Problem 48 - Precalculus: Mathematics for Calculus

Question

Use Cramer's Rule to solve the system. $$\left\\{\begin{aligned} 5 x-3 y+z &=& 6 \\ 4 y-6 z &=& 22 \\ 7 x+10 y &=&-13 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The solution is \(x = 1\), \(y \approx -0.7337\), \(z \approx -5.0\).

Step 1: Write the System in Matrix Form

The system of equations can be written in matrix form as \(A\mathbf{x} = \mathbf{b}\) where \(A\) is the coefficient matrix, \(\mathbf{x}\) is the column matrix of variables, and \(\mathbf{b}\) is the constant matrix. For the given system:\[A = \begin{bmatrix} 5 & -3 & 1 \ 0 & 4 & -6 \ 7 & 10 & 0 \end{bmatrix}, \quad \mathbf{x} = \begin{bmatrix} x \ y \ z \end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix} 6 \ 22 \ -13 \end{bmatrix}\]

Step 2: Calculate the Determinant of A

To use Cramer's Rule, first calculate the determinant of the matrix \(A\). The determinant \(D\) is calculated as:\[D = \begin{vmatrix} 5 & -3 & 1 \ 0 & 4 & -6 \ 7 & 10 & 0 \end{vmatrix} = 5(4\cdot0 - (-6)\cdot10) + 3(0\cdot0 - (-6)\cdot7) + 1(0\cdot10 - 4\cdot7)\]Calculate:\[D = 5(60) + 3(42) + 1(-28) = 300 + 126 - 28 = 398\]

Step 3: Create Matrices and Calculate Determinants for x, y, z

Cramer's Rule requires finding determinants of modified versions of matrix \(A\):\(A_x\), \(A_y\), and \(A_z\):- \(A_x\): Replace the first column of \(A\) with \(\mathbf{b}\):\[A_x = \begin{bmatrix} 6 & -3 & 1 \ 22 & 4 & -6 \ -13 & 10 & 0 \end{bmatrix}\]\[D_x = \begin{vmatrix} 6 & -3 & 1 \ 22 & 4 & -6 \ -13 & 10 & 0 \end{vmatrix} = 6(4\cdot0 - (-6)\cdot10) + 3(22\cdot0 - (-6)\cdot(-13)) + 1(22\cdot10 - 4\cdot(-13))\]Calculate:\[D_x = 6(60) + 3(0 - 78) + 1(220 + 52) = 360 - 234 + 272 = 398\]- \(A_y\): Replace the second column of \(A\) with \(\mathbf{b}\):\[A_y = \begin{bmatrix} 5 & 6 & 1 \ 0 & 22 & -6 \ 7 & -13 & 0 \end{bmatrix}\]\[D_y = \begin{vmatrix} 5 & 6 & 1 \ 0 & 22 & -6 \ 7 & -13 & 0 \end{vmatrix} = 5(22\cdot0 - (-6)\cdot(-13)) - 6(0\cdot0 - (-6)\cdot7) + 1(0\cdot(-13) - 22\cdot7)\]Calculate:\[D_y = 5(-78) + 6(42) + 1(-154) = -390 + 252 - 154 = -292\]- \(A_z\): Replace the third column of \(A\) with \(\mathbf{b}\):\[A_z = \begin{bmatrix} 5 & -3 & 6 \ 0 & 4 & 22 \ 7 & 10 & -13 \end{bmatrix}\]\[D_z = \begin{vmatrix} 5 & -3 & 6 \ 0 & 4 & 22 \ 7 & 10 & -13 \end{vmatrix} = 5(4\cdot(-13) - 22\cdot10) + 3(0\cdot(-13) - 22\cdot7) + 6(0\cdot10 - 4\cdot7)\]Calculate:\[D_z = 5(-52 - 220) + 3(0 - 154) + 6(-28) = 5(-272) + 3(-154) + 6(-28) = -1360 - 462 - 168 = -1990\]

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

Cramer's Rule states that \(x = \frac{D_x}{D}\), \(y = \frac{D_y}{D}\), and \(z = \frac{D_z}{D}\).Substitute the determinants found:\[x = \frac{398}{398} = 1\,\]\[y = \frac{-292}{398}\,\approx -0.7337,\]\[z = \frac{-1990}{398}\,\approx -5.0\]Thus, the solution to the system is \(x = 1\), \(y \approx -0.7337\), and \(z \approx -5.0\).

Key Concepts

Determinant Calculation

In matrix algebra, the determinant is a special number that can provide useful information about a matrix. For solving systems of linear equations using Cramer's Rule, calculating the determinant is a critical first step. The determinant tells us if the system has a unique solution. If it is non-zero, the system has a unique solution; otherwise, it doesn't have a unique solution or may have no solution at all.

To calculate the determinant of a 3x3 matrix, one can use the formula based on cross-multiplication of specific elements and subtracting particular products. For a matrix:\[\begin{vmatrix} a & b & c \ d & e & f \ g & h & i \end{vmatrix} = a(ei - fh) - b(di - fg) + c(dh - eg)\]This method involves multiplying along diagonals in the matrix and applying specific operands to reach the final result.

When using Cramer's Rule, we perform similar calculations but apply them to modified matrices where one column is replaced with the constants from the system of equations.

System of Linear Equations

A system of linear equations is a collection of two or more equations involving the same set of variables. For example, in this problem, the system consists of three equations with three variables: \(x\), \(y\), and \(z\).

Such systems can represent real-world situations where different conditions or constraints need to be met simultaneously. The main goal is to find the values of the variables that satisfy all the equations at once.

• In the matrix form, these equations are represented compactly and can be solved using matrices.
• Cramer's Rule is specifically useful for solving small systems, typically up to three variables, due to its reliance on determinant calculations, which can become complex for larger systems.

This approach leverages the mathematical properties of matrices to efficiently handle these kinds of problems.

Matrix Algebra

Matrix algebra provides a systematic way of manipulating and solving linear equations. In this context, matrices offer a structured and organized approach to handling systems of linear equations, where coefficients and variables are neatly arranged.

Key elements in matrix algebra include:

• Matrix Representation: This involves arranging the coefficients of the variables in the form of a matrix, termed the coefficient matrix.
• Transpose and Inverse: Though not directly needed for Cramer's Rule, these operations are core to matrix algebra and used in other solving techniques.
• Determinants: These are pivotal when using Cramer's Rule, offering insights into the properties of the matrix.

Understanding matrix operations aids greatly in efficiently solving and interpreting systems of equations, laying a foundation for more advanced mathematical concepts.

Precalculus Problem Solving

Precalculus serves as a bridge between basic algebra and calculus, introducing concepts like matrix operations and system solutions that are foundational for higher math. Solving systems with Cramer's Rule is a typical precalculus problem that combines algebraic techniques and matrix theory.

In precalculus,

• Students apply theoretical concepts to practical problems, such as systems of equations that model real-life scenarios.
• Equations are handled in structured forms, leading to systematic problem-solving approaches like using determinants and matrix transformations.

Mastering these techniques prepares students for more complex calculus problems by building a deep understanding of mathematical systems and their interrelations.