Problem 43 - Precalculus

Question

Apply Cramer's rule to solve each system of equations, if possible. $$\begin{aligned} &x+y-z=0\\\ &x-y+z=4\\\ &x+y+z=10 \end{aligned}$$

Step-by-Step Solution

Answer

The solution is \( x = 9.5 \), \( y = 3 \), \( z = -6.25 \).

Step 1: Write the System in Matrix Form

The system of equations can be represented in matrix form as \(A \mathbf{x} = \mathbf{b}\) where:\[ A = \begin{pmatrix} 1 & 1 & -1 \ 1 & -1 & 1 \ 1 & 1 & 1 \end{pmatrix}, \quad \mathbf{x} = \begin{pmatrix} x \ y \ z \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 0 \ 4 \ 10 \end{pmatrix} \]

Step 2: Calculate Determinant of Coefficient Matrix

Calculate the determinant of the matrix \(A\):\[ \det(A) = \begin{vmatrix} 1 & 1 & -1 \ 1 & -1 & 1 \ 1 & 1 & 1 \end{vmatrix} \]Using expansion by minors, we find:\[ \det(A) = 1(-1 - 1) - 1(1 - 1) - 1(1 + 1) = -2 - 0 - 2 = -4 \]

Step 3: Calculate Determinant for Each Variable

Since \( \det(A) eq 0 \), we can proceed to find determinants for each variable using Cramer's Rule:\( \det(A_x) \): Replace the first column of \(A\) with \(\mathbf{b}\), then calculate the determinant:\[ A_x = \begin{pmatrix} 0 & 1 & -1 \ 4 & -1 & 1 \ 10 & 1 & 1 \end{pmatrix} \]\[ \det(A_x) = 0(-1 - 1) - 1(4 - 10) - 1(4 + 40) = 0 + 6 - 44 = -38 \]\( \det(A_y) \): Replace the second column of \(A\) with \(\mathbf{b}\), then calculate the determinant:\[ A_y = \begin{pmatrix} 1 & 0 & -1 \ 1 & 4 & 1 \ 1 & 10 & 1 \end{pmatrix} \]\[ \det(A_y) = 1(4 - 10) - 0(1 - 1) - 1(10 - 4) = -6 - 6 = -12 \]\( \det(A_z) \): Replace the third column of \(A\) with \(\mathbf{b}\), then calculate the determinant:\[ A_z = \begin{pmatrix} 1 & 1 & 0 \ 1 & -1 & 4 \ 1 & 1 & 10 \end{pmatrix} \]\[ \det(A_z) = 1(-10 - 4) - 1(1 - 40) + 0 = -14 + 39 = 25 \]

Step 4: Solve for Variables Using Cramer's Rule

Now, use Cramer's rule to solve for each variable:\[ x = \frac{\det(A_x)}{\det(A)} = \frac{-38}{-4} = 9.5 \]\[ y = \frac{\det(A_y)}{\det(A)} = \frac{-12}{-4} = 3 \]\[ z = \frac{\det(A_z)}{\det(A)} = \frac{25}{-4} = -6.25 \]

Step 5: Conclusion

The solution to the system of equations using Cramer's Rule is \( x = 9.5 \), \( y = 3 \), and \( z = -6.25 \).

Key Concepts

Understanding the System of Equations

In mathematics, a **system of equations** is a set of two or more equations with the same variables. Solving such a system means finding value pairs (in 2D) or triples (in 3D) that satisfy all equations simultaneously. In our example, we are solving for three variables: \(x\), \(y\), and \(z\). Consider the given equations:

• \(x + y - z = 0\)
• \(x - y + z = 4\)
• \(x + y + z = 10\)

These equations can be visualized as three surfaces in a three-dimensional space, and finding the solution means finding their intersection point. This intersection, if it exists, corresponds to the values of \(x\), \(y\), and \(z\) that solve the system. Systems can be classified as consistent or inconsistent, and can have one solution, many solutions, or no solution at all. The method we are using here, Cramer's Rule, is particularly useful for finding a unique solution when the system has exactly as many equations as unknowns.

Exploring the Matrix Determinant

The **matrix determinant** is a special number calculated from a square matrix. It provides significant information about the matrix, especially concerning solvability of linear equations. A determinant can tell us whether a matrix is invertible, or if a system of linear equations can be uniquely solved.
For a matrix \(A\), its determinant is denoted as \(\det(A)\). In the context of the given system of equations, the matrix of coefficients is:
\[ A = \begin{pmatrix} 1 & 1 & -1 \ 1 & -1 & 1 \ 1 & 1 & 1 \end{pmatrix} \]The determinant is calculated using specific arithmetic operations on the elements of the matrix, employing techniques like cofactor expansion (expansion by minors) for larger matrices.
For a 3x3 matrix, like in our example, the calculation is explicit and involves the elements of the matrix:

• Expand along one row or column using minors.
• Each minor is formed by deleting the row and column of a chosen element.

In our solution, the determinant \(\det(A) = -4\). A non-zero determinant indicates that the matrix is invertible, meaning there is a unique solution for the system of equations.

Steps of Cramer's Rule

**Cramer's Rule** is an algebraic method for solving a system of linear equations with an equal number of equations and unknowns. Let's look at how to apply it:

• Step 1: Write the system in matrix form, \(A \mathbf{x} = \mathbf{b}\), where \(A\) is the matrix of coefficients, \(\mathbf{x}\) is the column of variables, and \(\mathbf{b}\) is the column of constants.
• Step 2: Calculate the determinant of the coefficient matrix \(A\), ensuring \(\det(A) eq 0\) for the rule to apply.
• Step 3: For each variable, replace their corresponding column in matrix \(A\) with \(\mathbf{b}\), forming matrices \(A_x\), \(A_y\), and \(A_z\).
• Step 4: Calculate the determinants of these new matrices: \(\det(A_x)\), \(\det(A_y)\), \(\det(A_z)\).
• Step 5: Find the variables using \(x = \frac{\det(A_x)}{\det(A)}\), \(y = \frac{\det(A_y)}{\det(A)}\), \(z = \frac{\det(A_z)}{\det(A)}\).

This structured approach delves into matrix algebra and makes solving systems systematic and straightforward once the determinant is known as non-zero.

Decoding Matrix Algebra

**Matrix Algebra** involves a set of rules and operations applied to matrices that helps in dealing with linear equations, transformations, and more complex mathematical concepts. In the context of Cramer's Rule and our exercise, matrix algebra aids in simplifying the representation and solution of systems of equations.

• **Matrix Formulation:** Writing systems of equations in matrix form \(A \mathbf{x} = \mathbf{b}\) not only simplifies notation but also sets the stage for computer algorithms.
• **Matrix Operations:** Include addition, subtraction, multiplication, and determinant calculation, which help in various transformations and solving techniques.
• **Determinant:** A crucial concept; as we’ve seen, the determinant reflects the system's solvability and is central to Cramer's Rule.

Practicing matrix algebra strengthens one’s mathematical foundation, especially in fields that require computational solutions to large systems, such as engineering and economics.