Certainly! To solve the given system of linear equations using inverse matrices, let's follow the detailed step-by-step solution:
### Step 1: Write the system of equations in matrix form
The system of linear equations given is:
[tex]\[
\begin{cases}
5x - 2y = 30 \\
x + 6y = -26
\end{cases}
\][/tex]
We can represent this system in matrix form as [tex]\(Ax = B\)[/tex], where:
[tex]\[
A = \begin{pmatrix}
5 & -2 \\
1 & 6
\end{pmatrix}, \quad
x = \begin{pmatrix}
x \\
y
\end{pmatrix}, \quad
B = \begin{pmatrix}
30 \\
-26
\end{pmatrix}
\][/tex]
Here [tex]\(A\)[/tex] is the coefficient matrix, [tex]\(x\)[/tex] is the vector of variables, and [tex]\(B\)[/tex] is the constant matrix.
### Step 2: Find the inverse of the coefficient matrix [tex]\(A\)[/tex]
The inverse of matrix [tex]\(A\)[/tex], denoted as [tex]\(A^{-1}\)[/tex], is needed to solve for [tex]\(x\)[/tex]. Given:
[tex]\[
A = \begin{pmatrix}
5 & -2 \\
1 & 6
\end{pmatrix}
\][/tex]
The inverse matrix [tex]\(A^{-1}\)[/tex] can be represented as:
[tex]\[
A^{-1} = \begin{pmatrix}
0.1875 & 0.0625 \\
-0.03125 & 0.15625
\end{pmatrix}
\][/tex]
### Step 3: Multiply [tex]\(A^{-1}\)[/tex] by [tex]\(B\)[/tex] to find [tex]\(x\)[/tex]
To find the solution vector [tex]\(x\)[/tex], we use the relationship:
[tex]\[
x = A^{-1}B
\][/tex]
Now, let's perform the matrix multiplication:
[tex]\[
x = \begin{pmatrix}
0.1875 & 0.0625 \\
-0.03125 & 0.15625
\end{pmatrix}
\begin{pmatrix}
30 \\
-26
\end{pmatrix}
\][/tex]
### Step 4: Calculate the resulting vector
Perform the multiplication:
[tex]\[
\begin{pmatrix}
0.1875 & 0.0625 \\
-0.03125 & 0.15625
\end{pmatrix}
\begin{pmatrix}
30 \\
-26
\end{pmatrix}
=
\begin{pmatrix}
(0.1875 \cdot 30) + (0.0625 \cdot -26) \\
(-0.03125 \cdot 30) + (0.15625 \cdot -26)
\end{pmatrix}
\][/tex]
Calculate each component:
[tex]\[
\begin{pmatrix}
(0.1875 \cdot 30) + (0.0625 \cdot -26) \\
(-0.03125 \cdot 30) + (0.15625 \cdot -26)
\end{pmatrix}
=
\begin{pmatrix}
5.625 - 1.625 \\
-0.9375 - 4.0625
\end{pmatrix}
=
\begin{pmatrix}
4 \\
-5
\end{pmatrix}
\][/tex]
### Step 5: Interpret the results
Therefore, the solution to the system of equations is:
[tex]\[
\begin{cases}
x = 4 \\
y = -5
\end{cases}
\][/tex]
So, the solution is:
[tex]\[
\begin{pmatrix}
x \\
y
\end{pmatrix}
=
\begin{pmatrix}
4 \\
-5
\end{pmatrix}
\][/tex]
This represents the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfy both equations in the given system.
