Problem 53 - Precalculus: Mathematics for Calculus

Question

Use Cramer's Rule to solve the system. $$\left\\{\begin{aligned} 3 y+5 z &=4 \\ 2 x &-z=10 \\ 4 x+7 y &=0 \end{aligned}\right.$$

Step-by-Step Solution

Answer

\( x = \frac{213}{59}, y = \frac{-84}{59}, z = \frac{208}{59} \)

Step 1: Write the System in Matrix Form

The given system of equations can be represented in matrix form as \( AX = B \) where \( A \) is the coefficient matrix, \( X \) is the column matrix of variables, and \( B \) is the column matrix of constants. The matrices are:\[A = \begin{bmatrix} 0 & 3 & 5 \ 2 & 0 & -1 \ 4 & 7 & 0 \end{bmatrix}, \quad X = \begin{bmatrix} x \ y \ z \end{bmatrix}, \quad B = \begin{bmatrix} 4 \ 10 \ 0 \end{bmatrix}\]

Step 2: Find the Determinant of A

Calculate the determinant of matrix \( A \), \( \text{det}(A) \). Use the formula for the determinant of a 3x3 matrix:\[\text{det}(A) = a(ei − fh) − b(di − fg) + c(dh − eg)\]Substitute the respective values:\[\text{det}(A) = 0(0 + 7) - 3(0 - 16) + 5(14 - 0) = 0 + 48 + 70 = 118\]

Step 3: Determine D_x by Replacing Column 1 of A with B

Replace the first column of matrix \( A \) with matrix \( B \), yielding matrix \( D_x \):\[D_x = \begin{bmatrix} 4 & 3 & 5 \ 10 & 0 & -1 \ 0 & 7 & 0 \end{bmatrix}\]Calculate \( \text{det}(D_x) \) using the determinant formula:\[\text{det}(D_x) = 4(0 + 7) - 3(0 - 16) + 5(70 - 0) = 28 + 48 + 350 = 426\]

Step 4: Determine D_y by Replacing Column 2 of A with B

Replace the second column of matrix \( A \) with matrix \( B \), yielding matrix \( D_y \):\[D_y = \begin{bmatrix} 0 & 4 & 5 \ 2 & 10 & -1 \ 4 & 0 & 0 \end{bmatrix}\]Calculate \( \text{det}(D_y) \) using the determinant formula:\[\text{det}(D_y) = 0(0 + 0) - 4(-4 - 4) + 5(-40 - 0) = 0 + 32 - 200 = -168\]

Step 5: Determine D_z by Replacing Column 3 of A with B

Replace the third column of matrix \( A \) with matrix \( B \), yielding matrix \( D_z \):\[D_z = \begin{bmatrix} 0 & 3 & 4 \ 2 & 0 & 10 \ 4 & 7 & 0 \end{bmatrix}\]Calculate \( \text{det}(D_z) \) using the determinant formula:\[\text{det}(D_z) = 0(0 - 70) - 3(-80 - 40) + 4(14 - 0) = 0 + 360 + 56 = 416\]

Step 6: Solve for x, y, and z using Cramer's Rule

Using Cramer's Rule, solve for \( x \), \( y \), and \( z \):\[x = \frac{\text{det}(D_x)}{\text{det}(A)} = \frac{426}{118} = \frac{213}{59}\]\[y = \frac{\text{det}(D_y)}{\text{det}(A)} = \frac{-168}{118} = \frac{-84}{59}\]\[z = \frac{\text{det}(D_z)}{\text{det}(A)} = \frac{416}{118} = \frac{208}{59}\]

Key Concepts

Determinant

The concept of a determinant is crucial when solving systems of equations with Cramer's Rule. A determinant is a special number that can be calculated from a square matrix. This number gives important information about the matrix, such as whether it has an inverse. For a 3x3 matrix, the determinant can be found using a specific formula: \ \[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \] where each letter represents elements within the matrix. The determinant helps us understand properties of the matrix and is essential in calculating solutions using Cramer's Rule.
Calculating determinants can seem tricky at first, but once you understand the pattern, it becomes manageable. Remember, if the determinant of a matrix is zero, the matrix doesn't have an inverse, and thus Cramer's Rule cannot be applied.

Matrix

Matrices are a pivotal part of linear algebra and are instrumental when working with systems of equations. A matrix is essentially a rectangular array of numbers arranged in rows and columns.
In the context of solving systems of equations, you use matrices to represent the coefficients of the variables in compact form. For instance, in this exercise, the matrix \( A \) represents the coefficients of the equations, while matrix \( X \) represents variables, and \( B \) contains constants on the right-hand side.
The power of matrices lies in their ability to succinctly express linear operations and transformations. Learning to manipulate matrices, including finding their determinants or inverses, is fundamental when working with more complex algebraic expressions and systems.

System of Equations

A system of equations comprises multiple equations that you solve simultaneously. The goal is to find values for the variables that satisfy all the equations at the same time. These systems can be linear or non-linear, depending on the nature of the equations.
For this exercise, the system is linear, indicating that it consists of linear equations. Solving such systems reveals a point or points where the equations intersect, which represents the solution to the problem. Systems of equations are common in various fields like physics, engineering, and economics, where multiple factors interact simultaneously.
Cramer's Rule offers an efficient technique to solve such systems when the number of equations equals the number of unknowns, provided that the determinant of the coefficient matrix is non-zero.

Linear Algebra

Linear algebra is the branch of mathematics dealing with vectors, matrices, and systems of linear equations. It helps in understanding geometrical concepts and is foundational in advanced mathematics and various scientific fields.
It explores operations like matrix addition, multiplication, and the finding of determinants and inverses. Linear algebra simplifies complex systems into manageable formats through vector and matrix representation.
In relation to Cramer's Rule, linear algebra provides all necessary tools and concepts. Mastery of linear algebra enables solving problems involving multiple variables and equations, making it a subject worth understanding thoroughly for its broad applications in science and engineering.