`4x+6y=12`
`y=80-kx`
For what value of k does the system of equation above have no solution?

Options

A. `-(3)/(2)`
B. `0`
C. `(2)/(3)`
D. `4`

Answer

To determine the value of \( k \) for which the system of equations has no solution, we need to analyze the given equations:

1. **First Equation**: \( 4x + 6y = 12 \)
2. **Second Equation**: \( y = 80 - kx \)

### Step 1: Convert the first equation into slope-intercept form (y = mx + b)

To convert the first equation into the slope-intercept form, we need to isolate \( y \):

\[ 6y = 12 - 4x \]

Now, divide every term by 6:

\[ y = -\frac{4}{6}x + 2 \]

Simplifying the slope:

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

### Step 2: Identify the slope of the first equation

From the equation \( y = -\frac{2}{3}x + 2 \), we can see that the slope (m) of the first equation is:

\[ m_1 = -\frac{2}{3} \]

### Step 3: Identify the slope of the second equation

The second equation is already in slope-intercept form:

\[ y = 80 - kx \]

Here, the slope (m) of the second equation is:

\[ m_2 = -k \]

### Step 4: Set the slopes equal for parallel lines

For the system of equations to have no solution, the lines must be parallel. This occurs when their slopes are equal:

\[ -\frac{2}{3} = -k \]

### Step 5: Solve for \( k \)

To find \( k \), we can multiply both sides by -1:

\[ \frac{2}{3} = k \]

### Conclusion

Thus, the value of \( k \) for which the system of equations has no solution is:

\[ \boxed{\frac{2}{3}} \]

---