Question

Decide whether each equation has one solution, no solutions, or infinitely many solutions. . 1. 2(x - 3) = 2x .
. one solution .
B. no solutions .
C. infinitely many solutions . .
2. 3(y - 3) = 2y - 9 + y .
A. one solution .
B. no solutions .
C. infinitely many solutions . .
3. 10x - 2 - 6x = 3x - 2 + x.
A. one solution .
B. no solutions .
C. infinitely many solutions. .
4. 4(x + 3) + 2x = x - 8 .
A. one solution .
B. no solutions .
C. infinitely many solutions

Answer 1

(1)
2(x - 3) = 2x -- distribute through the parenthesis
2x - 6 = 2x -- add 6 to both sides
2x = 2x + 6 -- subtract 2x from both sides
2x - 2x = 6 0 = 6 (incorrect)

If you answer comes out not equal, then there is no solution

(2)
3(y - 3) = 2y - 9 + y -- distribute and combine
3y - 9 = 3y - 9 -- subtract 3y
3y - 3y - 9 = - 9 -- add 9
3y - 3y = -9 + 9
0 = 0 (correct)

If you answer comes out equal, then there is infinite solutions
(3)
10x - 2 - 6x = 3x - 2 + x -- combine like terms 4x - 2 = 4x - 2
same thing here...it is gonna come out to 0 = 0 making it have infinite solutions

(4) 
4(x + 3) + 2x = x - 8
4x + 12 + 2x = x - 8
6x + 12 = x - 8
6x - x = -8 - 12
5x = - 20
x = -4

ONE SOLUTION