The phrase infinitely many solutions signals a linear identity: the equation must be true for every value of . Distribute on both sides, then match the coefficients of and the constants. Matching only the -terms is a trap; if the constants differ, the equation has no solutions instead.
Hints
- Hint 1
For infinitely many solutions, the two sides must become the same expression. Start by distributing each factor so you can see the terms multiplying and the terms without .
- Hint 2
The coefficient of is the amount multiplying it. Match the coefficients on the two sides to find , but don't stop there: what must also be true of the terms without ?
- Hint 3
Once you've found , check the constants. If the -terms cancel but the constants differ, you'd be left with a false statement, not infinitely many solutions.
Step-by-step
Match both parts of the equation
Step 1Simplify the left side
Distribute each factor on the left:
Combine the terms with and the terms without it:
- Step 2
Simplify the right side
Distribute on the right, including the minus sign from multiplying by :
Combine like terms:
- Step 3
Match the coefficients of x
For infinitely many solutions, both the -terms and the constants must match. A coefficient is the amount multiplying , so match with :
Subtract :
Add : . Matching coefficients alone isn't enough; check the constants next.
- Step 4
Check that the entire sides match
Divide by : . Type , then and into Desmos. Both constants print . Graph the two sides of the given equation shown below; their lines overlap, so the matching coefficients and constants make the equation true for every . The value of is . Choice B.