Question 569·Hard·Algebra

The sum-of-roots rule gives the requested sum without solving for each value.

A solution is a point on both lines. To have no solution, the lines must be parallel but distinct. In equations of the form $Ax+By=C$, parallel lines have proportional variable coefficients, so their cross-products match:

Multiply the brackets, combining $-k+2k$ into $+k$:

Subtract $12$ from both sides to put the quadratic equation in standard form:

For a quadratic $ak^2+bk+c=0$, a positive discriminant, $b^2-4ac$, means it has two real roots. Here $a=1$, $b=1$, and $c=-14$. Type $1^2-4(1)(-14)$ into Desmos; it prints $57$, so:

If the lines overlapped, multiplying the first equation by $\frac{9}{12}=\frac34$ would reproduce the second in every term. Match the $x$ coefficients:

Divide by $\frac34$:

Subtract $2$:

The same multiplier must also match the $y$ coefficients:

Multiply:

Add $1$:

One value of $k$ cannot be both $\frac{10}{3}$ and $\frac{13}{4}$. So neither candidate makes the lines overlap; both give no solution.

For a quadratic $ak^2+bk+c=0$, the sum of its roots is $-\frac ba$. In $k^2+k-14=0$, $a=1$ and $b=1$, so the two no-solution values sum to $-\frac11=-1$. The sum of all possible values of $k$ is $-1$. Choice B.