A linear equation asking how many solutions it has calls for a comparison of the terms multiplying . If those terms always match, they cancel, leaving only the constants to check. You can use a Desmos regression at to find the value of that makes those constants match. The trap is to think canceled -terms automatically mean no solutions.
Hints
- Hint 1
Distribute the fraction on the left. A coefficient is the number multiplying . How does the left-hand coefficient compare with the one on the right, regardless of ?
- Hint 2
Once the -terms match, changing changes both sides by the same amount. Since at least one value works, you can test the constants by putting into the equation.
- Hint 3
After finding , check the plain numbers on both sides. If they match along with the -coefficients, every value of makes the equation true.
Step-by-step
Compare the -terms, then check the constants
Step 1Show that the -terms always match
Distribute on the left:
The coefficient, or number multiplying , is on both sides for every . So the equation cannot single out exactly one value of .
- Step 2
Find the value that makes the constants match
Because the -terms match, if any works, then works too. Put in for in the given equation and change to for a regression. Desmos reports under PARAMETERS.
- Step 3
Check what the matching terms mean
Type the left-hand constant below the regression. Desmos prints , matching the right-hand constant . The -terms match as well, so both sides are the same expression for every . Every real value of solves the equation, giving infinitely many solutions. Choice D.