To determine the value of [tex]\( x \)[/tex], let's solve the system of linear equations given:
[tex]\[ 2x + y = 2 \][/tex]
[tex]\[ x + 2y = -2 \][/tex]
We will solve this system using the method of substitution or elimination (also known as the addition method). Let's proceed step by step.
### Step 1: Solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex] in the first equation
The first equation is:
[tex]\[ 2x + y = 2 \][/tex]
Isolate [tex]\( y \)[/tex] by moving [tex]\( 2x \)[/tex] to the other side:
[tex]\[ y = 2 - 2x \][/tex]
### Step 2: Substitute [tex]\( y \)[/tex] in the second equation
The second equation is:
[tex]\[ x + 2y = -2 \][/tex]
Substitute [tex]\( y = 2 - 2x \)[/tex] into the second equation:
[tex]\[ x + 2(2 - 2x) = -2 \][/tex]
### Step 3: Simplify and solve for [tex]\( x \)[/tex]
Simplify the expression on the left-hand side:
[tex]\[ x + 4 - 4x = -2 \][/tex]
[tex]\[ x - 4x + 4 = -2 \][/tex]
[tex]\[ -3x + 4 = -2 \][/tex]
Isolate [tex]\( x \)[/tex] by first moving the constant term to the other side of the equation:
[tex]\[ -3x = -2 - 4 \][/tex]
[tex]\[ -3x = -6 \][/tex]
Now, divide by -3:
[tex]\[ x = \frac{-6}{-3} \][/tex]
[tex]\[ x = 2 \][/tex]
Thus, the value of [tex]\( x \)[/tex] is [tex]\( \boxed{2} \)[/tex].
