SAT Desmos Trick: Find the Number of Solutions Instantly
“How many solutions does this system have?” is a classic Digital SAT trap. Desmos makes it trivial: graph both equations and count the intersections.
Step by step
- 1
Graph both equations
In Bluebook, open the Desmos calculator and type each equation on its own line, exactly as given — no rearranging needed.
- 2
Count where the graphs cross
Each crossing point is one solution. Two lines crossing once = one solution; a line and a parabola can cross twice = two solutions.
- 3
Spot the 'no solution' case
If the graphs never touch — for example, two parallel lines — the system has no solution.
- 4
Spot the 'infinitely many' case
If the two graphs land exactly on top of each other, every point is shared — infinitely many solutions.
- 5
Zoom out to be sure
Scroll or pinch to zoom out so you don't miss a crossing that sits off the visible screen.
Pro tip
For two straight lines you can often answer without graphing: same slope + different intercept = no solution; same slope + same intercept = infinitely many; different slopes = exactly one solution.
Try it yourself
Work the example right here in a live Desmos calculator — no Bluebook needed.
How many solutions does the equation x² = 4x − 4 have?
Graph both and count where they meet — that's the number of solutions.
Loading interactive calculator…
Show the answer
Answer: Exactly one
Graph y = x² and y = 4x − 4. The parabola just touches the line at a single point, (2, 4), so there is exactly one solution. (Algebra check: x² − 4x + 4 = 0 → (x − 2)² = 0 → one repeated root.)
Put the trick to work on a real test
Full-length adaptive SAT mocks · real Bluebook-style timing · detailed score reports