Answer:
Step-by-step explanation:To find a value of \( k \) for which the system has only one solution, we need to ensure that the two equations represent lines that intersect at exactly one point.
The first equation is \( 2x + y = 7 \) and the second equation is \( y - kx = 3 \).
We can solve this system by substituting \( y \) from the second equation into the first equation:
From the second equation, rearrange to solve for \( y \):
\[ y = 3 + kx \]
Now, substitute this expression for \( y \) into the first equation:
\[ 2x + (3 + kx) = 7 \]
\[ 2x + 3 + kx = 7 \]
\[ (2 + k)x + 3 = 7 \]
Now, subtract 3 from both sides:
\[ (2 + k)x = 4 \]
Divide both sides by \( (2 + k) \):
\[ x = \frac{4}{2+k} \]
For the system to have a unique solution, the denominator \( (2 + k) \) cannot be equal to zero. So, we need to find the value of \( k \) that makes \( 2 + k \neq 0 \).
\[ 2 + k \neq 0 \]
\[ k \neq -2 \]
So, any value of \( k \) except \( -2 \) will result in a system with only one solution.
