1. 2(x - 3) = 2x

a) 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

Y=2X-3

Y=-x+3

1. b

2. C

3. C
4. A

To determine the number of solutions for each equation, we need to solve them step by step and analyze the results.

1. 2(x - 3) = 2x

To solve this equation, let's distribute the 2 on the left side of the equation:

2 * x - 2 * 3 = 2x
2x - 6 = 2x

Now, subtract 2x from both sides of the equation to isolate the unknown variable:

2x - 2x - 6 = 2x - 2x
-6 = 0

Since -6 is not equal to 0, the equation becomes:

0 = 0

This means that the equation is always true, regardless of the value of x. Therefore, there are infinitely many solutions. So, the answer is:

c) infinitely many solutions

2. 3(y - 3) = 2y - 9 + y

Similarly, distribute the 3 on the left side of the equation:

3 * y - 3 * 3 = 2y - 9 + y
3y - 9 = 2y - 9 + y

Combine like terms:

3y - 9 = 3y - 9

Next, subtract 3y from both sides of the equation:

3y - 3y - 9 = 3y - 3y - 9
-9 = -9

Again, -9 is equal to -9, meaning the equation is always true. Thus, there are infinitely many solutions. The answer is:

c) infinitely many solutions

3. 10x - 2 - 6x = 3x - 2 + x

Combine like terms:

(10x - 6x) - 2 = (3x + x) - 2
4x - 2 = 4x - 2

Subtract 4x from both sides of the equation:

4x - 4x - 2 = 4x - 4x - 2
-2 = -2

Again, -2 is equal to -2, which means the equation is always true. Thus, there are infinitely many solutions. The answer is:

c) infinitely many solutions

4. 4(x + 3) + 2x = x - 8

Simplify the left side of the equation:

4x + 12 + 2x = x - 8
6x + 12 = x - 8

Subtract x from both sides of the equation:

6x - x + 12 = x - x - 8
5x + 12 = -8

Now, subtract 12 from both sides of the equation:

5x + 12 - 12 = -8 - 12
5x = -20

Finally, divide both sides of the equation by 5 to solve for x:

5x / 5 = -20 / 5
x = -4

Now, substitute the value of x back into the equation to check if it is true:

4(x + 3) + 2x = x - 8
4(-4 + 3) + 2(-4) = -4 - 8
4(-1) - 8 = -12
-4 - 8 = -12
-12 = -12

Since the equation is true, there is one solution. The answer is:

a) one solution