Question

-9+6x=-3(3-2x)
show the properties you used in order at the end
solve with 4 steps

Answer 1

well then what would we ill with 3 and 4?

Answer 2

To solve the equation \(-9 + 6x = -3(3 - 2x)\), we'll follow these four steps:

### Step 1: Distribute on the right side
Distribute \(-3\) into the parenthesis:
\[
-9 + 6x = -3 \cdot 3 + (-3)(-2x)
\]
This simplifies to:
\[
-9 + 6x = -9 + 6x
\]

### Step 2: Combine like terms
Now, we can add \(9\) to both sides to eliminate \(-9\) on both sides:
\[
-9 + 9 + 6x = -9 + 9 + 6x
\]
This simplifies to:
\[
6x = 6x
\]

### Step 3: Analyze the equation
Since both sides of the equation are equal (\(6x = 6x\)), this tells us that the equation is true for any value of \(x\). Thus, \(x\) can be any real number.

### Step 4: State the solution
The solution to the equation is:
\[
\text{All real numbers}
\]

### Properties Used:
1. **Distributive Property**: \(-3(3 - 2x) = -9 + 6x\)
2. **Addition Property of Equality**: Adding \(9\) to both sides to isolate terms.
3. **Identity Property**: Recognizing that \(6x = 6x\) implies all values are valid.
4. **General Solution Concept**: Understanding that an equation \(A = A\) (like \(6x = 6x\)) means infinite solutions, or all real numbers.

Answer 3

It seems there was a misunderstanding in my previous response regarding the solution and properties. Let's clarify and provide a concise breakdown:

### Re-solving the Equation:
Given the equation:
\[
-9 + 6x = -3(3 - 2x)
\]

### Step 1: Distribute
Distribute \(-3\) on the right side:
\[
-9 + 6x = -9 + 6x
\]

### Step 2: Examine the Equation
At this point, we see:
\[
-9 + 6x = -9 + 6x
\]

Both sides are identical, which tells us something.

### Step 3: Conclusion
Since both sides of the equation are equal, this means the equation holds true for all values of \(x\). Thus, the solution is:
\[
\text{All real numbers}
\]

### Properties We Used:
1. **Distributive Property**: To remove parentheses by distributing \(-3\).
2. **Identity Property of Equality**: To see that the two sides of the equation simplify to the same expression.
3. **Infinite Solutions Concept**: Recognizing that when any variable expression equals itself, it indicates infinite solutions, meaning all real numbers satisfy the equation.

Steps 3 and 4 from my previous response could be considered redundant since they reached the same conclusion that the original equation does not restrict \(x\) in any way. Hence, all \(x\) values are valid solutions.