In the equation below, 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. Match the coefficients, the numbers multiplying , then use a Desmos regression to find the unknown constant. Check the plain numbers too: matching coefficients alone isn't enough. If those numbers also match, every works.
Hints
- Hint 1
For no solution, the terms containing must cancel and leave a false statement, like . What must be true about the numbers multiplying on the two sides?
- Hint 2
The left coefficient is , and the right coefficient is twice that quantity. Set the two coefficients equal, then subtract the left one from both sides. What condition does that leave?
- Hint 3
After the -terms cancel, compare the constants, the terms without . If they match, every works; if they differ, no works. Could they match under your coefficient condition?
Step-by-step
Match the coefficients, then check the constants
Step 1Set the condition for the x-terms to cancel
The coefficient of is the number multiplying it. For no solution, the coefficients must match so the -terms cancel, while the constants must differ. Match the coefficients given here:
Matching them alone doesn't yet tell you whether the result is false or true for every .
- Step 2
Simplify the coefficient condition
Subtract from both sides to find what must be zero:
- Step 3
Rule out matching constants
The constants would match if . Divide by :
But at , , not . So a value that makes the -terms cancel cannot also make the constants match; the equation will be false rather than true for every .
- Step 4
Find the value of n
Type in Desmos. The subscript tells Desmos to find , and tells it to fit the equality. Under PARAMETERS, Desmos reports . This makes the -terms cancel, while step 3 shows the constants cannot match, so the original equation has no solution. Choice D.