Two linear equations have infinitely many shared solutions only when they are equivalent equations: they draw the same line. Find the multiplier that matches one variable term, then apply it to the entire equation. The same multiplier must work for both variable terms and the constant. Matching the coefficients before scaling can make two different lines look alike.
Hints
- Hint 1
An infinite-solution system describes the same line twice, so every point on one line must also be on the other. What multiplier changes the first equation’s into the second equation’s ?
- Hint 2
To keep an equation equivalent, multiply both sides by that multiplier. Once the terms and constants match, what must happen to the coefficients of ?
Step-by-step
Make the equations identical
Step 1Find the multiplier
For infinitely many solutions, the two equations must draw the same line. Compare the terms and constants:
Both take a multiplier of , so that multiplier must work on the term too.
- Step 2
Scale the first equation
Multiply every term of the first equation by :
The term and constant now match the second equation. Matching only those two pieces isn’t enough: the term must match as well.
- Step 3
Match the remaining coefficients
A coefficient is the number multiplying a variable. Set the two coefficients equal:
Don’t set by comparing the original terms. The first equation has been multiplied by , so its new coefficient is .
- Step 4
Find the value and check the lines
Divide by in Desmos: type , followed by the two original equations. Desmos shows and two overlapping lines. So every point on that line solves both equations, and the required value is . Choice B.