Use the inverse matrix method to solve the following: 2p - 4q = -14 2q - 3p = 13 Describe each of the following types of sets: disjoint

Steps to Solve

1. Rewrite the equations in standard form

Rearrange the second equation to match the order of variables in the first equation. The original equations are: $2p - 4q = -14$ $2q - 3p = 13$ Rewriting the second equation: $-3p + 2q = 13$

2. Express the system of equations in matrix form

The system can be written as $AX = B$, where $A = \begin{bmatrix} 2 & -4 \ -3 & 2 \end{bmatrix}$, $X = \begin{bmatrix} p \ q \end{bmatrix}$, and $B = \begin{bmatrix} -14 \ 13 \end{bmatrix}$.

3. Calculate the determinant of matrix $A$

The determinant of a 2x2 matrix $\begin{bmatrix} a & b \ c & d \end{bmatrix}$ is $ad - bc$. So, $\det(A) = (2)(2) - (-4)(-3) = 4 - 12 = -8$.

4. Calculate the inverse of matrix A

The inverse of a 2x2 matrix $\begin{bmatrix} a & b \ c & d \end{bmatrix}$ is $\frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}$. Therefore, $A^{-1} = \frac{1}{-8} \begin{bmatrix} 2 & 4 \ 3 & 2 \end{bmatrix} = \begin{bmatrix} -1/4 & -1/2 \ -3/8 & -1/4 \end{bmatrix}$.

5. Solve for $X$ using $X = A^{-1}B$

Multiply the inverse of A by B to find the values of p and q. $X = \begin{bmatrix} -1/4 & -1/2 \ -3/8 & -1/4 \end{bmatrix} \begin{bmatrix} -14 \ 13 \end{bmatrix}$ $X = \begin{bmatrix} (-1/4)(-14) + (-1/2)(13) \ (-3/8)(-14) + (-1/4)(13) \end{bmatrix} = \begin{bmatrix} 14/4 - 13/2 \ 42/8 - 13/4 \end{bmatrix} = \begin{bmatrix} 7/2 - 13/2 \ 21/4 - 13/4 \end{bmatrix} = \begin{bmatrix} -6/2 \ 8/4 \end{bmatrix} = \begin{bmatrix} -3 \ 2 \end{bmatrix}$ Therefore, $p = -3$ and $q = 2$.

6. Define disjoint sets

Disjoint sets are sets that have no elements in common. In other words, their intersection is the empty set.