Problem 68 - A Graphical Approach to Precalculus with Limits

Question

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to complete the solution. $$\begin{array}{l}x+y+z=4 \\\2 x-y+3 z=4 \\\4 x+2 y-z=-15\end{array}$$

Step-by-Step Solution

Answer

The solution is \(x = 3\), \(y = 3\), \(z = 5\).

Step 1: Write the system of equations in matrix form

The given system of equations is: \[\begin{aligned} x + y + z &= 4, \ 2x - y + 3z &= 4, \ 4x + 2y - z &= -15. \end{aligned}\] Set this system up in the form \(AX = B\), where \(A\) is the coefficient matrix, \(X = \begin{bmatrix} x \ y \ z \end{bmatrix}\) is the variable matrix, and \(B = \begin{bmatrix} 4 \ 4 \ -15 \end{bmatrix}\) is the constant matrix.

Step 2: Determine the coefficient matrix A

The coefficient matrix \(A\) for the system of equations is \[ A = \begin{bmatrix} 1 & 1 & 1 \ 2 & -1 & 3 \ 4 & 2 & -1 \end{bmatrix} . \]

Step 3: Compute the determinant of A (D)

To find the determinant \(D\) of matrix \(A\), use the formula for a 3x3 matrix: \[ D = 1\left((-1)(-1) - 3(2)\right) - 1\left(2(-1) - 3(4)\right) + 1\left(2\times2 - (-1)\times4\right). \] Simplifying gives \[ D = 1(1 - 6) - 1(-2 - 12) + 1(4 + 4) = -5 + 14 + 8 = 17. \] Thus, \(D = 17\).

Step 4: Calculate the determinant for each variable's matrix

To find \(x\), replace the first column of \(A\) with \(B\), and compute the determinant: \[ D_x = \begin{vmatrix} 4 & 1 & 1 \ 4 & -1 & 3 \ -15 & 2 & -1 \end{vmatrix} = 4((-1)(-1) - 3 \times 2) - 1(4 \times (-1) - 3 \times (-15)) + 1(4 \times 2 - (-1) \times 15). \] Simplifying gives \( D_x = 4(1 - 6) - 1(-4 + 45) + 1(8 + 15) \), which equals \(-20 + 49 + 23 = 52.\)

Step 5: Apply Cramer's Rule to find x

Using Cramer's Rule \(x = \frac{D_x}{D}\), we find \(x = \frac{52}{17} = 3.\)

Step 6: Calculate Dy and Dz to find y and z

For \(y\), replace the second column of \(A\) with \(B\): \[ D_y = \begin{vmatrix} 1 & 4 & 1 \ 2 & 4 & 3 \ 4 & -15 & -1 \end{vmatrix} = 1(4 \times (-1) - 3 \times -15) - 4(2 \times (-1) - 3 \times 4) + 1(2 \times -15 - 4 \times 4). \] Simplifying gives \( D_y = 1(-4 + 45) - 4(-2 - 12) + 1(-30 -16) = 41 + 56 - 46 = 51. \)Similarly, find \(D_z\) replacing the third column by \(B\): \[ D_z = \begin{vmatrix} 1 & 1 & 4 \ 2 & -1 & 4 \ 4 & 2 & -15 \end{vmatrix} = 1((-1 \times -15) - (4 \times 2)) - 1(2 \times -15 - 4 \times 4) + 4(2 \times 2 - (-1) \times 4). \] This evaluates to \( 1(15 - 8) - 1(-30 -16) + 4(4 + 4) = 7 + 46 + 32 = 85. \)

Step 7: Solve for y and z using Cramer's Rule

Using Cramer's Rule, solve for \(y\) and \(z\):- \(y = \frac{D_y}{D} = \frac{51}{17} = 3.\)- \(z = \frac{D_z}{D} = \frac{85}{17} = 5.\)

Key Concepts

Determinant

The determinant is a fundamental concept in matrix algebra. It is a value that can be calculated from a square matrix and provides crucial insights into the properties of the matrix. For any square matrix, the determinant is a scalar value that can tell us if the matrix has an inverse, and it is vital in solving systems of equations using methods such as Cramer's Rule.

To calculate the determinant of a 3x3 matrix, like in the given exercise, you can apply the formula which involves combinations of the elements in the matrix. For example, if the matrix is:

• \[A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix}\]

The determinant, \(D\), can be calculated by:

• \[D = a(ei - fh) - b(di - fg) + c(dh - eg)\]

This results in a single number, which in the context of the exercise is used to determine whether the system of equations has a unique solution (when \(D eq 0\)). If the determinant is zero, the system may have either no solutions or an infinite number of solutions.

Systems of Equations

A system of equations is simply a collection of two or more equations with shared variables. In this exercise, the system is expressed through the equations:

• \(x + y + z = 4\)
• \(2x - y + 3z = 4\)
• \(4x + 2y - z = -15\)

The goal is to find the values of the variables \(x, y,\) and \(z\) that satisfy all the equations simultaneously.

To solve such systems, you can employ methods like substitution, elimination, or Cramer's Rule—a technique relying on the determinants of matrices, useful when you have as many equations as unknowns. In this approach, you first need to convert the system into matrix form, so the given system is represented by matrices, where a unique solution is found if the determinant of the coefficient matrix is non-zero. This particular method is efficient for smaller systems with three variables as shown, offering precise solutions.

Matrix Algebra

Matrix algebra is a branch of mathematics dealing with matrices and the operations that can be performed on them. It's central to solving systems of equations and plays a key role in concepts like Cramer's Rule.

A matrix is essentially a rectangular array of numbers arranged in rows and columns. The exercise demonstrates matrix algebra by converting the system of linear equations into matrix form:

• Matrix \(A\) as the coefficient matrix
• Matrix \(X\) for the variables
• Matrix \(B\) for the constants

Using these matrices, matrix equations are written in the form \(AX = B\), and you solve them using operations on matrices, such as finding determinants or performing matrix multiplication.

Learning matrix algebra is crucial as it provides the tools to work with multiple equations and unknowns, turning complex algebraic problems into organized solvable tasks. It forms the backbone of many algebraic computations essential in fields like engineering, physics, computer graphics, and more.