Problem 34 - College Algebra

Question

Use Cramer's rule to solve each system. $$ \begin{array}{r}x-y+2 z=3 \\\2 x+3 y+z=9 \\\\-x-y+3 z=11\end{array} $$

Step-by-Step Solution

Answer

The solutions for the system of equations are: x = 2, y = 2, z = 4.

Step 1: Setup the main determinant from the system (D)

Create a 3x3 matrix using the coefficients of the variables of each equation: \[D= \begin{bmatrix} 1 & -1 & 2 \ 2 & 3 & 1 \ -1 & -1 & 3 \end{bmatrix}\]

Step 2: Calculate determinant D

The determinant of a 3x3 matrix can be calculated by subtracting the sum of the products of the diagonals going from right to left from the sum of the products of the diagonals going from left to right. Therefore, the main determinant D can be calculated as: \[D = (1.3.3) + (-1.1.-1) + (2.2.-1) - {(-1.3.-1) + (-1.2.3) + (2.1.1)} = 19\]

Step 3: Construct determinant Dx replacing x coefficients with the constants from the right side

Replace the first column of the matrix D with the constants from the right side of the equations: \[D_x= \begin{bmatrix} 3 & -1 & 2 \ 9 & 3 & 1 \ 11 & -1 & 3 \end{bmatrix}\]

Step 4: Calculate determinant Dx

Compute the determinant Dx using the same method as in Step 2. The determinant Dx can be calculated as: \[Dx = (3.3.3) + (-1.9.-1) + (2.11.-1) - {(-1.3.11) + (-1.9.3) + (2.3.1)} = 38\]

Step 5: Construct determinant Dy replacing y coefficients with constants from the right side

Replace the second column of the matrix D with the constants from the right side of the equations: \[D_y= \begin{bmatrix} 1 & 3 & 2 \ 2 & 9 & 1 \ -1 & 11 & 3 \end{bmatrix}\]

Step 6: Calculate determinant Dy

Compute the determinant Dy using the same method as in Step 2. The determinant Dy can be calculated as: \[Dy = (1.9.3) + (3.1.-1) + (2.2.11) - {(3.9.-1) + (3.2.3) + (2.1.1)} = 38\]

Step 7: Construct determinant Dz replacing z coefficients with constants from the right side

Replace the third column of the matrix D with the constants from the right side of the equations: \[D_z= \begin{bmatrix} 1 & -1 & 3 \ 2 & 3 & 9 \ -1 & -1 & 11 \end{bmatrix}\]

Step 8: Calculate determinant Dz

Compute the determinant Dz using the same method as in Step 2. The determinant Dz can be calculated as: \[Dz = (1.3.11) + (-1.9.-1) + (3.2.-1) - {(-1.3.-1) + (-1.9.11) + (3.2.3)} = 76\]

Step 9: Solve for the Variables Using Cramer’s Rule

Using Cramer's rule, each variable for the system of equations is solved by dividing the determinant created by replacing the coefficients of that variable in the main determinant by the main determinant. Here the solutions would be: \[x=\frac{Dx}{D}, y=\frac{Dy}{D}, z=\frac{Dz}{D}\]

Key Concepts

Determinants

Determinants are a vital tool in solving systems of linear equations, especially when Cramer's Rule is employed. A determinant is a unique number assigned to a square matrix. It serves as a scalar value which can tell us many things about the matrix, including whether the matrix is invertible, which is crucial when attempting to find unique solutions to a system.

In the context of the exercise, the determinant for the main matrix (D) comes from the coefficients of the variables. Calculating it is slightly intricate: involve both the addition and subtraction of products of diagonal elements. To obtain D we traverse both diagonally from top-left to bottom-right and top-right to bottom-left across the matrix, multiplying as we go. The difference between these sums gives us the determinant's value.

For a 3x3 matrix like in our exercise, it would look like this: \[ D = a(ei − fh) − b(di − fg) + c(dh − eg) \] where \( a, b, c, \) etc., are the elements of the matrix. By finding the determinant for the main matrix and its variants (Dx, Dy, Dz), we are capable of solving for x, y, and z in the equation system.

System of Linear Equations

A system of linear equations consists of two or more equations that share a set of variables and are solved simultaneously. In our given exercise, there are three such equations, all linked by the same set of variables: x, y, and z. The solutions to these equations are the values of x, y, and z that satisfy all equations at once, resulting in what is known as the 'solution set'.

In many cases, these systems can be represented in matrix form, making them easier to manipulate and solve, especially when using techniques such as Cramer's Rule.

The Importance of Non-Singular Matrices

One critical aspect when solving linear systems using determinants is ensuring that the main matrix is non-singular, meaning its determinant is not zero. A zero determinant indicates that no unique solution exists for the system, which might translate into either having infinitely many solutions (the system is dependent) or no solution at all (the system is inconsistent). In our exercise, as the determinant D is 19 (not zero), we can be assured of a unique solution.

Algebraic Matrices

Matrices are a fundamental part of algebra that enable us to handle systems of equations efficiently. An algebraic matrix is essentially a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. For linear systems, matrices can encapsulate all the coefficients conveniently, which simplifies many operations.

Matrices allow us to perform calculations that would otherwise be cumbersome. They are used, for example, to rotate or scale objects in space in computer graphics, to represent and solve linear equations in algebra, or even to model complex systems in economics.

Using Matrices to Solve the Given System

In the context of Cramer's Rule, we use matrices to express the system and then manipulate them to find individual variables. As shown in the exercise, we create a main matrix with coefficients and then alternating matrices (Dx, Dy, Dz) by replacing columns with the equations’ constant terms. By evaluating the determinants of these matrices, the solution to the system becomes clear after applying the rule: the variable equals its corresponding determinant (Dx, Dy, or Dz) divided by the main determinant (D).