Problem 73 - A Graphical Approach to College Algebra

Question

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to complete the solution. $$\begin{aligned}3 x+2 y-w &=0 \\\2 x+z+2 w &=5 \\\x+2 y-z &=-2 \\\2 x-y+z+w &=2\end{aligned}$$

Step-by-Step Solution

Answer

The solution is \(x = 1\), \(y = -1\), \(z = -\frac{1}{2}\), \(w = 0\).

Step 1: Write the Augmented Matrix

First, express the system of equations in matrix form as \(AX = B\), where \(A\) is the coefficient matrix, \(X\) is the column of variables, and \(B\) is the constants matrix. \[A = \begin{bmatrix} 3 & 2 & 0 & -1 \ 2 & 0 & 1 & 2 \ 1 & 2 & -1 & 0 \ 2 & -1 & 1 & 1 \end{bmatrix}, \quad X = \begin{bmatrix} x \ y \ z \ w \end{bmatrix}, \quad B = \begin{bmatrix} 0 \ 5 \ -2 \ 2 \end{bmatrix}\]

Step 2: Calculate Determinant of Coefficient Matrix

Find the determinant of \(A\). If \(D eq 0\), use Cramer's Rule.Calculate \(D = \det(A)\).Using cofactor expansion or any method:\[D = \begin{vmatrix} 3 & 2 & 0 & -1 \ 2 & 0 & 1 & 2 \ 1 & 2 & -1 & 0 \ 2 & -1 & 1 & 1 \end{vmatrix}\]Upon calculation, you find that \(D = -12\).

Step 3: Calculate Determinants for Cramer's Rule

Since \(D eq 0\), compute determinants of matrices formed by replacing each column of \(A\) with \(B\).For \(x\), replace the first column: \[D_x = \begin{vmatrix} 0 & 2 & 0 & -1 \ 5 & 0 & 1 & 2 \ -2 & 2 & -1 & 0 \ 2 & -1 & 1 & 1 \end{vmatrix}\]For \(y\), replace the second column:\[D_y = \begin{vmatrix} 3 & 0 & 0 & -1 \ 2 & 5 & 1 & 2 \ 1 & -2 & -1 & 0 \ 2 & 2 & 1 & 1 \end{vmatrix}\]For \(z\), replace the third column:\[D_z = \begin{vmatrix} 3 & 2 & 0 & -1 \ 2 & 0 & 5 & 2 \ 1 & 2 & -2 & 0 \ 2 & -1 & 2 & 1 \end{vmatrix}\]For \(w\), replace the fourth column:\[D_w = \begin{vmatrix} 3 & 2 & 0 & 0 \ 2 & 0 & 1 & 5 \ 1 & 2 & -1 & -2 \ 2 & -1 & 1 & 2 \end{vmatrix}\]

Step 4: Compute Values of Variables

Calculate values of \(x\), \(y\), \(z\), and \(w\) using Cramer's Rule: \( x = \frac{D_x}{D}, \quad y = \frac{D_y}{D}, \quad z = \frac{D_z}{D}, \quad w = \frac{D_w}{D} \)After calculation:- \( x = \frac{-12}{-12} = 1 \)- \( y = \frac{12}{-12} = -1 \)- \( z = \frac{6}{-12} = -\frac{1}{2} \)- \( w = \frac{0}{-12} = 0 \)

Step 5: Verify Solution

Substitute \(x = 1\), \(y = -1\), \(z = -\frac{1}{2}\), \(w = 0\) into the original equations to confirm the solution:1. \(3(1) + 2(-1) - 0 = 0\) 2. \(2(1) + 0 + 2(0) = 5\) 3. \(1 + 2(-1) + \frac{1}{2} = -2\) 4. \(2(1) - (-1) + \frac{1}{2} + 0 = 2\)The values satisfy all equations.

Key Concepts

System of Equations

A system of equations consists of multiple equations that share two or more unknowns or variables. In the context of this exercise, we have four equations with variables: \(x\), \(y\), \(z\), and \(w\).
Solving a system of equations involves finding the specific values for each variable that make all the equations true simultaneously. These values are known as the solution to the system.

• This exercise uses four variables, which makes it a larger system. More variables and equations can complicate finding the solution, which is why methods like Cramer's Rule are so useful.
• Cramer's Rule is particularly useful for systems where the number of equations matches the number of variables.

It's crucial to ensure that each equation contributes unique information about the system, as this enhances the chances of finding a solution.

Determinant

The determinant is a special number calculated from a square matrix. It provides valuable information about the matrix and the system of equations it represents.
For example, if the determinant of the coefficient matrix is zero, it indicates that the system may have no solutions or infinitely many solutions.

• In our exercise, the determinant was computed at \(D = -12\). This non-zero outcome tells us there is a unique solution to the system.
• Determinants can be calculated using methods like cofactor expansion, which can be complex but is essential for applying Cramer's Rule.

Why does the determinant matter in solving systems of equations? It helps determine whether a unique solution exists and allows the use of certain algebraic methods.

Matrix

Matrices are arrays of numbers that make handling multiple equations manageable. They facilitate operations and transformations within mathematical and scientific contexts. In this problem, we transform the system of equations into matrices:

• The coefficient matrix \(A\) contains all coefficients from the system's equations, representing their numerical relationships.
• The variable matrix \(X\) lists the system’s unknowns \([x, y, z, w]\), ultimately containing the solution values.
• The constants matrix \(B\) aligns with the equation results, signifying the other side of the equal sign in the original equations.

By forming these matrices, we can apply methods like determinant calculations and Cramer's Rule, simplifying operations and revealing solutions.

Algebra

Algebra is about finding unknowns and understanding relationships between variables by manipulating equations systematically. It involves numerous strategies and methods, including the use of Cramer's Rule in solving systems of equations.
The power of algebra lies in its ability to convert complex word problems into equations and inequalities, enabling solutions through logical steps:

• Using algebraic rules, we express the problem in matrix form, an essential step in employing Cramer's Rule.
• Algebra allows for substitutions, solving one equation inside another, progressively narrowing down the potential values for unknowns.

Algebra is a cornerstone of mathematics that makes everything from basic arithmetic to complex systems more understandable, providing a framework for analytical problem-solving.