Question

Tamara, Payton, Krish, and Haruki are analyzing the solution sets to the following linear equations. Tamara says that Equation A and Equation C have the same solution. Payton says Equation B and Equation C have the same solution. Krish says all three have the same solution. Haruki says that none of the equations have the same solution. Identify who is correct about the linear equations.

Equation A: 3(4−5h)−7=−12+2h
3
(
4
−
5
ℎ
)
−
7
=
−
12
+
2
ℎ


Equation B: 2(p−3)−1=6p+142
2
(
𝑝
−
3
)
−
1
=
6
𝑝
+
14
2


Equation C: 23v−3+83=−1
2
3
𝑣
−
3
+
8
3
=
−
1


(1 point)
Responses

Tamara
Tamara

Payton
Payton

Haruki
Haruki

Krish

Answer 1

Solve the inequality 45d≤24
4
5
𝑑
≤
24
.(1 point)
The solution to the inequality is d≤
𝑑
≤
.

Answer 2

To determine who is correct about the solution sets to the given equations, we need to solve each equation.

### Equation A:
\[ 3(4 - 5h) - 7 = -12 + 2h \]

1. Distribute the left side:
\[ 12 - 15h - 7 = -12 + 2h \]
2. Simplify:
\[ 5 - 15h = -12 + 2h \]
3. Move terms involving h to one side:
\[ 5 + 12 = 15h + 2h \]
\[ 17 = 17h \]
4. Solve for h:
\[ h = 1 \]

### Equation B:
\[ 2(p - 3) - 1 = 6p + 14 \]

1. Distribute:
\[ 2p - 6 - 1 = 6p + 14 \]
2. Simplify:
\[ 2p - 7 = 6p + 14 \]
3. Move terms involving p to one side:
\[ -7 - 14 = 6p - 2p \]
\[ -21 = 4p \]
4. Solve for p:
\[ p = -\frac{21}{4} \]

### Equation C:
\[ 23v - 3 + 83 = -1 \]

1. Combine like terms:
\[ 23v + 80 = -1 \]
2. Move the constant:
\[ 23v = -1 - 80 \]
\[ 23v = -81 \]
3. Solve for v:
\[ v = -\frac{81}{23} \]

### Summary of Solutions:
- Equation A: \( h = 1 \)
- Equation B: \( p = -\frac{21}{4} \)
- Equation C: \( v = -\frac{81}{23} \)

### Analysis of Claims:
- **Tamara**: Claims Equation A and Equation C have the same solution (FALSE)
- **Payton**: Claims Equation B and Equation C have the same solution (FALSE)
- **Krish**: Claims all three have the same solution (FALSE)
- **Haruki**: Claims that none of the equations have the same solution (TRUE)

### Conclusion:
**Haruki is correct**; none of the equations have the same solution.

Answer 3

To solve the inequality \( 45d \leq 24 \), follow these steps:

1. **Divide both sides by 45** to isolate \( d \):
\[
d \leq \frac{24}{45}
\]

2. **Simplify the fraction**:
\[
\frac{24}{45} = \frac{8}{15} \quad (\text{both the numerator and the denominator can be divided by 3})
\]

Thus, the solution to the inequality is:
\[
d \leq \frac{8}{15}
\]

In conclusion, the final answer is:
\( d \leq \frac{8}{15} \).