Problem 1 - Algebra for College Students

Question

For Problems \(1-32\), use Cramer's rule to find the solution set for each system. If the equations are dependent, simply indicate that there are infinitely many solutions. \(\left(\begin{array}{rl}2 x-y & =-2 \\ 3 x+2 y & =11\end{array}\right)\)

Step-by-Step Solution

Answer

The solution is \((x, y) = (1, 4)\).

Step 1: Identify the System of Equations

We start by identifying the system of equations given in the matrix form. The system is: 1. \(2x - y = -2\) 2. \(3x + 2y = 11\).

Step 2: Define the Matrices and Determinants

For Cramer's Rule, the coefficient matrix is \(A = \begin{pmatrix} 2 & -1 \ 3 & 2 \end{pmatrix}\). The determinants needed are: 1. \(\Delta = \begin{vmatrix} 2 & -1 \ 3 & 2 \end{vmatrix}\) 2. \(\Delta_x = \begin{vmatrix} -2 & -1 \ 11 & 2 \end{vmatrix}\) 3. \(\Delta_y = \begin{vmatrix} 2 & -2 \ 3 & 11 \end{vmatrix}\).

Step 3: Calculate the Determinant \(\Delta\)

The determinant \(\Delta\) is calculated as follows:\[\Delta = (2)(2) - (-1)(3) = 4 + 3 = 7\]

Step 4: Calculate the Determinant \(\Delta_x\)

The determinant \(\Delta_x\) is calculated as:\[\Delta_x = (-2)(2) - (-1)(11) = -4 + 11 = 7\]

Step 5: Calculate the Determinant \(\Delta_y\)

The determinant \(\Delta_y\) is calculated as:\[\Delta_y = (2)(11) - (-2)(3) = 22 + 6 = 28\]

Step 6: Apply Cramer's Rule to Find \(x\) and \(y\)

Cramer's Rule states that:\[ x = \frac{\Delta_x}{\Delta} = \frac{7}{7} = 1\]\[ y = \frac{\Delta_y}{\Delta} = \frac{28}{7} = 4\].

Step 7: Conclusion

The solution of the system of equations using Cramer's Rule is \((x, y) = (1, 4)\).

Key Concepts

System of Equations

A system of equations is a collection of two or more equations with a common set of variables. Solving these equations involves finding the values of the variables that satisfy each equation simultaneously.

In our exercise, we are presented with the system of equations:

• \(2x - y = -2\)
• \(3x + 2y = 11\)

These equations must be solved together to discover the values of \(x\) and \(y\). In systems like these, various methods such as substitution, elimination, or matrix approaches like Cramer's Rule can be employed.

Understanding how to identify and set up the system correctly is crucial in determining the solution. The goal is always to find the unique solution, determine if there are infinitely many solutions, or conclude the system has no solution at all.

Determinants

Determinants are scalar values that are computed from the elements of a square matrix. They provide important properties of matrices and are essential when applying Cramer's Rule for solving systems of equations.

In our example, the determinant \(\Delta\) of the coefficient matrix \(A\) is calculated as shown in the steps provided:\[\Delta = \begin{vmatrix} 2 & -1 \ 3 & 2 \end{vmatrix} = (2)(2) - (-1)(3) = 4 + 3 = 7\] A non-zero determinant, such as \(7\), indicates that the matrix is invertible and that a unique solution exists for the system of equations.

Similarly, we find specific determinants \(\Delta_x\) and \(\Delta_y\) obtained by replacing one column in the coefficient matrix with the constants from the equations. These substituted determinants are key to finding the specific values for each variable using Cramer's Rule.

Matrix Algebra

Matrix algebra is an essential tool when working with systems of equations. It employs matrices to represent and manipulate data efficiently. In the context of our exercise, it is crucial for applying Cramer's Rule, which uses matrices to derive solutions.

Consider the coefficient matrix \(A\), comprised of the coefficients of the variables:\[A = \begin{pmatrix} 2 & -1 \ 3 & 2 \end{pmatrix}\] This matrix represents the system of equations in a compact form. Manipulating matrices and using operations like determinants allows for more streamlined calculations and solutions to complex systems.

Utilizing matrix algebra not only helps to solve systems but also aids in understanding deeper linear transformations and the behavior of linear models.

Solution Set

The solution set of a system of equations is a collection of all ordered pairs that satisfy the whole system. In simple terms, solving a system of equations involves finding precise values for each variable.

In our example, this involves finding the values of \(x\) and \(y\). By applying Cramer's Rule using the determinants \(\Delta\), \(\Delta_x\), and \(\Delta_y\), we arrive at our unique solution:

• \(x = \frac{\Delta_x}{\Delta} = \frac{7}{7} = 1\)
• \(y = \frac{\Delta_y}{\Delta} = \frac{28}{7} = 4\)

This gives us a solution set \((x, y) = (1, 4)\).

In Cramer's Rule, the presence of a unique solution such as this suggests that the lines represented by each equation intersect at exactly one point in the plane, highlighting the intersections of solutions in two-dimensional space. If the determinant \(\Delta\) were zero, we would need to reconsider, as that scenario indicates either a dependent system with infinite solutions or no solution altogether.