Problem 10 - Algebra 2

Question

Use Cramer’s Rule to solve each system of equations. \(3 a+5 b=33\) \(5 a+7 b=51\)

Step-by-Step Solution

Answer

The solution is \( a = 6 \) and \( b = 3 \).

Step 1: Write the system in matrix form

The given system of equations is: \( 3a + 5b = 33 \) and \( 5a + 7b = 51 \). Write this system in matrix form, \( AX = B \), where \(A = \begin{bmatrix} 3 & 5 \ 5 & 7 \end{bmatrix}\),\(X = \begin{bmatrix} a \ b \end{bmatrix}\), and\(B = \begin{bmatrix} 33 \ 51 \end{bmatrix}\).

Step 2: Calculate the determinant of matrix A

To use Cramer's Rule, we first find the determinant of matrix \( A \), denoted as \( \text{det}(A) \). Calculate it using the formula \( \text{det}(A) = ad - bc \). Here, \( a = 3 \), \( b = 5 \), \( c = 5 \), and \( d = 7 \):\[\text{det}(A) = 3 \times 7 - 5 \times 5 = 21 - 25 = -4\]

Step 3: Calculate the determinant of A with the first column replaced (Determinant for a)

Replace the first column of \( A \) with \( B \), forming matrix \( A_a \):\[A_a = \begin{bmatrix} 33 & 5 \ 51 & 7 \end{bmatrix}\]Calculate \( \text{det}(A_a) \) using the same formula:\[\text{det}(A_a) = 33 \times 7 - 5 \times 51 = 231 - 255 = -24\]

Step 4: Calculate the determinant of A with the second column replaced (Determinant for b)

Replace the second column of \( A \) with \( B \), forming matrix \( A_b \):\[A_b = \begin{bmatrix} 3 & 33 \ 5 & 51 \end{bmatrix}\]Calculate \( \text{det}(A_b) \):\[\text{det}(A_b) = 3 \times 51 - 33 \times 5 = 153 - 165 = -12\]

Step 5: Solve for a and b using Cramer's Rule

According to Cramer's Rule, the solutions for \( a \) and \( b \) are:\[a = \frac{\text{det}(A_a)}{\text{det}(A)} = \frac{-24}{-4} = 6\]\[b = \frac{\text{det}(A_b)}{\text{det}(A)} = \frac{-12}{-4} = 3\]

Step 6: Conclusion: Check your solution

Plug the values \( a = 6 \) and \( b = 3 \) back into the original equations to verify. The first equation \( 3a + 5b = 33 \) becomes \( 3\times6 + 5\times3 = 18 + 15 = 33 \), and the second equation \( 5a + 7b = 51 \) becomes \( 5\times6 + 7\times3 = 30 + 21 = 51 \). Both equations hold true, confirming the solution's correctness.

Key Concepts

Determinant of a Matrix

The determinant of a matrix is a special number associated with a square matrix. It gives us important information about the matrix, such as whether it is invertible. For a 2x2 matrix \[ A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \]it is calculated using the formula: \[ \text{det}(A) = ad - bc \]A nonzero determinant implies that the matrix has an inverse and thus, solutions exist for the related system of equations. In Cramer's Rule, this determinant helps us compute the solution for each variable by using modified matrices.

System of Equations

A system of equations consists of multiple equations that are solved together. Each solution must satisfy all equations in the system. For example, the system:

• \(3a + 5b = 33\)
• \(5a + 7b = 51\)

is a pair of linear equations. The solution to this system is the pair of values \(a\) and \(b\) that makes both equations true. Systems of equations are fundamental in various mathematical applications, from simple problems to complex modeling in mathematics and engineering.

Matrix Algebra

Matrix algebra involves computations using matrices, which are arrays of numbers arranged in rows and columns. It is a cornerstone of linear algebra. In solving systems of equations, we represent the equations in matrix form: \[ AX = B \]Here,

• \(A\) is the coefficient matrix.
• \(X\) is the column matrix of variables.
• \(B\) is the constant column matrix.

The operations on matrices like addition, multiplication, and finding determinants are essential in manipulating and solving equations. Matrix algebra simplifies complex systems into manageable computations.

Linear Equations

Linear equations are equations of the first degree, meaning they have variables raised to the power of one. They graph as straight lines and can be written in the form \(ax + by = c\). In the context of a system of linear equations, like in our example, the goal is to find values for the variables that satisfy every equation in the system. Linear equations are among the most basic types of equations yet form the foundation for more advanced topics in algebra and calculus due to their straightforward solving techniques and representation of real-world scenarios like speed, distance, and economics.