A linear equation has infinite solutions if the left and right sides are the same. These equations can be manipulated into the expression 0=0, which is true no matter which value we choose for x, so they have infinite solution.
It has no solutions if the terms involving x cancel out, but the constant terms are different. These equations can be turned into an expression like [tex] a=b [/tex], where a and b are actually different numbers, and so it can never be true.
In all other cases, you have exactly one solution.
1
[tex] 2x+2x+2=4x+2 \iff 4x+2=4x+2 \iff [/tex] infinite solutions
2
[tex] 3(x-1)=2x+9 \iff 3x-3 = 2x+9 \iff x=12 [/tex]
3
[tex] 2x+8=2(x+4) \iff 2x+8=2x+8\iff[/tex] infinite solutions
4
[tex] 8(x+2)=2x+16 \iff 8x+16=2x+16 \iff 6x=0 \iff x=0 [/tex]
5
[tex] 4x+1 = 2(2x+3) \iff 4x+1 = 4x+6 \iff 1=6 \iff [/tex] no solutions
6
[tex] -2(x+1) = 2(x-1) \iff -2x-2=2x-2 \iff 4x=0 \iff x=0 [/tex]
7
[tex] 3x+1=3(x-1)+4 \iff 3x+1=3x-3+4 \iff 3x+1=3x+1\iff[/tex] infinite solutions
8
[tex] \dfrac{1}{2}(2-4x)+2x=13 \iff 1-2x+2x=13 \iff 1=13 \iff [/tex] no solutions.
