Ben correctly solves a system of two linear equations and finds that the system has an infinite number of solutions. If one of the two equations is `3(x+y)=6-x`, which could be the other equation in this system?

Options

A. `y=(3)/(4)x+2`
B. `y=(4)/(3)x`
C. `y=-(4)/(3)x+2`
D. `y=-(4)/(3)x+6`

Answer

To solve the problem step by step, we need to find another equation that, when combined with the given equation, results in a system of equations with an infinite number of solutions. The given equation is:

\[ 3(x + y) = 6 - x \]

### Step 1: Convert the given equation into standard form

First, we simplify the equation:

1. Distribute the 3 on the left side:
\[ 3x + 3y = 6 - x \]

2. Move \(x\) to the left side:
\[ 3x + x + 3y = 6 \]

3. Combine like terms:
\[ 4x + 3y = 6 \]

Now, we have the equation in standard form:
\[ 4x + 3y = 6 \]

### Step 2: Identify the condition for infinite solutions

For a system of two linear equations to have an infinite number of solutions, the two equations must be equivalent. This means that the ratios of the coefficients must be equal:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \]
where \(a_1, b_1, c_1\) are the coefficients from the first equation and \(a_2, b_2, c_2\) are from the second equation.

From our equation \(4x + 3y = 6\), we have:
- \(a_1 = 4\)
- \(b_1 = 3\)
- \(c_1 = 6\)

### Step 3: Analyze the options

Now we need to convert each of the given options into standard form and check which one satisfies the condition for infinite solutions.

1. **Option 1:** \(y = \frac{3}{4}x + 2\)
- Rearranging gives:
\[ 3x - 4y = -8 \quad \text{(or } 3x - 4y = 2\text{)} \]

2. **Option 2:** \(y = \frac{4}{3}x\)
- Rearranging gives:
\[ 4x - 3y = 0 \]

3. **Option 3:** \(y = -\frac{4}{3}x + 2\)
- Rearranging gives:
\[ 4x + 3y = 6 \]

4. **Option 4:** \(y = -\frac{4}{3}x + 6\)
- Rearranging gives:
\[ 4x + 3y = 18 \]

### Step 4: Compare the equations

Now we compare the coefficients:

- For **Option 1**: \(3x - 4y = 2\) does not match.
- For **Option 2**: \(4x - 3y = 0\) does not match.
- For **Option 3**: \(4x + 3y = 6\) matches exactly with our original equation.
- For **Option 4**: \(4x + 3y = 18\) does not match.

### Conclusion

The only option that satisfies the condition for infinite solutions is **Option 3**:

\[ y = -\frac{4}{3}x + 2 \]