Cramer's Rule, which uses determinants, to solve for variables in a linear system.
• Matrix operations, such as addition, multiplication, and finding inverse matrices.

In practice, methods from linear algebra like those used in this problem, help in modeling real-world phenomena. From economic forecasts to engineering designs, using the principles of linear algebra provides clarity to complex systems.

Matrices

Matrices are rectangular arrays of numbers, symbols, or expressions, arranged in rows and columns. They are powerful tools in representing and solving systems of linear equations.
The matrix form consists of:

• Rows that correspond to individual equations in the system.
• Columns corresponding to each variable’s coefficients.

In the exercise, we see a 3x3 matrix for the coefficients of \(x\), \(y\), and \(z\). By manipulating matrices through operations like finding the determinant or substituting columns, we can solve equations effectively. Matrices also highlight the power of systematic solutions, bringing order to the complex interaction of numbers. Whether it's in computers changing graphics or systems predicting weather, matrices are central to applied mathematics.