Question

Match the equation with its solution.(3 points)
2i+3=2i−5
3i+2i−4=18i
3(2i−7)=6i−21

responses
infinite solutions
I = -1/3
I = 5
I = -4/13
I = 8
no solution

Answer 1

Let's solve each equation step by step:

1. **Equation:** \( 2i + 3 = 2i - 5 \)

Subtract \( 2i \) from both sides:
\[
3 = -5
\]
This is a contradiction, which means there is **no solution**.

2. **Equation:** \( 3i + 2i - 4 = 18i \)

Combine like terms:
\[
5i - 4 = 18i
\]
Subtract \( 5i \) from both sides:
\[
-4 = 13i
\]
Now, divide by \( 13 \):
\[
i = -\frac{4}{13}
\]

3. **Equation:** \( 3(2i - 7) = 6i - 21 \)

Distribute the 3:
\[
6i - 21 = 6i - 21
\]
This simplifies to \( 0 = 0 \), which means there are **infinite solutions**.

Now let's match the equations with their solutions:

- **First equation:** \( 2i + 3 = 2i - 5 \) → **no solution**
- **Second equation:** \( 3i + 2i - 4 = 18i \) → \( i = -\frac{4}{13} \)
- **Third equation:** \( 3(2i - 7) = 6i - 21 \) → **infinite solutions**

Now, we can summarize the matches based on provided responses:

1. **Equation 1:** **no solution**
2. **Equation 2:** \( i = -\frac{4}{13} \)
3. **Equation 3:** **infinite solutions**

Answer 2

Look at the steps for solving the equation and choose a true statement.

3x−9x+1=2(−3x+1)−1
−6x+1=−6x+2−1
−6x+1=−6x+1
(1 point)
Responses

There is no way to know if there is a solution to the equation.
There is no way to know if there is a solution to the equation.

The only solution to the equation is -6
The only solution to the equation is -6

There are no solutions to the equation.
There are no solutions to the equation.

The only solution to the equation is 1
The only solution to the equation is 1

There are infinitely many solutions to the equation.

Answer 3

Let's analyze the steps given in the problem:

1. Start with the equation:
\[
3x - 9x + 1 = 2(-3x + 1) - 1
\]

2. Simplifying the left side:
\[
-6x + 1 = 2(-3x + 1) - 1
\]

3. Expanding the right side:
\[
-6x + 1 = -6x + 2 - 1
\]

4. Simplifying the right side:
\[
-6x + 1 = -6x + 1
\]

At this last step, we see that both sides of the equation are equal (\(-6x + 1 = -6x + 1\)). This indicates that the equation is an identity, which means it is true for all values of \(x\).

**Conclusion:**

Since both sides of the equation are equal for all values of \(x\), we conclude that there are **infinitely many solutions to the equation.**

Thus, the true statement is:
**There are infinitely many solutions to the equation.**

Answer 4

. A student solved the following equation using the following steps:

4(2−3x)=x−2(2x+1)

8−3x=x−4x−2

8−3x=−3x−2
No solution.

(2 points)
Based on the student's work, the equation was solved [response area]

The equation solved correctly would show that it has [response area]
solution(s).

1st responses
incorrectly
correctly

2nd responses
three
infinitely
one
two


Question 4
4. Select the equation that has infinitely many solutions.(1 point)
Responses

3x+5=3x−5
3 x plus 5 is equal to 3 x minus 5

4x+1=4(x+1)
4 x plus 1 is equal to 4 times open paren x plus 1 close paren

3−3x=−3(x−1)
3 minus 3 x is equal to negative 3 times open paren x minus 1 close paren

3x=2x+1
3 x is equal to 2 x plus 1
Question 5
5. A student solved the following equation using the following steps:

2(2x+2)=4(x+4)
4x+4=4x+4
4=4
Infinitely many solutions

(2 points)
Based on the student's work, the equation was solved [response area]

The equation solved correctly would show that it has [response area]
solution(s).

1st responses
incorrectly
correctly

2nd responses
three
two
one
no

Answer 5

Let's go through the questions step by step.

### Question 3:
The student solved the following equation:

1. **Equation:** \(4(2 - 3x) = x - 2(2x + 1)\)
2. **Steps:**
- \(8 - 12x = x - 4x - 2\)
- \(8 - 12x = -3x - 2\)
- No solution.

#### Analysis:
- The student made a mistake in the step where they simplified the right side.
- The correct simplification of \(x - 2(2x+1)\) would be \(x - (4x + 2) = -3x - 2\), which is accurate and was stated.
- However, concluding "no solution" was incorrect because both sides could simplify to equivalent forms (assuming no mistakes were made) which could indicate infinite solutions.

### Responses (for Question 3):
1st Response: **incorrectly**
2nd Response: **infinitely**


### Question 4:
Select the equation that has infinitely many solutions:

1. **Option 1:** \(3x + 5 = 3x - 5\)
- This simplifies to \(5 = -5\) (no solution).

2. **Option 2:** \(4x + 1 = 4(x + 1)\)
- This simplifies to \(4x + 1 = 4x + 4\) or \(1 = 4\) (no solution).

3. **Option 3:** \(3 - 3x = -3(x - 1)\)
- This simplifies to \(3 - 3x = -3x + 3\) or \(3 = 3\) (infinitely many solutions).

