Problem 3 - Algebra for College Students

Question

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. \(\left(\begin{array}{l}5 x+2 y=5 \\ 3 x-4 y=29\end{array}\right)\)

Step-by-Step Solution

Answer

The solution is \(x = 3\) and \(y = -5\).

Step 1: Write the system of equations in matrix form

The given system of linear equations is: \[\begin{align*} 5x + 2y &= 5 \ 3x - 4y &= 29 \end{align*}\]We can represent these equations in matrix form as \(AX = B\), where:\[A = \begin{pmatrix} 5 & 2 \ 3 & -4 \end{pmatrix}, \quad X = \begin{pmatrix} x \ y \end{pmatrix}, \quad B = \begin{pmatrix} 5 \ 29 \end{pmatrix}\]

Step 2: Calculate the determinant of matrix A

The determinant of matrix \(A\) is calculated as follows:\[ \text{det}(A) = \begin{vmatrix} 5 & 2 \ 3 & -4 \end{vmatrix} = (5)(-4) - (2)(3) = -20 - 6 = -26 \]

Step 3: Calculate the determinant for x-coordinates

To find the determinant for the \(x\)-values (\(D_x\)), replace the first column of \(A\) with the matrix \(B\):\[D_x = \begin{vmatrix} 5 & 2 \ 29 & -4 \end{vmatrix} = (5)(-4) - (2)(29) = -20 - 58 = -78\]

Step 4: Calculate the determinant for y-coordinates

To find the determinant for the \(y\)-values (\(D_y\)), replace the second column of \(A\) with the matrix \(B\):\[D_y = \begin{vmatrix} 5 & 5 \ 3 & 29 \end{vmatrix} = (5)(29) - (5)(3) = 145 - 15 = 130\]

Step 5: Solve for x and y using Cramer's rule

Using Cramer's rule, the solutions for \(x\) and \(y\) are given by:\[x = \frac{D_x}{\text{det}(A)} = \frac{-78}{-26} = 3\]\[y = \frac{D_y}{\text{det}(A)} = \frac{130}{-26} = -5\]

Key Concepts

System of Linear Equations

A system of linear equations is a set of equations where each equation is linear, which means it graphs as a straight line. These are expressed in the form of variables and constants. The primary aim of solving such systems is to find the values of the variables that satisfy all given equations simultaneously.

For instance, in our problem, the system is:

• \(5x + 2y = 5\)
• \(3x - 4y = 29\)

This means we're looking for values of \(x\) and \(y\) that work for both equations at the same time. A common scenario involves analyzing how the lines intersect:

• If there is one intersection point, there's one solution.
• If they're parallel with no intersection, there are no solutions.
• If they're the same line, there are infinitely many solutions.

Recognizing these conditions is essential because they dictate how you approach finding a solution.

Determinants in Algebra

Determinants play a crucial role in algebra, particularly in the analysis of matrices when dealing with systems of linear equations. The determinant of a square matrix is a scalar attribute essential in solving systems using methods like Cramer's Rule.

For a 2x2 matrix: \(\begin{pmatrix} a & b \c & d \ \end{pmatrix}\), the determinant is calculated as:

• \(ad - bc\)

In our example, the determinant of our coefficient matrix \(A\) is \(-26\), which we found by calculating: \((5)(-4) - (2)(3) = -20 - 6\).

The determinant helps determine whether a unique solution exists:

• If the determinant is zero, it indicates dependency or no unique solution.
• If non-zero, a unique solution is possible.

This property of the determinant informs us that our system can be solved when applying Cramer's Rule.

Matrix Representation

Representing a system in matrix form is a powerful technique. This approach condenses the system of equations into a form that is easier to analyze and solve, especially with computer algorithms or applying algebraic rules like Cramer's Rule.

The typical representation is:

• \(AX = B\)

where \(A\) is the coefficient matrix, \(X\) is the variable matrix, and \(B\) is the constant matrix.

In our system, we have:

• \(A = \begin{pmatrix} 5 & 2 \ 3 & -4 \end{pmatrix}\)
• \(X = \begin{pmatrix} x \ y \end{pmatrix}\)
• \(B = \begin{pmatrix} 5 \ 29 \end{pmatrix}\)

This representation simplifies visualization of interactions between variables and constants, making application of rules like determinants straightforward.

Solving Equations

There are several methods to solve systems of linear equations, with Cramer's Rule being a popular one when dealing with small systems where determinant calculation is feasible. For larger systems, other methods like Gaussian elimination or matrix inverses might be more efficient.

Using Cramer's Rule involves several steps:

• Calculate the determinant of the coefficient matrix.
• Calculate determinants for each variable by replacing their respective columns with the constant matrix.
• Use the formula \(x = \frac{D_x}{\text{det}(A)}\) and \(y = \frac{D_y}{\text{det}(A)}\).

In our case:

• The solution for \(x\) came out to be 3.
• Similarly, \(y\) came out to be -5.

Cramer's Rule is efficient and straightforward when applicable, giving you direct solutions by leveraging determinants. This makes understanding its use critical for algebra practitioners.