A linear system with infinitely many solutions has two equations that describe the same line. Find the multiplier that takes the first equation’s and coefficients to the second equation’s coefficients. Then apply that multiplier to the constants too. The trap is matching to while overlooking the in .
Hints
- Hint 1
For two equations to have infinitely many solutions, they must describe the same line. Multiplying every term of one equation by a single nonzero number keeps its line unchanged. What multiplier takes to ?
- Hint 2
Check that the multiplier also takes to . Then apply it to the entire first equation, including . The resulting constant must equal the second equation’s .
- Hint 3
The question asks for , not the multiplier. Use the relationship you found between and , and keep the attached to as you isolate the requested ratio.
Step-by-step
Match the equations term by term
Step 1Find the multiplier
No Desmos needed. Infinitely many solutions means the equations draw the same line, so compare their coefficients exactly. Divide the second coefficient by the first to find the multiplier:
Multiply both variable coefficients by to check the match:
- Step 2
Match the constants
For the equations to draw the same line, the multiplier must apply to the entire equation, including its right side. Match the resulting constant to the second equation’s :
- Step 3
Put the requested ratio into the equation
The target is . Divide both sides by , which must be nonzero for that ratio to exist:
- Step 4
Isolate the ratio
Multiply both sides by to leave the requested ratio on its own:
So the system requires . Choice B.