The equation , where and are constants, has infinitely many solutions.
Which of the following must be true?
I.
II.
III.
An identity is an equation that stays true for every value of its variable. That’s the cue when a linear equation has infinitely many solutions. Two linear expressions that agree at two distinct inputs agree everywhere, so a Desmos regression can find the missing constants. The -terms and the constant terms must both match; matching only the -terms isn’t enough.
Hints
- Hint 1
Two different lines can cross only once. If both sides of an equation are linear, making them agree at two inputs, such as and , makes them agree for every . How could you test both inputs at once?
- Hint 2
A Desmos regression uses to fit unknown constants. Give the list , then replace with in the equation. Check all three statements after Desmos finds the constants.
Step-by-step
Approach 1: Fit two inputs in Desmos
Step 1Choose inputs that pin down the lines
Both sides are linear in . If they agree at two different inputs, they’re the same line: different lines can meet only once. Use and to find constants that make the equation true for every .
- Step 2
Fit the constants and check the statements
Type , then . The tells Desmos to find and that make the equation work at both inputs. Under PARAMETERS, it gives and . So I and II must be true, while III contradicts . Choice C.
Approach 2: Match the terms by hand
Step 1Distribute on the right
Distribute to both terms inside the parentheses:
- Step 2
Match the coefficients of x
A coefficient is the number multiplying . For every to work, the coefficients must match: the left has and the right has , so .
- Step 3
Match the constant terms
The constant terms are the parts without . They must match too, so . Substitute :
Multiply: . Statement III says the opposite, so I and II must be true. Choice C.