Write both equations as $a_1x + b_1y = c_1$ and $a_2x + b_2y = c_2$. For the lines to be parallel, how should $a_1,a_2$ and $b_1,b_2$ be related?

Use the condition for equal slopes to write an equation involving $k$ (a proportion between $a_1,a_2$ and $b_1,b_2$), then solve that equation for all possible values of $k$.

After you find the $k$-values that make the slopes equal, plug them back in and check whether the entire equations (including the constants) are multiples of each other, or only the left-hand sides are.

In Desmos, type k = 0 and tap the slider icon to create a slider for $k$. Then enter the two equations (k+1)x - 2y = 4 and 3x + (k-4)y = k. You will see two lines that change as you move the $k$ slider.

Move the $k$ slider and look for values where the two lines have the same slope (they never cross or they overlap). Note the $k$-values where the lines are parallel.

For each $k$ you found, check visually whether the lines are exactly on top of each other (infinitely many solutions) or are distinct parallel lines (no solution). Collect the $k$-values that give distinct parallel lines, then add those values outside Desmos to get the final answer.

For a system of two linear equations in $x$ and $y$:

• One solution: lines intersect once.
• No solution: lines are parallel but different (same slope, different intercepts).
• Infinitely many solutions: lines are exactly the same (all coefficients proportional).

So we want values of $k$ that make the two equations represent parallel, distinct lines.

Write the system in the form $a_1x + b_1y = c_1$ and $a_2x + b_2y = c_2$:

• First equation: $(k+1)x - 2y = 4$ → $a_1 = k+1$, $b_1 = -2$, $c_1 = 4$.
• Second equation: $3x + (k-4)y = k$ → $a_2 = 3$, $b_2 = k-4$, $c_2 = k$.

For the lines to be parallel, the ratios of the $x$ and $y$ coefficients must match:

This equation will give the $k$-values where the lines have the same slope.

Solve

Cross-multiply:

Expand the left side:

So

Factor the quadratic:

Thus the candidate values of $k$ that make the lines parallel are $k=1$ and $k=2$.

Now check whether, for each candidate $k$, the entire equations (including constants) are multiples of each other.

• If all three ratios $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$, and $\frac{c_1}{c_2}$ are equal → infinitely many solutions (same line).
• If only the first two match, but $\frac{c_1}{c_2}$ is different → no solution (parallel, distinct lines).

Check $k=1$:

• First equation: $(1+1)x - 2y = 4$ → $2x - 2y = 4$.
• Second equation: $3x + (1-4)y = 1$ → $3x - 3y = 1$.

Compare $(a_1,b_1)$ and $(a_2,b_2)$:

• $\frac{a_1}{a_2} = \frac{2}{3}$, $\frac{b_1}{b_2} = \frac{-2}{-3} = \frac{2}{3}$ (slopes match).
• But $\frac{c_1}{c_2} = \frac{4}{1} = 4$, which is not $\frac{2}{3}$.

So for $k=1$, the lines are parallel and different → no solution.

Check $k=2$:

• First equation: $(2+1)x - 2y = 4$ → $3x - 2y = 4$.
• Second equation: $3x + (2-4)y = 2$ → $3x - 2y = 2$.

Here $(a_1,b_1) = (3,-2)$ and $(a_2,b_2) = (3,-2)$, so the ratios of $a$ and $b$ are equal (both 1), but

So again, the lines are parallel but different → no solution for $k=2$ as well.

The system has no solution when $k=1$ or $k=2$.

We are asked for the sum of all possible values of $k$:

So the correct answer is 3.