Asked by I_EAT_EVERYDAY

Solve the equation justifying each step with the correct reasoning.
2(x + 8) = 2x + 8

Step 1: _______ Property to get Response area simplified equation

Step 2: _______ Property to get Response area simplified equation

lastly: For this equation, there is/are ____________

properties and reasons:
1. subtraction of equality
2. Multiplication of equality
3. combine like terms
4. division of equality
5. addition of equality
6. distributive

Equation simplified:
1. One solution
2. 16 = 8
3. 2x + 16 = 2x + 8
4. 2x + 8 = 2x + 8
5. No solution
6. x = -8
7. x + 16 = 8
8. 2x + 8 = 8
9. x = 2
10. x + 8 = 8

Answered by I_EAT_EVERYDAY
Match the description of the one variable equation with the number of solutions it will have.

1. x − 7 = 7 − x

2. 3(x + 5) = 3x + 5

3. 10 − x = 25

4. 2(x + 4) = 2x + 5 + 3

options:
1. one solution
2. Two solutions
3. No solution
4. Infinite solutions
Answered by I_EAT_EVERYDAY
A student wants to purchase some new school supplies. He wants to buy a calculator that costs $24 and some notebooks for school. Each notebook costs $2. The student only has $37 to spend.

Let n represent the number of notebooks that he buys.

Which inequality describes this scenario?

A). 24n + 2 ≥ 37

B). 37 > 2n + 24

C). 37 ≥ 2n + 24

D). 37 < 2n + 24
Answered by I_EAT_EVERYDAY
Solve for b in the following equation: A = 1 over 2 (a + b)

A). b = 1 over 2A − a

B). b = 2A + a

C). b = 1 over 2A + a

D). b = 2A − a
Answered by I_EAT_EVERYDAY
A student solves the following equation:

Problem: 2(x−3)+3x=19

Step 1: 2x − 6 + 3x = 19

Step 2: (2x + 3x) − 6 = 19

Step 3: 5x − 6 = 19

Step 4: 5x − 6 + 6 = 19 + 6

Step 5: 5x = 25

Step 6: x = 5

What property justifies going from step 3 to step 4?

A). Substitution Property

B). Combine Like Terms

C). Addition Property of Equality

D). Distributive Property

E). Commutative Property of Addition
Answered by I_EAT_EVERYDAY
A student solved the equation: x + 2(x + 1) = 17

Step 1: x + 2x + 2 = 17

Step 2: 3x + 2 = 17

Step 3: 3x = 15

Step 4: x = 45

1. Identify the property used to arrive at Step 1: _____________

2. What step includes a mistake made from the previous step? ___________

3. What should have been the correct answer for x ? ____________
Answered by I_EAT_EVERYDAY
what's the last answer again?
Answered by I_EAT_EVERYDAY
so x = 5
Answered by GPT-4o mini
Let's solve the equation step by step, justifying each with the correct reasoning.

**Given equation:**
\[ 2(x + 8) = 2x + 8 \]

**Step 1:** Use the **Distributive Property** to simplify the left side of the equation.
\[ 2(x + 8) = 2 \cdot x + 2 \cdot 8 \]
This simplifies to:
\[ 2x + 16 = 2x + 8 \]

**Step 2:** Subtract \( 2x \) from both sides using the **Subtraction of Equality Property**.
\[ 2x + 16 - 2x = 2x + 8 - 2x \]
This simplifies to:
\[ 16 = 8 \]

This statement, \( 16 = 8 \), is clearly false. Therefore, we conclude that the original equation has no solution.

Lastly, for this equation, there is/are **no solution**.

### Summary of Properties and Reasoning:
- Step 1: **Distributive Property** to simplify the equation to \( 2x + 16 = 2x + 8 \).
- Step 2: **Subtraction of Equality Property** to derive \( 16 = 8 \).

### Conclusion:
The equation simplifies to \( 16 = 8 \), which is not true. Therefore, there is no solution to the equation.
Answered by GPT-4o mini
Let's analyze each equation to determine how many solutions it has.

1. **Equation:** \( x - 7 = 7 - x \)
**Solution:**
Adding \( x \) to both sides:
\( x + x - 7 = 7 \)
\[ 2x - 7 = 7 \]
Adding 7 to both sides:
\[ 2x = 14 \]
Dividing by 2:
\[ x = 7 \]
**Number of solutions:** **1 (one solution)**

2. **Equation:** \( 3(x + 5) = 3x + 5 \)
**Solution:**
Distributing on the left:
\( 3x + 15 = 3x + 5 \)
Subtracting \( 3x \) from both sides:
\( 15 = 5 \)
This statement is false.
**Number of solutions:** **3 (no solution)**

3. **Equation:** \( 10 - x = 25 \)
**Solution:**
Adding \( x \) to both sides:
\( 10 = x + 25 \)
Subtracting 25 from both sides:
\( 10 - 25 = x \)
\( -15 = x \)
Thus, \( x = -15 \).
**Number of solutions:** **1 (one solution)**

4. **Equation:** \( 2(x + 4) = 2x + 5 + 3 \)
**Solution:**
Distributing on the left:
\( 2x + 8 = 2x + 8 \)
This simplifies to:
\( 2x + 8 - 2x = 8 \)
Which gives us \( 8 = 8 \), a true statement.
This indicates that the two sides are always equal for any value of \( x \).
**Number of solutions:** **4 (infinite solutions)**

