A system with infinitely many solutions has two equations that describe the same line. Compare the and coefficients to find their shared multiplier, then apply it to the constants. A Desmos regression can solve for the hidden value. The trap is matching only the variable terms: that can give parallel lines instead of one line.
Hints
- Hint 1
Infinitely many solutions means every point on one line also lies on the other. What would have to happen to the entire first equation to turn it into the second?
- Hint 2
The coefficient goes from to , so the multiplier is . Check whether doubling the whole coefficient, , gives the second equation’s coefficient.
- Hint 3
An equation stays true when you multiply both sides by the same number. If the first equation’s left side doubles, what must its right side become?
Step-by-step
Match the whole equation
Step 1Find the multiplier on the variable terms
The coefficient goes from to , so the second left side must be twice the first. That works for the term too:
- Step 2
Make the constants match
For infinitely many solutions, the multiplier must work on both sides of the equation. Twice the first right side is , so the second right side must equal :
. Matching only the variable terms would give lines with the same direction, not necessarily the same line.
- Step 3
Solve the matching condition in Desmos
Type , using as a stand-in for . The tells Desmos to find a value that fits the equation; using instead of prevents a slider. Under PARAMETERS, Desmos shows .
- Step 4
Read the exact value
Type on the next line and tap its fraction button. Desmos displays . Since stood for , the system has infinitely many solutions when . Choice B.