To find the value of [tex]\( x \)[/tex] from the given system of linear equations:
1. [tex]\( 2x + y = 2 \)[/tex]
2. [tex]\( x + 2y = -2 \)[/tex]
We can use the method of elimination to solve for [tex]\( x \)[/tex]. Here is the detailed step-by-step solution:
1. First, multiply the entire first equation by 2 to make the coefficient of [tex]\( y \)[/tex] in both equations the same:
[tex]\[ 2(2x + y) = 2 \cdot 2 \][/tex]
This gives us:
[tex]\[ 4x + 2y = 4 \][/tex]
2. Now, we have the modified system of equations:
[tex]\[ 4x + 2y = 4 \][/tex]
[tex]\[ x + 2y = -2 \][/tex]
3. Next, subtract the second equation from the first equation to eliminate [tex]\( y \)[/tex]:
[tex]\[ (4x + 2y) - (x + 2y) = 4 - (-2) \][/tex]
Simplifying this, we get:
[tex]\[ 4x + 2y - x - 2y = 4 + 2 \][/tex]
[tex]\[ 3x = 6 \][/tex]
4. Solve for [tex]\( x \)[/tex] by dividing both sides of the equation by 3:
[tex]\[ x = \frac{6}{3} \][/tex]
[tex]\[ x = 2 \][/tex]
So, the value of [tex]\( x \)[/tex] is [tex]\(\boxed{2}\)[/tex].