### Summary of Matches:

1. \( x - 7 = 7 - x \) → **1 (one solution)**
2. \( 3(x + 5) = 3x + 5 \) → **3 (no solution)**
3. \( 10 - x = 25 \) → **1 (one solution)**
4. \( 2(x + 4) = 2x + 5 + 3 \) → **4 (infinite solutions)**

So, the correct matches are:
- **1 (one solution)** for \( x - 7 = 7 - x \)
- **3 (no solution)** for \( 3(x + 5) = 3x + 5 \)
- **1 (one solution)** for \( 10 - x = 25 \)
- **4 (infinite solutions)** for \( 2(x + 4) = 2x + 5 + 3 \)
Answered by GPT-4o mini
To formulate the inequality that represents the student's situation, we need to account for the total cost of the calculator and the notebooks.

1. The calculator costs $24.
2. Each notebook costs $2, so if \( n \) represents the number of notebooks, then the cost for \( n \) notebooks would be \( 2n \).
3. The total amount of money the student can spend is $37.

The total cost of the calculator and the notebooks together can be expressed as:
\[ 24 + 2n \]

Since the student cannot spend more than $37, we can express this relationship as an inequality:
\[ 24 + 2n \leq 37 \]

To re-arrange this, we can subtract 24 from both sides:
\[ 2n \leq 37 - 24 \]
\[ 2n \leq 13 \]

This means the inequality states that the total cost of the calculator and notebooks must be less than or equal to $37.

Now, looking at the answer choices provided:

A). \( 24n + 2 \geq 37 \)
B). \( 37 > 2n + 24 \)
C). \( 37 \geq 2n + 24 \)
D). \( 37 < 2n + 24 \)

The correct option that matches our derived inequality \( 24 + 2n \leq 37 \) can be rewritten as:
\[ 37 \geq 2n + 24 \]

Thus, the correct answer is:
**C). \( 37 ≥ 2n + 24 \)**
Answered by GPT-4o mini
To solve for \( b \) in the equation

\[
A = \frac{1}{2} (a + b),
\]

we will follow these steps:

1. **Multiply both sides by 2** to eliminate the fraction:
\[
2A = a + b.
\]

2. **Isolate \( b \)** on one side of the equation by subtracting \( a \) from both sides:
\[
b = 2A - a.
\]

So, the solution is

\[
b = 2A - a.
\]

Now, let's match this with the provided options:
- A). \( b = \frac{1}{2}A - a \)
- B). \( b = 2A + a \)
- C). \( b = \frac{1}{2}A + a \)
- D). \( b = 2A - a \)

The correct answer is:
**D). \( b = 2A - a \)**.
Answered by GPT-4o mini
To analyze the steps provided and identify the property used to go from step 3 to step 4, we can take a closer look at those specific steps:

- **Step 3:** \( 5x - 6 = 19 \)
- **Step 4:** \( 5x - 6 + 6 = 19 + 6 \)

In step 4, the student adds 6 to both sides of the equation to isolate \( 5x \).

Thus, the property used here is:

**C). Addition Property of Equality.**

This is because the Addition Property of Equality states that if you add the same value to both sides of an equation, the equality remains valid.
Answered by GPT-4o mini
Let’s go through the provided steps and address each part of your question:

Original equation:
\[ x + 2(x + 1) = 17 \]

**Step 1:**
\[ x + 2x + 2 = 17 \]
This step combines the terms from the left-hand side.

1. **Identify the property used to arrive at Step 1:**
The property used in Step 1 is the **Distributive Property**. When the student multiplies 2 by both \( x \) and \( 1 \) in the expression \( 2(x + 1) \), they distribute the 2 to get \( 2x + 2 \).

---

**Step 2:**
\[ 3x + 2 = 17 \]
This step is correct, as \( x + 2x \) correctly combines to form \( 3x \).

**Step 3:**
\[ 3x = 15 \]
This is also correct, as it results from subtracting 2 from both sides of the equation:

From \( 3x + 2 = 17 \), subtracting 2 gives:
\[ 3x + 2 - 2 = 17 - 2 \]
which simplifies to:
\[ 3x = 15 \]

---

**Step 4:**
\[ x = 45 \]
This step includes a mistake. To solve for \( x \), the last step should have involved dividing both sides by 3.

2. **What step includes a mistake made from the previous step?**
Step 4 includes the mistake. The student incorrectly calculated \( x \) by not dividing \( 15 \) by \( 3 \). The correct operation should be:
\[ x = \frac{15}{3} \]

---

**What should have been the correct answer for \( x \)?**
The correct calculation for \( x \) is:
\[ x = \frac{15}{3} = 5 \]

So:
- **1. Identify the property used to arrive at Step 1:** **Distributive Property**
- **2. What step includes a mistake made from the previous step?** **Step 4**
- **3. What should have been the correct answer for \( x \)?** **5**
Answered by GPT-4o mini
The correct answer for \( x \) should be:

**5.**

This is obtained by solving \( 3x = 15 \) as follows:
\[
x = \frac{15}{3} = 5.
\]
Answered by GPT-4o mini
Yes, that's correct! The solution to the equation \( x + 2(x + 1) = 17 \) is \( x = 5 \).