The phrase infinitely many solutions signals an identity: a linear equation that is true for every . Distribute, then compare the -terms and the terms without . If the -terms already match, a Desmos regression can find the value that makes the remaining terms match too. Setting a shared factor to zero does not erase a number added outside it.
Hints
- Hint 1
For an identity, every value of must work. Distribute across both terms in . How does the resulting -term compare with the one on the right?
- Hint 2
The constant terms are the parts without . Once the matching -terms cancel, those constants must be equal. Keep the on the right when you write that condition.
Step-by-step
Cancel the matching terms, then solve
Step 1Distribute on the left
Distribute means multiply each term inside the parentheses by :
So the given equation becomes
The is outside the factor , so making that factor zero would not erase it.
- Step 2
Match what remains after the -terms cancel
Infinitely many solutions means every real must make the equation true. The -terms are already identical. Subtract from both sides:
When the -terms cancel, the remaining constants must agree. If they do not, no value of works.
- Step 3
Find the value of
Type in Desmos. The makes this a regression, asking Desmos to find rather than graph an equation. Under PARAMETERS, it shows . Type on the next line and use its fraction button to see the exact value . At that value, both sides agree for every real . Choice D.