Problem 59 - Intermediate Algebra

Question

Use Cramer's rule to solve each system of equations. If a system is inconsistent or if the equations are dependent, so indicate. $$ \left\\{\begin{array}{l} 4 x-3 y=1 \\ 6 x-8 z=1 \\ 2 y-4 z=0 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \( x = \frac{1}{4} \), \( y = -\frac{2}{13} \), and \( z = -\frac{1}{13} \).

Step 1: Identify the Coefficient Matrix

The given system of equations can be written in terms of matrices. Identify the coefficient matrix \( A \) based on the variables \( x, y, \) and \( z \). The coefficient matrix is:\[A = \begin{bmatrix} 4 & -3 & 0 \ 6 & 0 & -8 \ 0 & 2 & -4 \end{bmatrix}\]

Step 2: Find the Determinant of the Coefficient Matrix

Calculate the determinant of matrix \( A \). Use the formula for a 3x3 matrix determinant:\[\det(A) = 4(0\cdot(-4)-(-8)\cdot2) - (-3)(6\cdot(-4)-0\cdot(-8)) + 0(6\cdot2-0\cdot0)\]This simplifies to:\[\det(A) = 32 + 72 + 0 = 104\]

Step 3: Check the Determinant

Since the determinant of the coefficient matrix \( A \) is not zero (\( \det(A) = 104 \)), the system of equations has a unique solution and is not dependent or inconsistent.

Step 4: Apply Cramer's Rule to Find \( x \)

Replace the first column of \( A \) with the constants from the equations to form matrix \( A_x \):\[A_x = \begin{bmatrix} 1 & -3 & 0 \ 1 & 0 & -8 \ 0 & 2 & -4 \end{bmatrix}\]Calculate \( \det(A_x) \):\[\det(A_x) = 1(0\cdot(-4)-(-8)\cdot2) - (-3)(1\cdot(-4)-0\cdot(-8))\]This gives \(\det(A_x) = 16 + 12 = 28\)Thus, \( x = \frac{\det(A_x)}{\det(A)} = \frac{28}{104} = \frac{1}{4} \).

Step 5: Apply Cramer's Rule to Find \( y \)

Replace the second column of \( A \) with the constants to form matrix \( A_y \):\[A_y = \begin{bmatrix} 4 & 1 & 0 \ 6 & 1 & -8 \ 0 & 0 & -4 \end{bmatrix}\]Calculate \( \det(A_y) \):\[\det(A_y) = 4(1\cdot(-4)) + 0 + 0 = -16\]Thus, \( y = \frac{\det(A_y)}{\det(A)} = \frac{-16}{104} = -\frac{4}{26} = -\frac{2}{13} \).

Step 6: Apply Cramer's Rule to Find \( z \)

Replace the third column of \( A \) with the constants to form matrix \( A_z \):\[A_z = \begin{bmatrix} 4 & -3 & 1 \ 6 & 0 & 1 \ 0 & 2 & 0 \end{bmatrix}\]Calculate \( \det(A_z) \):\[\det(A_z) = 4(0\cdot1 - 1\cdot2) - (-3)(6\cdot0 - 1\cdot0)\]This gives \(\det(A_z) = -8\)Thus, \( z = \frac{\det(A_z)}{\det(A)} = \frac{-8}{104} = -\frac{2}{26} = -\frac{1}{13} \).

Step 7: Conclusion

The unique solution of the system is \( x = \frac{1}{4} \), \( y = -\frac{2}{13} \), and \( z = -\frac{1}{13} \).

Key Concepts

System of Equations

A system of equations is a collection of two or more equations with the same set of variables. Solving a system of equations entails finding values for the variables that satisfy all the equations simultaneously. For instance, in the exercise you are dealing with a system of three equations with variables \( x \), \( y \), and \( z \):

• 4x - 3y = 1
• 6x - 8z = 1
• 2y - 4z = 0

In practical terms, solving such systems helps in finding points where all equations intersect. These systems can be solved using various algebraic methods, such as substitution, elimination, and matrix techniques like Cramer's Rule, which is used in this exercise.

Determinant

The determinant is a special number that can be calculated from a square matrix. It's a critical concept in linear algebra, as it provides important insights into the properties of the matrix.The determinant is particularly useful when working with systems of linear equations. It helps in determining if a unique solution exists.

• If the determinant of the coefficient matrix is zero, the system of equations doesn't have a unique solution — it might be dependent or inconsistent.
• If it's not zero, as in this exercise where \( \det(A) = 104 \), the system has a unique solution.

In the context of Cramer's Rule, the determinant is instrumental in solving for variable values by replacing columns of the matrix with constant terms and recalculating the determinant.

Unique Solution

A unique solution means there is only one set of values for the variables that satisfies all the equations in the system.For a linear system to have a unique solution, the determinant of its coefficient matrix must be non-zero. This condition ensures that the lines or planes represented by the equations intersect at only one point.In this example, the determinant \( \det(A) = 104 \) confirms that the system is neither parallel nor identical at any points other than the unique solution:

• \( x = \frac{1}{4} \)
• \( y = -\frac{2}{13} \)
• \( z = -\frac{1}{13} \)

Thus, the calculations reveal the specific intersection point in the three-dimensional space defined by the system.

Dependent Equations

Dependent equations in a system mean that one equation is a multiple of another, indicating that they describe the same line or plane in geometric terms.When equations are dependent, they do not provide new information — they merely reinforce existing relationships.
A determinant of zero signifies this dependency, leading either to infinite solutions (if they represent the same geometric line or plane) or no solutions (if they are parallel without intersections). However, because the determinant in this problem is \( 104 \), these equations are independent, guaranteeing a single solution rather than dependency. Understanding dependency is crucial because it tells you whether adding more equations adds more information or merely repeats what's already known.