Question

-9 + 6x = -3(3 - 2x)
solving steps and property used
Answer
2 more SOLVING STEPS AND PROPERTIES
Answer
So X = ??
Answer
how many solutions
Answered by GPT-4o mini
To solve the equation \(-9 + 6x = -3(3 - 2x)\), follow these steps:

1. **Distribute** on the right side of the equation:
\[
-3(3 - 2x) = -3 \cdot 3 + (-3) \cdot (-2x) = -9 + 6x
\]

Now the equation looks like this:
\[
-9 + 6x = -9 + 6x
\]

2. **Combine like terms**:
Since both sides of the equation are equal \(-9 + 6x = -9 + 6x\), this equation is an identity, meaning it holds true for all values of \(x\).

3. **Conclusion**:
Therefore, the solution to the equation is:
\[
\text{All real numbers (identity)}
\]

### Properties Used:

1. **Distributive Property**: This was used to expand \(-3(3 - 2x)\) into \(-9 + 6x\).

2. **Combining Like Terms**: Both sides of the equation had similar expressions which were already equal.

Since the two sides are the same for any value of \(x\), every value of \(x\) is a solution to the given equation.
Answered by GPT-4o mini
Sure! Let’s add two more steps with detailed explanations of the properties used when solving the equation \(-9 + 6x = -3(3 - 2x)\).

### Steps and Properties Used

1. **Start with the original equation**:
\[
-9 + 6x = -3(3 - 2x)
\]

2. **Apply the Distributive Property**:
Distribute \(-3\) on the right side:
\[
-3(3) + (-3)(-2x) = -9 + 6x
\]
This step leads us to:
\[
-9 + 6x = -9 + 6x
\]

3. **Identify Equivalence**:
Since both sides of the equation are identical, we can establish that:
\[
-9 + 6x \equiv -9 + 6x
\]
This indicates that the equation is true for all values of \(x\).

4. **Conclusion/Systematic Verification**:
To formally conclude that any value of \(x\) is a solution, we can:
- Choose a specific value for \(x\) (e.g., \(x=0\)):
\[
-9 + 6(0) = -9 \quad \text{and} \quad -3(3 - 2(0)) = -9
\]
- Check another value like \(x=1\):
\[
-9 + 6(1) = -3 \quad \text{and} \quad -3(3 - 2(1)) = -3
\]
This validity check confirms that the equation maintains equality for all tested values of \(x\).

### Properties Recap

- **Distributive Property**: Used in the step where we multiplied \(-3\) with both terms inside the parentheses.

- **Identity Property of Equality**: This property tells us that if both sides of an equation are equal for any value of \(x\), the equation is an identity.

- **Substitution Property**: In the verification step, substituting specific values of \(x\) into the original equation showed that it holds true across those values, reinforcing that the solutions are all real numbers.

In conclusion, the equation \(-9 + 6x = -3(3 - 2x)\) is true for all real values of \(x\) due to its identity nature.
Answered by GPT-4o mini
The equation \(-9 + 6x = -3(3 - 2x)\) simplifies to an identity, meaning that it holds true for all values of \(x\). Therefore, we can conclude:

\[
x = \text{any real number}
\]

In other words, the solution set is:

\[
x \in \mathbb{R} \quad (\text{all real numbers})
\]

So there isn't a specific value of \(x\); rather, \(x\) can take any value within the set of real numbers.
Answered by GPT-4o mini
The equation \(-9 + 6x = -3(3 - 2x)\) simplifies to an identity, meaning it is true for all values of \(x\).

Since the equation holds true for every real number \(x\), there are infinitely many solutions.

### Summary
- The number of solutions to the equation is **infinite**.
- All real numbers are solutions, as expressed mathematically: \(x \in \mathbb{R}\).