Question

Solve each system of equations using Cramer's Rule if is applicable. If Cramer's Rule is not applicable, write, "Not applicable" \(\left\\{\begin{array}{l}4 x-6 y=-42 \\ 7 x+4 y=-1\end{array}\right.\)

Short answer

x = -3, y = 5

Step-by-step solution

Write the system of equations in matrix form

The given system of equations is: (1) \(4x - 6y = -42\) (2) \(7x + 4y = -1\) We can write this system in the form AX = B, where A = \begin{bmatrix} 4 & -6 \ 7 & 4 \end{bmatrix}, X = \begin{bmatrix} x \ y \end{bmatrix}, B = \begin{bmatrix} -42 \ -1 \end{bmatrix}

Calculate the determinant of matrix A

The determinant of matrix A, denoted as Det(A), is calculated as follows: Det(A) = \(4 \cdot 4 - (-6) \cdot 7\), or written out Det(A) = \(16 + 42\) = 58

Determine if Cramer’s Rule is applicable

Since Det(A) \(\e\) 0, Cramer's Rule is applicable.

Calculate \(x\) using Cramer's Rule

To find \(x\), we replace the first column of A with B and then calculate the determinant: A_x = \begin{bmatrix} -42 & -6 \ -1 & 4 \end{bmatrix} Det(A_x) = \left(-42\cdot4 - (-6)\cdot(-1)\right) = \left(-168 - 6\right) = -174 Thus, \(x = \frac{Det(A_x)}{Det(A)} = \frac{-174}{58} = -3\).

Calculate \(y\) using Cramer's Rule

To find \(y\), we replace the second column of A with B and then calculate the determinant: A_y = \begin{bmatrix} 4 & -42 \ 7 & -1 \end{bmatrix} Det(A_y) = \left(4\cdot(-1) - (-42)\cdot7\right)= \left(-4 + 294\right) = 290 Thus, \(y = \frac{Det(A_y)}{Det(A)} = \frac{290}{58} = 5\).

Key concepts

System of Equations

A system of equations is a collection of two or more equations with the same set of variables. In this case, we have two equations with variables x and y:
\(4x - 6y = -42 \) and \(7x + 4y = -1\)
Each equation represents a line, and the solution to the system is the point(s) where these lines intersect. For linear systems, there are three possibilities:

• One solution (lines intersect at a single point).
• No solution (lines are parallel).
• Infinite solutions (lines overlap).

Here, we are looking for the unique solution (if it exists) using Cramer's Rule.

Determinant

The determinant of a matrix is a special number that can be calculated from its elements. It is useful for checking whether a matrix is invertible and for solving systems of linear equations. For a 2x2 matrix \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant is calculated as: \[ \text{Det}(A) = ad - bc \]
In this exercise, we have matrix
\(A = \begin{bmatrix} 4 & -6 \ 7 & 4 \end{bmatrix}\), and its determinant is:
\[ \text{Det}(A) = (4)(4) - (-6)(7) = 16 + 42 = 58 \] Since the determinant is non-zero, Cramer's Rule can be used.

Matrix Algebra

Matrix algebra involves various operations with matrices, such as addition, subtraction, and multiplication. In solving systems of equations, we often convert the system into matrix form.
Given the system: \(4x - 6y = -42 \) and \(7x + 4y = -1\), we can write it in matrix form as \Ax = B\ where:
\A = \begin{bmatrix} 4 & -6 \ 7 & 4 \end{bmatrix}, \ X = \begin{bmatrix} x \ y \end{bmatrix}, \ B = \begin{bmatrix} -42 \ -1 \end{bmatrix}\
This conversion helps in applying methods like Cramer's Rule to find the solution.

Solving Linear Systems

Solving linear systems involves finding the values of the variables that satisfy all equations simultaneously. One common method is to use Cramer's Rule, which uses determinants to solve for each variable individually.
With Cramer's Rule, we follow these steps:

• Calculate the determinant of matrix A.
• Check if the determinant is non-zero; if so, proceed.
• Replace columns of A with B to form matrices \A_x\ (replace first column) and \A_y\ (replace second column).
• Calculate determinants of \A_x\ and \A_y\.
• Divide these determinants by \Det(A)\ to find the variables.

For x:
\[ x = \frac{ \text{Det}(A_x) }{ \text{Det}(A) } = \frac{ -174 }{ 58 } = -3 \]
For y:
\[ y = \frac{ \text{Det}(A_y) }{ \text{Det}(A) } = \frac{ 290 }{ 58 } = 5 \] So, the solution to the system is x = -3 and y = 5.