Decide whether each equation has one, no solutions, or infanatly many solutions

1.) 2(x - 3) = 2x

One solution
no solutions
many solutions
infinatly many solutions

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

One solution
no solutions
many solutions
infinatly many solutions

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

One solution
no solutions
infinatly many solutions

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

One solution
no solutions
infinatly many solutions

MY ANSWERS

1. D
2. A
3. B
4. C

1. Actually it's B.

Expanding,
2(x - 3) = 2x
2x - 6 = 2x
-6 = 0
Note that if you get an answer like this (constant = zero), there is no solution.

2. It's D. (btw I don't know why you put question mark beside 3)
Expanding,
3(y - 3) = 2y - 9 + y
3y - 9 = 3y - 9
or
0 = 0
Note that if you get an answer like this (zero = zero), there are infinite solutions.

3. It's C.
Expanding,
10x - 2 - 6x = 3x -2 + x
4x - 2 = 4x - 2
0 = 0
(infinite solutions)

4. It's A.
Expanding,
4(x + 3) + 2x = x - 8
4x + 12 + 2x = x - 8
6x + 12 = x - 8
5x = -20
x = -4
(one solution)

I guess you have to study these topics.
Hope this helps :3

Check your answers again.

1. -3 = 0?

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

3. 4x-2 = 4x-2 (identity)

4. 6x+12 = x-8

5x = -20

Thank you Jai, I understand it all perfectly now.

7x+3=7x+b

1)B

2)C
3)C
4)A

For the students that do not have 4 options for answers.

To determine if an equation has one solution, no solutions, or infinitely many solutions, we need to simplify the equation and analyze the resulting expression.

1.) 2(x - 3) = 2x

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

2x - 6 = 2x

Now, let's simplify by subtracting 2x from both sides:

-6 = 0

This equation results in an inconsistency. The left side is not equal to the right side (-6 ≠ 0). Therefore, this equation has no solutions. So the correct answer is B.

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

To solve this equation, first distribute the 3 on the left side:

3y - 9 = 2y - 9 + y

Next, combine the like terms on the right side:

3y - 9 = 3y - 9

Simplify by subtracting 3y from both sides:

-9 = -9

This equation is an identity because the left side is equal to the right side (-9 = -9). Therefore, this equation has infinitely many solutions. So the correct answer is D.

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

Start by combining like terms on both sides:

4x - 2 = 4x - 2

Now, subtract 4x from both sides:

-2 = -2

This equation is also an identity because the left side is equal to the right side (-2 = -2). Therefore, it has infinitely many solutions. So the correct answer is D.

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

To solve this equation, let's distribute the 4 on the left side:

4x + 12 + 2x = x - 8

Combine the like terms on both sides:

6x + 12 = x - 8

Next, subtract x from both sides:

5x + 12 = -8

Finally, subtract 12 from both sides:

5x = -20

Divide both sides by 5:

x = -4

This equation has exactly one solution, which is x = -4. So the correct answer is A.

Your answers are as follows:
1. B
2. D
3. D
4. A