Question

Solve each system of equations using Cramer's Rule if is applicable. If Cramer's Rule is not applicable, write, "Not applicable" \(\left\\{\begin{array}{r}x+3 y=5 \\ 2 x-3 y=-8\end{array}\right.\)

Short answer

The solution is \(x = -1\) and \(y = 2\).

Step-by-step solution

Identify the coefficient matrix

Write down the coefficients of the variables from the system of equations:o\(A = \begin{pmatrix} 1 & 3 \ 2 & -3 \end{pmatrix} \)

Calculate the determinant of the coefficient matrix

Calculate the determinant of matrix \(A\):det\(A = \begin{vmatrix} 1 & 3 \ 2 & -3 \end{vmatrix} = 1(-3) - 3(2) = -3 - 6 = -9\)

Determine if Cramer's Rule is applicable

Since the determinant of the coefficient matrix \(det(A) = -9 e 0\), Cramer's Rule is applicable.

Formulate the matrices for x and y

Create matrices for solving for \(x\) and \(y\):oFor \(x\): \(A_x = \begin{pmatrix} 5 & 3 \ -8 & -3 \end{pmatrix}\)oFor \(y\): \(A_y = \begin{pmatrix} 1 & 5 \ 2 & -8 \end{pmatrix}\)

Calculate determinant of \(A_x\)

Calculate the determinant of matrix \(A_x\):det\(A_x = \begin{vmatrix} 5 & 3 \ -8 & -3 \end{vmatrix} = 5(-3) - 3(-8) = -15 + 24 = 9\)

Calculate determinant of \(A_y\)

Calculate the determinant of matrix \(A_y\):det\(A_y = \begin{vmatrix} 1 & 5 \ 2 & -8 \end{vmatrix} = 1(-8) - 5(2) = -8 - 10 = -18\)

Solve for \(x\) and \(y\)

Using Cramer's Rule, solve for \(x\) and \(y\):o\(x = \frac{det(A_x)}{det(A)} = \frac{9}{-9} = -1\)o\(y = \frac{det(A_y)}{det(A)} = \frac{-18}{-9} = 2\)

Key concepts

System of Equations

A system of equations is a set of two or more equations that share the same variables. In this problem, we have two equations:
\(x + 3y = 5\) and \(2x - 3y = -8\).

Our goal is to find values for \(x\) and \(y\) that satisfy both equations at the same time. Systems of equations are very common in mathematics, physics, and engineering.
They help us understand relationships between different quantities. There are multiple methods for solving such systems, including substitution, elimination, and matrix methods like Cramer's Rule.

Determinant

The determinant is a special number that can be calculated from a square matrix. For a 2x2 matrix, the determinant is given by:
\(det(A) = a*d - b*c \)
where \(A\) is the matrix \(\begin{pmatrix} a & b \ c & d \end{pmatrix}\).

In our problem, the coefficient matrix \(A\) is \(\begin{pmatrix} 1 & 3 \ 2 & -3 \end{pmatrix}\).

So, we calculate the determinant as:
\(det(A) = 1*(-3) - 3*(2) = -3 - 6 = -9\)

The determinant tells us if a system of equations has a unique solution. If the determinant is zero, Cramer's Rule cannot be applied, and the system may have no solution or infinitely many solutions. In our case, the determinant is \(-9\), so we can use Cramer's Rule to find a unique solution.

Matrix Algebra

Matrix algebra involves mathematical operations with matrices. Here, we use matrices to represent and solve systems of equations. The coefficient matrix \(A\) is:
\(A = \begin{pmatrix} 1 & 3 \ 2 & -3 \end{pmatrix}\)

We also form matrices to solve for each variable. For example:
For \(x\):
\(A_x = \begin{pmatrix} 5 & 3 \ -8 & -3 \end{pmatrix}\)

For \(y\):
\(A_y = \begin{pmatrix} 1 & 5 \ 2 & -8 \end{pmatrix}\)

Matrix algebra is powerful because it allows us to handle complex systems efficiently. We use determinants of these matrices to find specific solutions for the variables.

Solving Equations

To solve the system using Cramer's Rule, we need the determinants of our matrices:
1. Calculate \(det(A_x)\):
\(det(A_x) = 5*(-3) - 3*(-8) = -15 + 24 = 9\)

2. Calculate \(det(A_y)\):
\(det(A_y) = 1*(-8) - 5*2 = -8 - 10 = -18\)

Finally, we solve for \(x\) and \(y\) using:

• \(x = \frac{det(A_x)}{det(A)} = \frac{9}{-9} = -1\)
• \(y = \frac{det(A_y)}{det(A)} = \frac{-18}{-9} = 2\)

So, the solution to the system of equations is \(x = -1\) and \(y = 2\). This step-by-step process helps us clearly see how matrix operations can solve complex systems efficiently.