Certainly! To solve for [tex]\( x \)[/tex] given the matrix equation
[tex]\[
\begin{pmatrix}
2 & x \\
-5 & 1
\end{pmatrix}
\begin{pmatrix}
4 \\
7
\end{pmatrix}
=
\begin{pmatrix}
20 \\
-13
\end{pmatrix},
\][/tex]
we need to break down this equation into a system of linear equations.
### Step 1: Matrix Multiplication
The given matrix equation can be expressed as:
[tex]\[
\begin{pmatrix}
2 & x \\
-5 & 1
\end{pmatrix}
\begin{pmatrix}
4 \\
7
\end{pmatrix}
=
\begin{pmatrix}
2 \cdot 4 + x \cdot 7 \\
-5 \cdot 4 + 1 \cdot 7
\end{pmatrix}.
\][/tex]
So, we get:
[tex]\[
\begin{pmatrix}
2 \cdot 4 + x \cdot 7 \\
-5 \cdot 4 + 1 \cdot 7
\end{pmatrix}
=
\begin{pmatrix}
8 + 7x \\
-20 + 7
\end{pmatrix}.
\][/tex]
### Step 2: Form two equations from the resulting matrix
From this multiplication, equate the resulting vector components to the given vector components:
1. For the first component:
[tex]\[
8 + 7x = 20
\][/tex]
2. For the second component:
[tex]\[
-20 + 7 = -13
\][/tex]
### Step 3: Solve the equation that contains [tex]\( x \)[/tex]
#### Second equation verification:
Checking the second equation:
[tex]\[
-20 + 7 = -13
\][/tex]
[tex]\[
-13 = -13 \quad \text{(True)}
\][/tex]
The second equation is always true and serves as a verification step for our system.
#### Solve for [tex]\( x \)[/tex] from the first equation:
[tex]\[
8 + 7x = 20
\][/tex]
Subtract 8 from both sides:
[tex]\[
7x = 20 - 8
\][/tex]
Simplify:
[tex]\[
7x = 12
\][/tex]
Divide by 7:
[tex]\[
x = \frac{12}{7}
\][/tex]
So, the value of [tex]\( x \)[/tex] is:
[tex]\[
x = \frac{12}{7}
\][/tex]
Hence, the solution is:
[tex]\[
x = \frac{12}{7}
\][/tex]
