Problem 41 - Algebra and Trigonometry

Question

\(29-44\) 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

The solution is approximately \( x \approx 4.61 \), \( y \approx -0.78 \), \( z \approx 2.15 \).

Step 1: Write the System of Equations in Matrix Form

First, ensure all variables are on the left and constants on the right. This gives three equations: \(0x + 3y + 5z = 4\), \(2x - z = 10\), \(4x + 7y + 0z = 0\). Identify the coefficient matrix \(A\), the variable matrix \(X\), and the constant matrix \(B\): \[ A = \begin{bmatrix} 0 & 3 & 5 \ 2 & 0 & -1 \ 4 & 7 & 0 \end{bmatrix}, \quad X = \begin{bmatrix} x \ y \ z \end{bmatrix}, \quad B = \begin{bmatrix} 4 \ 10 \ 0 \end{bmatrix} \] So, \(AX = B\).

Step 2: Compute the Determinant of Matrix A

To apply Cramer's Rule, calculate \( \det(A) \). The determinant of matrix \(A\) is calculated as follows:\[ \det(A) = 0(0 - (-1)\cdot7) - 3(2\cdot0 - 4\cdot(-1)) + 5(2\cdot7 - 4\cdot0) \] This simplifies to \( \det(A) = 0 + 3\cdot4 + 5\cdot14 = 0 + 12 + 70 = 82 \).

Step 3: Formulate the Matrices for Cramer's Rule

We need to replace the columns of \(A\) with \(B\) to find each variable. - Replace the first column with \(B\) to get \(A_x\): \[ A_x = \begin{bmatrix} 4 & 3 & 5 \ 10 & 0 & -1 \ 0 & 7 & 0 \end{bmatrix} \] - Replace the second column with \(B\) to get \(A_y\): \[ A_y = \begin{bmatrix} 0 & 4 & 5 \ 2 & 10 & -1 \ 4 & 0 & 0 \end{bmatrix} \] - Replace the third column with \(B\) to get \(A_z\): \[ A_z = \begin{bmatrix} 0 & 3 & 4 \ 2 & 0 & 10 \ 4 & 7 & 0 \end{bmatrix} \]

Step 4: Calculate Determinants for Each Variable

Calculate \( \det(A_x) \), \( \det(A_y) \), and \( \det(A_z) \).- \( \det(A_x) = 4(0-(-1)\cdot7) - 3(10\cdot0-0\cdot(-1)) + 5(10\cdot7-0\cdot4) \) \(= 4\cdot7 + 0 + 350 = 28 + 350 = 378 \)- \( \det(A_y) = 0(0-0) - 4(2\cdot0-4\cdot(-1)) + 5(2\cdot0-4\cdot4) \) \(= 0 + 16 - 80 = -64 \)- \( \det(A_z) = 0(10\cdot0-0\cdot7) - 3(2\cdot0-4\cdot10) + 4(2\cdot7-4\cdot0) \) \(= 0 - 3(-40) + 56 = 120 + 56 = 176 \)

Step 5: Use Cramer's Rule to Solve for Variables

Cramer's Rule states \(x = \frac{\det(A_x)}{\det(A)}\), \(y = \frac{\det(A_y)}{\det(A)}\), and \(z = \frac{\det(A_z)}{\det(A)}\). Calculate the values for each variable:- \( x = \frac{378}{82} \approx 4.61 \)- \( y = \frac{-64}{82} = -\frac{32}{41} \approx -0.78 \)- \( z = \frac{176}{82} = \frac{88}{41} \approx 2.15 \)

Key Concepts

System of Equations

A system of equations is essentially a collection of two or more equations with a set of variables. In the problem at hand, we are dealing with three equations:

• 3y + 5z = 4
• 2x - z = 10
• 4x + 7y = 0

These equations must be solved simultaneously to find the values of the variables x, y, and z.
In mathematical contexts such as physics or economics, systems of equations are particularly important because they allow us to model and solve multiple relationships at once. The solutions to these systems give us values for variables that satisfy all given equations simultaneously.
An efficient way to solve these systems, especially when there are multiple equations and variables, is using matrix methods such as Cramer's Rule.

Matrix Determinant

The determinant is a special number that can be calculated from a square matrix. It provides essential insights into the properties of the matrix. For matrices involved in linear algebra, the determinant often indicates whether a system of linear equations has a unique solution.
If the determinant of a coefficient matrix, like our matrix A, is zero, the system may have no unique solution. This is because a zero determinant identifies a singular matrix, meaning it does not have an inverse.

• In our earlier example, the determinant of matrix A is 82, which isn't zero. This tells us the system of equations has a unique solution.

The determinant is calculated using a pattern of multiplication and subtraction based on the matrix's elements. For a 3x3 matrix, the calculation involves applying the rule of Sarrus and the properties of determinants, as demonstrated in the detailed solution above.

Linear Algebra

Linear algebra is the branch of mathematics that deals with vectors, vector spaces (also known as linear spaces), and linear transformations between these spaces. This subject is central to understanding systems of linear equations, matrices, and determinants.
In the context of Cramer's Rule, linear algebra provides the framework for understanding how a set of linear equations can be expressed in matrix form. Each equation corresponds to a row in the matrix, and each variable corresponds to a column. The solutions to the system are found by applying operations such as calculating determinants, which are all grounded in linear algebra concepts.

• Using matrices simplifies the handling of complex systems of equations, making computations more structured and the solutions more easily obtained.
• Linear algebra is foundational in fields such as computer science, physics, and engineering because it provides methods for solving problems involving linear relationships.

Solving Linear Systems

Solving linear systems can be approached in several ways, with Cramer's Rule being one of them. This method is particularly useful when dealing with systems of equations where the number of variables equals the number of equations, forming a square matrix.
Cramer's Rule leverages determinant calculations to find each variable individually, through specific matrices where columns are replaced by the constant terms from the equations.
Here's how it works:

• Calculate the determinant of the original coefficient matrix.
• Create matrices for each variable by replacing the corresponding column in the original matrix with the constant terms.
• Find the determinant for each of these new matrices.
• Finally, each variable is solved by dividing the determinant of its matrix by the determinant of the original matrix.

This method elucidates the solutions where every variable is distinctly calculated, showing the tangible power of determinants in solving systems analytically.