To solve the given system of linear equations using matrices, let's follow these necessary steps.
1. Write the system of linear equations in matrix form:
The given system of equations can be written as:
[tex]\[
\begin{pmatrix}
2 & 5 \\
1 & 3
\end{pmatrix}
\begin{pmatrix}
x \\
y
\end{pmatrix}
=
\begin{pmatrix}
7 \\
5
\end{pmatrix}
\][/tex]
2. Identify the coefficient matrix [tex]\(A\)[/tex], the variable matrix [tex]\( \mathbf{x} \)[/tex], and the constants matrix [tex]\( \mathbf{B} \)[/tex]:
[tex]\[
A = \begin{pmatrix}
2 & 5 \\
1 & 3
\end{pmatrix}
, \quad
\mathbf{x} = \begin{pmatrix}
x \\
y
\end{pmatrix}
, \quad
\mathbf{B} = \begin{pmatrix}
7 \\
5
\end{pmatrix}
\][/tex]
3. Find the inverse of the coefficient matrix [tex]\(A\)[/tex]:
The solution to the system of equations involves finding the inverse of matrix [tex]\(A\)[/tex]. If [tex]\(A^{-1}\)[/tex] is the inverse of [tex]\(A\)[/tex], we multiply both sides of the equation [tex]\(A \mathbf{x} = \mathbf{B}\)[/tex] by [tex]\(A^{-1}\)[/tex] to obtain:
[tex]\[
A^{-1} A \mathbf{x} = A^{-1} \mathbf{B}
\][/tex]
Since [tex]\(A^{-1} A\)[/tex] is the identity matrix [tex]\(I\)[/tex], we have:
[tex]\[
\mathbf{x} = A^{-1} \mathbf{B}
\][/tex]
After calculating the inverse of matrix [tex]\(A\)[/tex], we get:
[tex]\[
A^{-1} = \begin{pmatrix}
3 & -5 \\
-1 & 2
\end{pmatrix}
\][/tex]
4. Multiply the inverse of [tex]\(A\)[/tex] by [tex]\( \mathbf{B} \)[/tex] to find [tex]\( \mathbf{x} \)[/tex]:
[tex]\[
\mathbf{x} = A^{-1} \mathbf{B} = \begin{pmatrix}
3 & -5 \\
-1 & 2
\end{pmatrix}
\begin{pmatrix}
7 \\
5
\end{pmatrix}
\][/tex]
Perform the matrix multiplication:
[tex]\[
\mathbf{x} = \begin{pmatrix}
3 \cdot 7 + (-5) \cdot 5 \\
-1 \cdot 7 + 2 \cdot 5
\end{pmatrix}
= \begin{pmatrix}
21 - 25 \\
-7 + 10
\end{pmatrix}
= \begin{pmatrix}
-4 \\
3
\end{pmatrix}
\][/tex]
5. Extract the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] from the solution matrix [tex]\( \mathbf{x} \)[/tex]:
[tex]\[
x = -4 \quad \text{and} \quad y = 3
\][/tex]
Therefore, the solution to the system of equations is:
[tex]\[
x = -4 \quad \text{and} \quad y = 3
\][/tex]
