For a linear equation, a question about no solution asks you to make the two sides disagree for every value of . Match the coefficients, the numbers multiplying , so the -terms cancel. Then check that the constants, the numbers without , differ. A Desmos regression can find the value of the unknown constant that makes the coefficients match.
Hints
- Hint 1
A coefficient is the number multiplying . For no solution, the coefficients on the two sides must match; if they differ, you can solve for one value of . Which two expressions multiply here?
- Hint 2
Matching the -terms isn't enough. Once those terms cancel, the constants, the numbers without , must leave a false statement. What will be left here?
- Hint 3
Enter the equation you made from the coefficients as a regression in Desmos. Replace with and with , then read the value under PARAMETERS.
Step-by-step
Match the coefficients, then check the constants
Step 1Make the x-terms cancel
The coefficients of are and . If they differ, the equation has one solution for . For no solution, they must match:
- Step 2
Check what cancellation leaves
The constants are and . Subtract the matching -terms from both sides:
That's false, so matching the coefficients will give no solution, not infinitely many. After the -terms cancel, the constants decide whether any works.
- Step 3
Find the value of p
Type into Desmos. The tells Desmos to find the unknown value, and makes this a regression rather than a slider. Under PARAMETERS, Desmos shows . Since the constants still disagree, makes the original equation have no solution. Choice C.