Question

-6y + 8 = y +9 - 7y - 1
solving steps properties used

Answer 1

To solve the equation \(-6y + 8 = y + 9 - 7y - 1\), we'll follow a systematic approach using properties of equality and arithmetic. Here's a step-by-step breakdown:

1. **Start with the original equation:**
\[
-6y + 8 = y + 9 - 7y - 1
\]

2. **Simplify the right side of the equation:**
Combine like terms on the right side:
\[
y - 7y = -6y
\]
\[
9 - 1 = 8
\]
So, the equation simplifies to:
\[
-6y + 8 = -6y + 8
\]

3. **Isolate terms:**
To isolate \(y\), we notice that both sides of the equation are now identical:
\[
-6y + 8 = -6y + 8
\]

4. **Subtract \(-6y\) from both sides:**
\[
8 = 8
\]

Since we have arrived at a true statement that \(8 = 8\), we understand this means that the original equation is an identity (true for all \(y\)). Therefore, the solution set consists of all real numbers.

### Summary of properties used:
- **Properties of equality:** We used the principle that if two expressions are equal, you can manipulate both sides of the equation equally (subtracting \(-6y\)).
- **Combining like terms:** We simplified the equation by combining like terms on the right side.
- **Identity:** Realized the equation simplifies to a statement that is always true.

### Conclusion:
The equation \(-6y + 8 = y + 9 - 7y - 1\) holds true for all values of \(y\), meaning all real numbers are solutions.

Answer 2

so infinite solutions?

Answer 3

Yes, that's correct! Since we found that the equation simplifies to a statement that is always true (\(8 = 8\)), it means that the equation holds true for all values of \(y\).

In this case, we have an infinite number of solutions. The solution set can be expressed as:

\[
y \in \mathbb{R}
\]

This notation means that \(y\) can take any real number value. Thus, there are infinitely many solutions to the equation.