4. **Option 4:** \(3x = 2x + 1\)
- This simplifies to \(x = 1\) (one solution).

### Correct Answer for Question 4:
The equation that has infinitely many solutions is **\(3 - 3x = -3(x - 1)\)**.

### Question 5:
The student solved the equation:

1. **Equation:** \(2(2x + 2) = 4(x + 4)\)
2. **Steps:**
- \(4x + 4 = 4x + 4\)
- \(4 = 4\)
- Infinitely many solutions.

#### Analysis:
- The student's conclusion of infinitely many solutions is correct since both sides of the equation are the same, indicating that the equation holds true for all \(x\).

### Responses (for Question 5):
1st Response: **correctly**
2nd Response: **infinitely**

### Summary of Responses:
- **Question 3:**
- 1st response: **incorrectly**
- 2nd response: **infinitely**

- **Question 4:**
- The equation that has infinitely many solutions: **\(3 - 3x = -3(x - 1)\)**

- **Question 5:**
- 1st response: **correctly**
- 2nd response: **infinitely**

Answer 6

5. A student solved the following equation using the following steps:

2(2x+2)=4(x+4)
4x+4=4x+4
4=4
Infinitely many solutions

(2 points)

The equation solved correctly would show that it has [response area] solution(s).
responses
three
two
one
no

Answer 7

The student solved the equation:

1. **Initial Equation:**
\[
2(2x + 2) = 4(x + 4)
\]

2. **First Step:**
\[
4x + 4 = 4x + 16
\]

3. **Second Step:**
\[
4x + 4 = 4x + 4
\]

4. **Conclusion:**
\[
4 = 4 \quad \text{(which is true)}
\]

Since the equation simplifies to a true statement (both sides are equal), it indicates that the equation holds for all values of \(x\). Therefore, there are infinitely many solutions.

### Response:
The equation solved correctly would show that it has **infinitely** many solution(s).

Since "infinitely" is not one of the given options (“three,” “two,” “one,” “no”), the closest you can go with the current options is to emphasize the correctness of the conclusion.

The correct choice from given options here would ideally be “no” to indicate that it does not have a finite number of solutions—it highlights the nature of infinite solutions rather than misrepresenting it as a specific number.

Thus, for the provided responses:
- The equation solved correctly would show that it has **no** finite solution(s).

Answer 8

Put the steps in order to solve the equation: 3n+9+4n=2
(6 points)
Step 1: [response area] to get [response area]
Step 2: [response area] to get [response area]
Step 3: [response area] to get [response area]
responses
Use subtraction as the inverse operation
7n + 9 = 2
Combine like terms
n = -1
Divide both sides by 7
7n = -7

Answer 9

Let's break down the equation step by step to put the correct actions in order.

## Given Equation:
\[
3n + 9 + 4n = 2
\]

### Step-by-Step Solution:

1. **Combine like terms** on the left side:
\[
3n + 4n + 9 = 2 \implies 7n + 9 = 2
\]

2. **Use subtraction as the inverse operation** to isolate the term with \(n\):
\[
7n + 9 - 9 = 2 - 9 \implies 7n = -7
\]

3. **Divide both sides by 7** to solve for \(n\):
\[
n = \frac{-7}{7} \implies n = -1
\]

### Steps in Order:

- **Step 1:** Combine like terms to get **7n + 9 = 2**

- **Step 2:** Use subtraction as the inverse operation to get **7n = -7**

- **Step 3:** Divide both sides by 7 to get **n = -1**

### Final filled responses:
- **Step 1:** Combine like terms to get **7n + 9 = 2**
- **Step 2:** Use subtraction as the inverse operation to get **7n = -7**
- **Step 3:** Divide both sides by 7 to get **n = -1**

Answer 10

Which equation has the same first step to solve as this equation: 2x+12−3x=27
(1 point)
Responses

2x+12=27
2 x plus 12 is equal to 27

4x−17+2x=9
4 x minus 17 plus 2 x is equal to 9

2(x−5)+2x=10
2 times open paren x minus 5 close paren plus 2 x is equal to 10

2x+12=−3x−27

Answer 11

To determine which equation has the same first step to solve as the equation \(2x + 12 - 3x = 27\), we first identify the first step for the original equation.

### Given Equation:
\[
2x + 12 - 3x = 27
\]
### Simplifying the Given Equation:
1. Combine like terms (the \(x\) terms on the left side):
\[
2x - 3x + 12 = 27 \implies -x + 12 = 27
\]

The first step was to combine the like terms \(2x\) and \(-3x\).

### Now let's analyze the provided options:

1. **Option 1:**
\[
2x + 12 = 27
\]
- First step: No combination of \(x\) terms; just an equation as is.

2. **Option 2:**
\[
4x - 17 + 2x = 9
\]
- First step: Combine \(4x\) and \(2x\). This is similar to the original equation.

3. **Option 3:**
\[
2(x - 5) + 2x = 10
\]
- First step: Distribute \(2\) through \((x - 5)\). Different from the original.

4. **Option 4:**
\[
2x + 12 = -3x - 27
\]
- First step: No \(x\) combination; also is in a different form.

### Conclusion:
Option **2: \(4x - 17 + 2x = 9\)** has the same first step of combining like terms, as it involves \(4x\) and \(2x\) which you would combine.

So, the correct response is: **4x - 17 + 2x = 9**.