Question

Select the equation that has infinitely many solutions.(1 point) Responses 3x=2x+1 3 x is equal to 2 x plus 1 3−3x=−3(x−1) 3 minus 3 x is equal to negative 3 times open paren x minus 1 close paren 4x+1=4(x+1) 4 x plus 1 is equal to 4 times open paren x plus 1 close paren 3x+5=3x−5

Answer 1

the answer choices are based on the student's work, the equation was solve correctly or incorrectly. the equation solved correctly would show that it has infinitely many, one, two, three

Answer 2

the answer choices are based on the student's work, the equation was solve correctly or incorrectly. the equation solved correctly would show that it has none, one, two, three

Answer 3

To determine which equation has infinitely many solutions, we can solve each equation and see which one leads to a true statement regardless of the variable value.

1. **Equation 1:** \( 3x = 2x + 1 \)

Subtract \( 2x \) from both sides:
\[
3x - 2x = 1 \implies x = 1
\]
(One solution, not infinitely many.)

2. **Equation 2:** \( 3 - 3x = -3(x - 1) \)

Expand the right side:
\[
3 - 3x = -3x + 3
\]
Simplifying both sides gives:
\[
3 - 3x = 3 - 3x
\]
This statement is always true for any \( x \), thus it has infinitely many solutions.

3. **Equation 3:** \( 4x + 1 = 4(x + 1) \)

Expand the right side:
\[
4x + 1 = 4x + 4
\]
Subtract \( 4x \) from both sides:
\[
1 = 4
\]
(This is a false statement, no solutions.)

4. **Equation 4:** \( 3x + 5 = 3x - 5 \)

Subtract \( 3x \) from both sides:
\[
5 = -5
\]
(This is a false statement, no solutions.)

The equation that has infinitely many solutions is:

**\( 3 - 3x = -3(x - 1) \)**

Answer 4

Thank you for the clarification! Given your explanation about the equation being solved correctly or incorrectly, here is how the equations relate to the concept of solutions:

1. **Equation 1:** \( 3x = 2x + 1 \)

Solving yields:
\[
x = 1
\]
This has **one solution**.

2. **Equation 2:** \( 3 - 3x = -3(x - 1) \)

This simplifies to:
\[
3 - 3x = -3x + 3 \quad \Rightarrow \quad 3 - 3x = 3 - 3x
\]
This is true for all \( x \). Thus, it has **infinitely many solutions**.

3. **Equation 3:** \( 4x + 1 = 4(x + 1) \)

This simplifies to:
\[
4x + 1 = 4x + 4 \quad \Rightarrow \quad 1 = 4
\]
This is a false statement, leading to **no solutions**.

4. **Equation 4:** \( 3x + 5 = 3x - 5 \)

This simplifies to:
\[
5 = -5
\]
This is also a false statement, leading to **no solutions**.

Thus, based on the student's work:

- The equation that was solved **correctly** and has **infinitely many solutions** is:

**\( 3 - 3x = -3(x - 1) \)**.

Answer 5

Thank you for your clarification regarding the framework of your question. Let's classify each equation based on whether the student solved it **correctly** or **incorrectly**, and also identify how many solutions each equation actually has.

1. **Equation 1:** \( 3x = 2x + 1 \)

- Solving it correctly gives:
\[
3x - 2x = 1 \Rightarrow x = 1
\]
- **Solves correctly** showing **one solution**.

2. **Equation 2:** \( 3 - 3x = -3(x - 1) \)

- Simplifying gives:
\[
3 - 3x = -3x + 3
\]
- This simplifies to \( 3 = 3 \), which is always true for any \( x \).
- **Solves correctly** showing **infinitely many solutions** (this is usually considered as part of the set of "correct" types of solutions but not within the choices of none, one, two, or three).

3. **Equation 3:** \( 4x + 1 = 4(x + 1) \)

- This becomes:
\[
4x + 1 = 4x + 4 \quad \Rightarrow \quad 1 = 4
\]
- This is false.
- **Incorrect solution**, showing **no solution**.

4. **Equation 4:** \( 3x + 5 = 3x - 5 \)

- This becomes:
\[
5 = -5
\]
- This is also false.
- **Incorrect solution**, showing **no solution**.

### Summary of Each Equation's Result:

- **Equation 1:** **Correct; One solution**
- **Equation 2:** **Correct; Infinitely many solutions**
- **Equation 3:** **Incorrect; No solution**
- **Equation 4:** **Incorrect; No solution**

In the context of your format:

- If you are looking for which equations were solved correctly according to your grading criteria, that would pertain to Equations 1 and 2.

To match the criteria of the correct/incorrect solution types:

- **No solutions:** Equations 3 and 4
- **One solution:** Equation 1
- **Infinitely many solutions** (not typically categorized within "none, one, two, three"): Equation 2

If you need to determine which equations specifically fit within the disciplines of "none," "one," "two," "three" solution types while also noting correctness, please let me know!