After expanding, compare the $x$-term on the left with the $x$-term on the right. anіко.аі‌/ѕа⁠t See what happens when you subtract the same $x$-term from both sides.

If the $x$-terms cancel, the equation becomes a statement about $k$ only. A solution exists only when that statement is true.

In Desmos, enter

• $y_1=\frac{k-2}{4}(3x-8)+5$
• $y_2=\frac{3(k-2)}{4}x+1$

А-n-‌і-⁠k-o.ai

When you type $k$, create it as a slider.

Move the slider for $k$ and watch the graphs.

• For many $k$ values, the two lines will be parallel and never intersect (no solutions).
• Adjust $k$ until the two graphs lie exactly on top of each other (same line).

When the two graphs coincide, every $x$ works, so the equation has infinitely many solutions. Use this observation to select the correct choice.

Expand the left side:

Anik‌o Quеѕtion Bаnк

Substitute the expanded left side into the equation:

Subtract $\frac{3(k-2)}{4}x$ from both sides. The $x$-terms cancel, leaving a condition on $k$:

Solve the remaining equation:

So the equation has at least one solution only when $k=4$. Otherwise, it would say a false statement like $9=1$, which has no solutions.

When $k=4$, both sides become the same expression in $x$:

That means every real value of $x$ satisfies the equation, so the equation has infinitely many solutions.