Question

what are the answers to the following problems:
Problem 1: 3(2i+7)=6i+21
Problem 2: 3i+5=2i−7
Problem 3: 3i+2i−7=18
Problem 4: 3(2i−7)=15
Problem 5: 3i+5=3i+7

Answer 1

Let's solve each problem step by step.

**Problem 1:**
\[3(2i + 7) = 6i + 21\]

Distributing on the left side:
\[6i + 21 = 6i + 21\]

Both sides are equal, so the equation holds for all values of \( i \). The answer is **true** for all \( i \).

---

**Problem 2:**
\[3i + 5 = 2i - 7\]

Subtracting \(2i\) from both sides:
\[
3i - 2i + 5 = -7
\]
This simplifies to:
\[i + 5 = -7\]

Subtracting 5 from both sides:
\[i = -12\]

So, the solution is \(i = -12\).

---

**Problem 3:**
\[3i + 2i - 7 = 18\]

Combining like terms:
\[5i - 7 = 18\]

Adding 7 to both sides:
\[5i = 25\]

Dividing both sides by 5:
\[i = 5\]

So, the solution is \(i = 5\).

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**Problem 4:**
\[3(2i - 7) = 15\]

Distributing on the left side:
\[6i - 21 = 15\]

Adding 21 to both sides:
\[6i = 36\]

Dividing both sides by 6:
\[i = 6\]

So, the solution is \(i = 6\).

---

**Problem 5:**
\[3i + 5 = 3i + 7\]

Subtracting \( 3i \) from both sides:
\[5 = 7\]

This is not true, so the equation has **no solution**.

---

To summarize:
- **Problem 1:** True for all \( i \)
- **Problem 2:** \( i = -12 \)
- **Problem 3:** \( i = 5 \)
- **Problem 4:** \( i = 6 \)
- **Problem 5:** No solution.