Question

How can i tell if an equation like 2x + 3 = 2x + 7 has one solution, no solutions, or infinite solutions?

Answer 1

What equations would fall under that

Answer 2

what about 4x+22=8x+10

Answer 3

3 (4x + 8) = 4 (3x + 6)?

Answer 4

To determine whether an equation like \(2x + 3 = 2x + 7\) has one solution, no solutions, or infinite solutions, you can follow these steps:

1. **Rearrange the equation:** Start by simplifying both sides of the equation to see if you can isolate \(x\).

\[
2x + 3 = 2x + 7
\]

2. **Subtract \(2x\) from both sides:**

\[
3 = 7
\]

3. **Analyze the result:**
- If you end up with a true statement (like \(5 = 5\)), it means that the original equation is true for all values of \(x\), indicating **infinite solutions**.
- If you end up with a false statement (like \(3 = 7\)), it indicates that there are **no solutions** because no value of \(x\) can satisfy the equation.
- If you had been able to isolate \(x\) and found a specific value (like \(x = 2\)), then there would be **one solution**.

In the case of \(2x + 3 = 2x + 7\), since we found \(3 = 7\), which is false, we conclude that there are **no solutions** to this equation.

Answer 5

Equations that would lead to a false statement when simplified (indicating **no solutions**) usually have the same variable terms on both sides but different constant terms. Here are some examples:

1. **Example 1:**
\[
4x + 2 = 4x + 5
\]
Simplifying gives:
\[
2 = 5 \quad (\text{false})
\]

2. **Example 2:**
\[
3x - 1 = 3x + 4
\]
Simplifying gives:
\[
-1 = 4 \quad (\text{false})
\]

3. **Example 3:**
\[
5x + 8 = 5x + 10
\]
Simplifying gives:
\[
8 = 10 \quad (\text{false})
\]

4. **Example 4:**
\[
7 - 2y = 7 - 3y
\]
Simplifying gives:
\[
2y = 3 \quad\Rightarrow\quad 7 = 7 + 2 \quad\Rightarrow\quad 7 = 7(1) \quad (\text{false, and leads to } 0 = 1)
\]

### Equations with Infinite Solutions

Equations that yield a true statement after simplification often have the same variable terms and the same constant terms, leading to **infinite solutions**:

1. **Example 1:**
\[
2x + 4 = 2x + 4
\]
This simplifies to \(4 = 4\), which is always true.

2. **Example 2:**
\[
3(x + 1) = 3x + 3
\]
This simplifies to \(3 = 3\), which is always true.

3. **Example 3:**
\[
5x + 6 - 2 = 5x + 4
\]
This simplifies to \(4 = 4\), which is always true.

4. **Example 4:**
\[
8 = 3 + 5
\]
This simplifies to \(8 = 8\), which is always true.

### Equations with One Solution

Equations that result in a specific value of \(x\) once simplified yield exactly **one solution**:

1. **Example 1:**
\[
x + 3 = 7
\]
This simplifies to \(x = 4\).

2. **Example 2:**
\[
2x - 4 = 10
\]
This simplifies to \(2x = 14\) and \(x = 7\).

3. **Example 3:**
\[
3y + 2 = 11
\]
This simplifies to \(3y = 9\) and \(y = 3\).

4. **Example 4:**
\[
4z + 6 = 18
\]
This simplifies to \(4z = 12\) and \(z = 3\).

By understanding these types of equations, you can categorize them based on their solutions.

Answer 6

Let’s analyze the equation \(4x + 22 = 8x + 10\) to determine whether it has one solution, no solutions, or infinite solutions.

1. **Rearrange the equation:** Start by simplifying both sides to get all terms involving \(x\) on one side and constant terms on the other side.

\[
4x + 22 = 8x + 10
\]

2. **Subtract \(4x\) from both sides:**
\[
22 = 8x - 4x + 10
\]
This simplifies to:
\[
22 = 4x + 10
\]

3. **Next, subtract \(10\) from both sides:**
\[
22 - 10 = 4x
\]
This simplifies to:
\[
12 = 4x
\]

4. **Now, divide both sides by \(4\):**
\[
x = \frac{12}{4}
\]
Thus,
\[
x = 3
\]

### Conclusion
The equation \(4x + 22 = 8x + 10\) has **one solution**, which is \(x = 3\).

This is an example of an equation with one unique solution, as we isolated \(x\) and found a specific value for it.

Answer 7

Let's analyze the equation \(3(4x + 8) = 4(3x + 6)\) step by step to determine whether it has one solution, no solutions, or infinite solutions.

1. **Distribute both sides of the equation:**

\[
3(4x + 8) = 4(3x + 6)
\]

- Left side:
\[
3 \cdot 4x + 3 \cdot 8 = 12x + 24
\]

- Right side:
\[
4 \cdot 3x + 4 \cdot 6 = 12x + 24
\]

So the equation becomes:
\[
12x + 24 = 12x + 24
\]

2. **Reorganize the equation:**

Now, let's see what we have:
\[
12x + 24 = 12x + 24
\]

3. **Subtract \(12x\) from both sides:**

\[
24 = 24
\]

### Conclusion
The statement \(24 = 24\) is always true for any value of \(x\).

This means that the original equation has **infinite solutions**; any value of \(x\) will satisfy the equation.