Some system questions never ask you to solve anything. Instead they hand you two equations with an unknown constant in one of them and ask which value of that constant gives the system no solution. It is a graph question in disguise: no solution means the two lines are parallel and never cross.
Two linear equations describe two lines. If the lines cross once, the system has exactly one solution. If they lie on top of each other, every point works and there are infinitely many solutions. If they are parallel and separate, they never touch, so there is no solution at all. Every no-solution question is really asking you to force the parallel case.
Parallel means equal slopes. The fastest route is to put both equations in slope-intercept form and set the slopes equal. Take the system 3x plus 2y equals 8 and 9x plus ky equals 5. Solving the first for y gives y equals negative three halves x plus 4, so the slope is negative three halves. Solving the second gives y equals negative 9 over k times x plus 5 over k, so the slope is negative 9 over k. Setting negative three halves equal to negative 9 over k gives 3k equals 18, so k equals 6.
There is a faster version once you trust it. If you write both equations in the form ax plus by equals c, the lines are parallel exactly when the coefficient ratios match but the constant ratio does not. In the example, 9 divided by 3 is 3, so for the y coefficients to scale the same way you need k divided by 2 to equal 3, which again gives k equals 6. This coefficient-ratio move avoids fractions entirely and is usually the cleaner scratch work.
The last step is the one people skip. After you find the constant, check that the constants on the right side do not scale by the same factor. With k equal to 6 the second equation becomes 9x plus 6y equals 5. Tripling the first equation gives 9x plus 6y equals 24, and 24 is not 5, so the lines really are separate. If the right sides had matched, the two equations would describe the same line and the answer would be infinitely many solutions instead of none.
If the question gives you numbers with no unknown constant and asks how many solutions the system has, run the same comparison in reverse. Compare the two slopes first. Different slopes means one solution and you are done. Same slopes sends you to the constants, which split parallel from identical.
The trap
Finding the constant that makes the coefficients proportional and stopping there, without checking the constant terms, so an infinitely-many-solutions system gets called no solution.