Question

How many solutions? QuickChek
Which of the following is example of an equation with no solution
3X +4 equals 3X +4
4X +3 equals 3X +3
3X +4 equals 3X +3
3X +4 = 4X +3
Answered by GPT-4o mini
To determine which of the given equations has no solution, let's analyze each one:

1. **\(3X + 4 = 3X + 4\)**
This equation is true for all values of \(X\) because both sides are identical. Therefore, it has infinitely many solutions.

2. **\(4X + 3 = 3X + 3\)**
If we subtract \(3X\) from both sides, we get:
\[
4X - 3X + 3 = 3
\]
Simplifying yields:
\[
X + 3 = 3 \implies X = 0
\]
This equation has exactly one solution (X = 0).

3. **\(3X + 4 = 3X + 3\)**
Subtracting \(3X\) from both sides gives us:
\[
4 = 3
\]
This statement is false, indicating there are no values of \(X\) that satisfy this equation. Therefore, it has no solution.

4. **\(3X + 4 = 4X + 3\)**
Rearranging gives us:
\[
3X + 4 - 4X = 3 \implies -X + 4 = 3
\]
Simplifying yields:
\[
-X = -1 \implies X = 1
\]
This equation has exactly one solution (X = 1).

In summary, the equation **\(3X + 4 = 3X + 3\)** is the example of an equation with no solution.
Answer
Which of the following is an example of an equation with exactly one solution?
3X +4 = 4X +3
4X +3 = 4X +3
3X +4 = 3X +3
3X +4 = 3X +4
Answered by GPT-4o mini
To identify which of the given equations has exactly one solution, let's examine each one:

1. **\(3X + 4 = 4X + 3\)**
Rearranging gives:
\[
3X + 4 - 4X = 3 \implies -X + 4 = 3
\]
Simplifying this:
\[
-X = -1 \implies X = 1
\]
This equation has exactly **one solution**: \(X = 1\).

2. **\(4X + 3 = 4X + 3\)**
This equation is an identity; both sides are exactly the same. Thus, it has **infinitely many solutions**.

3. **\(3X + 4 = 3X + 3\)**
Subtracting \(3X\) from both sides gets us:
\[
4 = 3
\]
This is a contradiction, indicating that there are **no solutions**.

4. **\(3X + 4 = 3X + 4\)**
Similar to the second equation, this is also an identity and has **infinitely many solutions**.

In conclusion, the equation **\(3X + 4 = 4X + 3\)** has exactly one solution, which is \(X = 1\).
Answer
Solve X determine if the equation has one non-or infinite solutions.
3 (X +2) -6 = 3X
The equation has no solution
The equation has infinite solutions
The equation has one solution X = 3
The equation has one solution X = -3
Answered by GPT-4o mini
Let's solve the equation step by step to determine how many solutions it has.

**Given equation:**
\[
3(X + 2) - 6 = 3X
\]

1. **Distribute the 3 on the left side:**
\[
3X + 6 - 6 = 3X
\]
This simplifies to:
\[
3X = 3X
\]

2. **Now, we can subtract \(3X\) from both sides:**
\[
3X - 3X = 3X - 3X \implies 0 = 0
\]

This equation \(0 = 0\) is always true, meaning that all values of \(X\) satisfy the equation.

### Conclusion:
Since the equation is true for all values of \(X\), it has **infinite solutions**.

So the correct answer is:
- **The equation has infinite solutions.**
Answer
Solve for X determine if the equation has one none or infinite solutions
4X = 3 (X +5) -15
The equation has one solution X = -10
The equation has no solution
The equation has infinite solutions
The equation has one solution = 0
Answered by GPT-4o mini
Let's solve the equation step by step to determine how many solutions it has.

**Given equation:**
\[
4X = 3(X + 5) - 15
\]

1. **Expand the right side:**
\[
4X = 3X + 15 - 15
\]
Simplifying the right side, we get:
\[
4X = 3X + 0 \implies 4X = 3X
\]

2. **Subtract \(3X\) from both sides:**
\[
4X - 3X = 3X - 3X \implies X = 0
\]

### Conclusion:
The equation has one solution:
- **\(X = 0\)**.

So, the correct answer from your options is:
- **The equation has one solution \(X = 0\)**.
Answer
Solve for extra to determine if the equation is one none or infinite solutions
5(2x+2)-7X = 3X +6
The equation has infinite solutions
The equation has one solution, X = 10
The equation has no solution
The equation has one solution X = 6
Answered by GPT-4o mini
Let's solve the equation step by step to determine how many solutions it has.

**Given equation:**
\[
5(2x + 2) - 7X = 3X + 6
\]

1. **Distribute the 5 on the left side:**
\[
10x + 10 - 7x = 3x + 6
\]

2. **Combine like terms on the left side:**
\[
(10x - 7x) + 10 = 3x + 6
\]
This simplifies to:
\[
3x + 10 = 3x + 6
\]

3. **Subtract \(3x\) from both sides:**
\[
3x + 10 - 3x = 3x + 6 - 3x \implies 10 = 6
\]

This statement \(10 = 6\) is a contradiction, meaning there is no value of \(x\) that can satisfy the equation.

### Conclusion:
The equation has **no solution**.

So, the correct answer from your options is:
- **The equation has no solution.**