Question

For each linear equation, indicate whether the equation has no solution, one solution. or infinitely many solutions.

This problem requires you to show your work.

(3 point.
3(x+3)=7+3x+6−x
8(x+2)=5x+16+3x
6x+2−4x=2(x+2)

Answer 1

Let's go through each equation step by step.

### Equation 1: \(3(x + 3) = 7 + 3x + 6 - x\)

1. **Expand both sides:**
\[
3(x + 3) = 3x + 9
\]
\[
7 + 3x + 6 - x = 3x + 13 - x = 2x + 13
\]

So we rewrite the equation:
\[
3x + 9 = 2x + 13
\]

2. **Get all \(x\) terms on one side and constant terms on the other:**
\[
3x - 2x = 13 - 9
\]
\[
x = 4
\]

**Conclusion:** This equation has **one solution**, \(x = 4\).

---

### Equation 2: \(8(x + 2) = 5x + 16 + 3x\)

1. **Expand both sides:**
\[
8(x + 2) = 8x + 16
\]
\[
5x + 16 + 3x = 8x + 16
\]

So we rewrite the equation:
\[
8x + 16 = 8x + 16
\]

2. **Simplifying:**
\[
8x + 16 - 8x - 16 = 0
\]
\[
0 = 0
\]

**Conclusion:** This equation has **infinitely many solutions** (the two sides are identical).

---

### Equation 3: \(6x + 2 - 4x = 2(x + 2)\)

1. **Combine like terms on the left:**
\[
6x - 4x + 2 = 2x + 2
\]
\[
2x + 2 = 2x + 2
\]

2. **Simplifying:**
\[
2x + 2 - 2x - 2 = 0
\]
\[
0 = 0
\]

**Conclusion:** This equation also has **infinitely many solutions** (the two sides are identical).

---

### Summary of Solutions:
1. **First equation:** One solution \( (x = 4) \)
2. **Second equation:** Infinitely many solutions
3. **Third equation:** Infinitely many solutions

Answer 2

hiii

Answer 3

one has to be no answer