Given that
[tex]\[ \left(\begin{array}{cc} 2 & x \\ -5 & 1 \end{array}\right) \binom{4}{7} = \binom{20}{-13} \][/tex]
find the value of [tex]\( x \)[/tex].



Answer :

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]