Sara correctly solves a system of two lines equations and finds that the system has no solution. If one of the two equation is `(y)/(6)=(x)/(4)=1`, which could be the oher equation in this system?

Options

A. `y=(2)/(3)x+12`
B. `y=(3)/(2)x`
C. `y=-(3)/(2)x`
D. `y=(3)/(2)x+6`

Answer

To solve the problem, we need to find the other equation in the system that results in no solution when paired with the given equation. The given equation is:

\[ \frac{y}{6} = \frac{x}{4} = 1 \]

### Step 1: Rewrite the given equation
We can break down the given equation into two separate equations:

1. \(\frac{y}{6} = 1\)
2. \(\frac{x}{4} = 1\)

From these, we can solve for \(y\) and \(x\):

- From \(\frac{y}{6} = 1\), we multiply both sides by 6:
\[ y = 6 \]

- From \(\frac{x}{4} = 1\), we multiply both sides by 4:
\[ x = 4 \]

### Step 2: Identify the condition for no solution
For a system of two linear equations to have no solution, the lines represented by the equations must be parallel. This means they have the same slope but different y-intercepts.

### Step 3: Analyze the options
We need to check which of the provided options results in a line that is parallel to the line represented by the given equation.

The line from the given equation can be rewritten in slope-intercept form \(y = mx + b\). We already know that:

\[ y = 6 \quad \text{(a horizontal line)} \]

### Step 4: Check each option
We will check each option to see if it represents a line parallel to \(y = 6\):

1. **Option A:** \(y = 2/3 x + 12\)
- This has a slope of \(2/3\) (not parallel).

2. **Option B:** \(y = 3/2 x\)
- This has a slope of \(3/2\) (not parallel).

3. **Option C:** \(y = -1/2 x + 6\)
- This has a slope of \(-1/2\) (not parallel).

4. **Option D:** \(y = 6\)
- This has a slope of \(0\) (parallel to the given line).

### Conclusion
The only equation that results in no solution when paired with the given equation is:

\[ y = 6 \]