In the linear equation , is a constant. For which value of does the equation have no solution?
A linear equation has no solution when its -terms cancel but the remaining numbers disagree. If a question asks you to choose a constant that makes this happen, match the coefficients (the expressions multiplying ), then check the constants. You can graph each side in Desmos to confirm that the lines never meet. Matching coefficients alone is not enough: if the constants match too, every works.
Hints
- Hint 1
A coefficient is what multiplies . For a linear equation to have no solution, the coefficients on both sides must match so the -terms cancel. What are those coefficients here?
- Hint 2
Once you've matched the -coefficients, check the constant terms, the parts without . If those match too, every works; if they differ, no works.
- Hint 3
Graph each side as its own line after finding . Lines whose -coefficients match rise by the same amount for each increase of in . Do these lines have different starting heights?
Step-by-step
Match coefficients, then check the constants
Step 1Make the -terms cancel
The coefficient of is what multiplies it: on the left and on the right. For no solution, the -terms must cancel, so set those coefficients equal:
- Step 2
Find the candidate value
Subtract from both sides:
This makes the -terms cancel, but don't stop there. The constants decide whether no works or every works.
- Step 3
Check that the two sides never meet
Type , then graph each side on its own line: and . Desmos shows and parallel lines. Both have coefficient , but their -intercepts, their heights at , are and . Equal -coefficients and unequal constants mean no solution. So the required value is . Choice C.