Consider the system of linear equations
where is a constant.
For exactly one value of , the system has no solutions. What is that value of ?
A linear system with no solutions describes two lines that never cross. The lines must be parallel but distinct: one multiplier matches both variable coefficients, but that multiplier does not match the constants. Find the multiplier from the coefficients without a letter, use it to find the missing coefficient, and check the constants. Matching coefficients alone could give the same line.
Hints
- Hint 1
For parallel lines, one multiplier must turn both variable coefficients in one equation into those in the other. The coefficients are both known. What multiplier turns into ?
- Hint 2
Apply that same multiplier to the coefficient. It must turn into . Why would using a different multiplier for the term prevent the lines from being parallel?
- Hint 3
Matching variable terms isn't enough for no solutions. If the constants match under the same multiplier, the equations describe the same line and have infinitely many solutions.
Step-by-step
Match coefficients, then check the graph
Step 1Find the multiplier from the y terms
For no solution, the lines must be parallel: they run in the same direction and never meet. Find the multiplier from the terms in the two equations:
So multiply the second equation by to match the first equation's term.
- Step 2
Rule out the same line
Multiply the second equation's constant by that multiplier:
The first equation has , not . So if the variable terms match, these will be different parallel lines rather than the same line. Match variable coefficients, then check constants.
- Step 3
Find the missing x coefficient
Multiply the second equation's coefficient by the same multiplier so that becomes :
Using a positive multiplier here would undo the match between the terms.
- Step 4
Confirm that the lines never meet
The expression for gives . Type , then type both equations in Desmos. The graphs are separate parallel lines, with no intersection. Since the terms fixed the multiplier, no other value of can make the lines parallel. The system has no solutions when . Choice A.