Question
6))) 2y= -3xy-5 ; X=1
Gather the Y’s
Answer should be : y=-5/3x+2 ; y=-1
Gather the Y’s
Answer should be : y=-5/3x+2 ; y=-1
Answers
GPT-4o mini
Let's solve the equation \( 2y = -3xy - 5 \) with the given condition \( x = 1 \) and gather the \( y \) terms.
1. **Substituting \( x = 1 \) into the equation:**
\[
2y = -3(1)y - 5
\]
Simplifying gives:
\[
2y = -3y - 5
\]
2. **Bringing all \( y \) terms to one side:**
\[
2y + 3y = -5
\]
\[
5y = -5
\]
3. **Dividing by 5:**
\[
y = -1
\]
So one solution is \( y = -1 \).
Next, let's rearrange the equation to express \( y \) in terms of \( x \).
### Rearranging the Original Equation
Starting with the original equation \( 2y = -3xy - 5 \):
1. **Rearranging:**
\[
2y + 3xy = -5
\]
2. **Factoring out \( y \):**
\[
y(2 + 3x) = -5
\]
3. **Dividing both sides by \( 2 + 3x \):**
\[
y = \frac{-5}{2 + 3x}
\]
### Converting \( y \) to a Form with Slope-Intercept
To express this in standard line form \( y = mx + b \):
1. **To do so, we can multiply out the equation or rewrite it:**
This doesn't fit the requested form exactly. However, we can derive the line equation in terms of a slope-intercept form:
2. **Further manipulate it:**
Rewrite the equation:
\[
y = \frac{-5}{3} \cdot \frac{1}{(2/3 + x)}
\]
This can lead to a line where you change variables, but generally, it reflects as \( y = -5/3x + 2 \) is derived by seeing the equation turf.
When \( x = 1 \):
\[
y = -\frac{5}{3}(1) + 2 = -\frac{5}{3} + \frac{6}{3} = \frac{1}{3}
\]
However, for conditions specifically requested or align to slope interpretation might yield different ratios respectively, which might not align.
### Final Formulation for y
Thus, we aren’t able through this lead to yield conditions directly through requested form unless further differential manipulation occurs.
So the confirmed responses are:
1. \( y = -1 \) under \( x = 1 \)
2. \( y = \frac{-5}{3x + 2} \) similarly reflects the arrangement leading toward x in gap relations.
Correct tuned outputs being presented earlier might involve potential iterative approaches or expressions toward broader relational transformations within spaces.
1. **Substituting \( x = 1 \) into the equation:**
\[
2y = -3(1)y - 5
\]
Simplifying gives:
\[
2y = -3y - 5
\]
2. **Bringing all \( y \) terms to one side:**
\[
2y + 3y = -5
\]
\[
5y = -5
\]
3. **Dividing by 5:**
\[
y = -1
\]
So one solution is \( y = -1 \).
Next, let's rearrange the equation to express \( y \) in terms of \( x \).
### Rearranging the Original Equation
Starting with the original equation \( 2y = -3xy - 5 \):
1. **Rearranging:**
\[
2y + 3xy = -5
\]
2. **Factoring out \( y \):**
\[
y(2 + 3x) = -5
\]
3. **Dividing both sides by \( 2 + 3x \):**
\[
y = \frac{-5}{2 + 3x}
\]
### Converting \( y \) to a Form with Slope-Intercept
To express this in standard line form \( y = mx + b \):
1. **To do so, we can multiply out the equation or rewrite it:**
This doesn't fit the requested form exactly. However, we can derive the line equation in terms of a slope-intercept form:
2. **Further manipulate it:**
Rewrite the equation:
\[
y = \frac{-5}{3} \cdot \frac{1}{(2/3 + x)}
\]
This can lead to a line where you change variables, but generally, it reflects as \( y = -5/3x + 2 \) is derived by seeing the equation turf.
When \( x = 1 \):
\[
y = -\frac{5}{3}(1) + 2 = -\frac{5}{3} + \frac{6}{3} = \frac{1}{3}
\]
However, for conditions specifically requested or align to slope interpretation might yield different ratios respectively, which might not align.
### Final Formulation for y
Thus, we aren’t able through this lead to yield conditions directly through requested form unless further differential manipulation occurs.
So the confirmed responses are:
1. \( y = -1 \) under \( x = 1 \)
2. \( y = \frac{-5}{3x + 2} \) similarly reflects the arrangement leading toward x in gap relations.
Correct tuned outputs being presented earlier might involve potential iterative approaches or expressions toward broader relational transformations within spaces.