Solve kx + 6 = 3x + 4k for x, treating k as a parameter.

Step-by-step solution

1. 1

   Gather the $x$ terms on one side. Subtract $3x$ from both sides:

   $kx-3x+6=4k\quad\Longrightarrow\quad x(k-3)+6=4k$

   Factoring $x$ out immediately exposes $k-3$ as the coefficient that decides everything.

2. 2

   Move the constant across. Subtract $6$:

   $x(k-3)=4k-6$

3. 3

   Divide, subject to the coefficient being nonzero. For $k\neq 3$:

   $x=\frac{4k-6}{k-3}$

4. 4

   Examine $k=3$ separately. The coefficient of $x$ vanishes and the original equation becomes

   $3x+6=3x+12\quad\Longrightarrow\quad 6=12$

   which is false, so there is no solution when $k=3$. Note this is a contradiction rather than an identity, because $4k-6=6\neq 0$ at $k=3$.

5. 5

   Check the formula at two parameter values. At $k=0$: $x=\tfrac{-6}{-3}=2$, and the original reads $0+6=6+0$ $\checkmark$. At $k=5$: $x=\tfrac{14}{2}=7$, and $35+6=41$ while $21+20=41$ $\checkmark$.