`3(r+2s)=r+s`
If `(r, s)` is a solution to the equation above and `sne0`, what is the value of `(r)/(s)`?

Options

A. `-(5)/(2)`
B. `-(2)/(5)`
C. `(2)/(3)`
D. `(3)/(2)`

Answer

To solve the equation \( 3(r + 2s) = r + s \) and find the value of \( \frac{r}{s} \), we can follow these steps:

### Step 1: Expand the left side of the equation
Start by distributing the 3 on the left side:
\[ 3(r + 2s) = 3r + 6s \]
So, the equation becomes:
\[ 3r + 6s = r + s \]

### Step 2: Rearrange the equation
Next, we want to get all terms involving \( r \) on one side and all terms involving \( s \) on the other side. We can do this by subtracting \( r \) and \( s \) from both sides:
\[ 3r + 6s - r - s = 0 \]
This simplifies to:
\[ (3r - r) + (6s - s) = 0 \]
Which gives us:
\[ 2r + 5s = 0 \]

### Step 3: Isolate \( r \)
Now, we can isolate \( r \):
\[ 2r = -5s \]
Dividing both sides by 2 gives:
\[ r = -\frac{5}{2}s \]

### Step 4: Find \( \frac{r}{s} \)
Now, we can find \( \frac{r}{s} \):
\[ \frac{r}{s} = \frac{-\frac{5}{2}s}{s} \]
Since \( s \neq 0 \), we can cancel \( s \):
\[ \frac{r}{s} = -\frac{5}{2} \]

### Final Answer
Thus, the value of \( \frac{r}{s} \) is:
\[ \frac{r}{s} = -\frac{5}{2} \]

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