Question

Create a single variable linear equation that has no solution. Solve the equation algebraically to prove that it does not have a solution.
Create a single variable linear equation that has one solution. Solve the equation algebraically to prove that there is one distinct solution for the equation.
Create a single variable linear equation that has infinitely many solutions. Solve the equation algebraically to prove that there is an infinite number of solutions for the equation

Answer 1

can you make it less advanced

Answer 2

can you use less words

Answer 3

### 1. Linear Equation with No Solution

**Equation:**
\[ 2x + 3 = 2x - 4 \]

**Solving the Equation:**

1. Subtract \(2x\) from both sides:
\[
2x + 3 - 2x = 2x - 4 - 2x
\]
This simplifies to:
\[
3 = -4
\]

This statement is false, indicating that the original equation has **no solution**.

### 2. Linear Equation with One Solution

**Equation:**
\[ 3x - 5 = 4 \]

**Solving the Equation:**

1. Add \(5\) to both sides:
\[
3x - 5 + 5 = 4 + 5
\]
This simplifies to:
\[
3x = 9
\]

2. Divide by \(3\):
\[
x = \frac{9}{3}
\]
Thus,
\[
x = 3
\]

This demonstrates that the equation has **one solution**, which is \(x = 3\).

### 3. Linear Equation with Infinitely Many Solutions

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

**Solving the Equation:**

1. Expand the right side:
\[
4x + 2 = 4x + 2
\]

2. Subtract \(4x\) from both sides:
\[
4x + 2 - 4x = 4x + 2 - 4x
\]
This simplifies to:
\[
2 = 2
\]

This statement is always true, regardless of the value of \(x\). Therefore the equation has **infinitely many solutions**, meaning any value of \(x\) satisfies the equation.

Answer 4

Sure! Let's simplify the explanations so they're more straightforward.

### 1. Linear Equation with No Solution

**Equation:**
\[ 2x + 3 = 2x - 4 \]

**Solving the Equation:**

1. Start with the equation:
\[ 2x + 3 = 2x - 4 \]

2. Subtract \( 2x \) from both sides:
\[ 3 = -4 \]

This is clearly not true (3 does not equal -4). Since we ended up with a false statement, this equation has **no solution**.

---

### 2. Linear Equation with One Solution

**Equation:**
\[ 3x - 5 = 4 \]

**Solving the Equation:**

1. Start with the equation:
\[ 3x - 5 = 4 \]

2. Add \( 5 \) to both sides:
\[ 3x = 9 \]

3. Now, divide by \( 3 \):
\[ x = 3 \]

So, the equation has **one solution**, which is \( x = 3 \).

---

### 3. Linear Equation with Infinitely Many Solutions

**Equation:**
\[ 4x + 2 = 4x + 2 \]

**Solving the Equation:**

1. Start with the equation:
\[ 4x + 2 = 4x + 2 \]

2. Notice that both sides are exactly the same.

This means that the statement is always true—no matter what value \( x \) takes. Therefore, this equation has **infinitely many solutions** because any value for \( x \) will work!

Answer 5

Of course! Here are the simplified versions with fewer words:

### 1. Linear Equation with No Solution

**Equation:**
\[ 2x + 3 = 2x - 4 \]

**Solution:**

1. Subtract \( 2x \):
\[ 3 = -4 \] (false)

**Conclusion:** No solution.

---

### 2. Linear Equation with One Solution

**Equation:**
\[ 3x - 5 = 4 \]

**Solution:**

1. Add \( 5 \):
\[ 3x = 9 \]

2. Divide by \( 3 \):
\[ x = 3 \]

**Conclusion:** One solution: \( x = 3 \).

---

### 3. Linear Equation with Infinitely Many Solutions

**Equation:**
\[ 4x + 2 = 4x + 2 \]

**Solution:** Both sides are the same.

**Conclusion:** Infinitely many solutions. Any \( x \) works!