Problem 5 - Algebra 2

Question

Use Cramer’s Rule to solve each system of equations. \(2 x-y+3 z=5\) \(3 x+2 y-5 z=4\) \(x-4 y+11 z=3\)

Step-by-Step Solution

Answer

The system has no unique solution; \(|A| = 0\). Cramer's Rule cannot be used.

Step 1: Write the System in Matrix Form

First, express the system of equations in matrix form: \[A = \begin{bmatrix} 2 & -1 & 3 \ 3 & 2 & -5 \ 1 & -4 & 11 \end{bmatrix}, \quad X = \begin{bmatrix} x \ y \ z \end{bmatrix}, \quad B = \begin{bmatrix} 5 \ 4 \ 3 \end{bmatrix}\]The system can be expressed as \(AX = B\).

Step 2: Calculate Determinant of Matrix A

Calculate the determinant of matrix \(A\), denoted as \( |A| \):\[|A| = \begin{vmatrix} 2 & -1 & 3 \ 3 & 2 & -5 \ 1 & -4 & 11 \end{vmatrix}\]Use the formula for the determinant of a 3x3 matrix:\[|A| = 2(2\times11 - (-5)\times(-4)) - (-1)(3\times11 - (-5)\times1) + 3(3\times(-4) - 2\times1)\]Calculate each term:\[= 2(22 - 20) + 1(33 + 5) + 3(-12 - 2)\]\[= 2(2) + 1(38) + 3(-14)\]\[= 4 + 38 - 42\]\[= 0\]Since \(|A| = 0\), this indicates that the system does not have a unique solution.

Step 3: Conclusion Based on Determinant

Since the determinant \(|A| = 0\), Cramer's Rule cannot be applied. This generally indicates that the system either has no solution or has infinitely many solutions. We need another method to further explore the nature of the solutions if they exist.

Key Concepts

System of Equations

A system of equations involves multiple equations that share the same variables. Think of them as recipes needing the same ingredients.
In our problem, we're working with a system of 3 equations, each involving the variables \(x\), \(y\), and \(z\).
We want to find values for these variables that satisfy all the equations at once.

• The first equation is \(2x - y + 3z = 5\).
• The second equation is \(3x + 2y - 5z = 4\).
• The third equation is \(x - 4y + 11z = 3\).

We can solve systems of equations using various methods, and one popular technique is Cramer's Rule, which will be explained further.

Matrix Form

Expressing a system of equations in matrix form is a key step in simplifying the problem.
This involves organizing the coefficients of the variables into a matrix, known as matrix \(A\), while the variables themselves form a vector \(X\), and the constant terms form another matrix \(B\).

• For our exercise, matrix \(A\) is \(\begin{bmatrix}2 & -1 & 3 \3 & 2 & -5 \1 & -4 & 11 \end{bmatrix}\).
• The vector \(X\) is \(\begin{bmatrix} x \ y \ z \end{bmatrix}\).
• Matrix \(B\) is \(\begin{bmatrix}5 \4 \3 \end{bmatrix}\).

Putting them together, the system can be expressed as \(AX = B\).
This representation is a compact and systematic way of dealing with multiple equations.

Determinant

The determinant provides crucial information about a matrix. For a 3x3 matrix, it can indicate if a unique solution exists for a system of equations.
To find the determinant of matrix \(A\), apply the formula for a 3x3 determinant:\[|A| = a(ei - fh) - b(di - fg) + c(dh - eg)\]Where \(a, b, c, d, e, f, g, h, i\) are elements of matrix \(A\). For example, calculate the minor determinants and simplify:

• Our calculation shows \(|A| = 4 + 38 - 42 = 0\).

This result tells us something important: if the determinant is zero, then there isn’t a unique solution.

Unique Solution

In systems of equations, the uniqueness of a solution is determined by the determinant. A non-zero determinant indicates a unique solution.
However, in our example, the determinant \(|A|\) is zero.
This means the system does not have a unique solution. Instead, it might:

• Be consistent but have infinitely many solutions.
• Be inconsistent with no solution at all.

When dealing with such scenarios in systems of equations, other methods like Gaussian elimination might be necessary to determine the type of solutions the system has.