To solve the equation \( 2y = -3xy - 5 \) for \( y \) in terms of \( x \), we will rearrange the equation to isolate \( y \).
Starting with the given equation:
\[ 2y = -3xy - 5 \]
We will first move all terms involving \( y \) to one side of the equation. To do this, we can add \( 3xy \) to both sides:
\[ 2y + 3xy = -5 \]
Now, we can factor out \( y \) from the left side:
\[ y(2 + 3x) = -5 \]
Next, we can solve for \( y \) by dividing both sides by \( (2 + 3x) \):
\[ y = \frac{-5}{2 + 3x} \]
From this form, we can further manipulate it to express \( y \) in the form \( y = mx + b \).
To rewrite \( y = \frac{-5}{2 + 3x} \) in a linear form, we can perform long division or simply express it in a different way:
To create a function of the form \( y = mx + b \), we can simplify the expression. However, since your request suggests you want it in a specific form, we can focus on the desired form of the equation.
Assuming you want to represent it using the slope-intercept form and there's a misalignment with the output expression you noted, we can break it down further but it won't match with \( y = -\frac{5}{3} x + 2 \) if we solve correctly as per the algebraic process.
Next, since you mentioned plugging in \( x = 1\) later, let’s substitute \( x = 1 \):
\[ y = \frac{-5}{2 + 3(1)} = \frac{-5}{2 + 3} = \frac{-5}{5} = -1 \]
Thus, when \( x = 1 \), \( y = -1 \), which is consistent with your problem context where you evaluate the function and find \( (1, -1) \).
However, to clarify, the equation simplifies to \( y = -\frac{5}{3}x + 2 \) does not apply directly from our rearrangement here. If we specifically are resolving to the \( y = -\frac{5}{3}x + 2\), additional steps or context may modify this understanding.
To conclude, the correct expression holds:
\[ y = \frac{-5}{2 + 3x} \]
And when \( x = 1 \):
\[ y = -1 \]
If your expectation of getting explicitly \( y = -\frac{5}{3}x + 2 \) arises from a context of transformation or mismatch, it affirms that the calculated output doesn't directly align with prior expectations of alternative transformations.