Desmos for the SAT · Technique 6 of 15
Check if Two Expressions Are Equivalent
Graph both, and if they land on the exact same curve, they are equal.
Some SAT questions hand you a messy expression and ask which of the choices is equivalent to it. The by-hand way is to factor, expand, and cancel until you match a choice, and every one of those moves is a spot to drop a sign or cancel a term you should not have.
There is a faster check: graph the original expression as one curve, graph your candidate answer as a second curve, and see if they land exactly on top of each other. If two expressions are equal for every x, their graphs are the same curve. A wrong option peels away onto a different curve the moment you graph it.
One catch: this only works for single-variable expressions, the kind you can graph as y = (expression). A two-variable expression is a surface, not a 2D curve, so this move does not apply there.
Nova
Here's a move that turns 'which one is equivalent?' into a quick eye test: graph the original, graph your guess, and see if they land on the same curve. When they overlap, you're done. No factoring, no canceling, no traps to fall for.
The algebra way vs the Desmos way
This technique sidesteps distributing and combining like terms by hand.
The algebra way
- Distribute the 3:
3(x + 4) = 3x + 12. - Combine like terms with the
2x:3x + 12 + 2x = 5x + 12, then compare to the candidate. - Drop the
+2xor mishandle a sign and you match a trap choice instead.
The Desmos way
- Type
y = (original expression)andy = (candidate answer)as two lines. - Check whether the two curves land exactly on top of each other.
The payoff: Graphing both and comparing sidesteps every distribute-and-combine slip the trap answers are built to catch.
See the move, live
That same example, already loaded in the real calculator, the one built into Bluebook on test day. Try the move yourself.
Step by step
Nova
Follow along one step at a time, hit Next as you go. I'll flag the spots where people slip.
Type the original expression as its own line, y = (original expression).
Nova: Copy it in exactly as the problem gives it. No simplifying first, that is the whole point, let Desmos do the checking.
Now try it on a real SAT question
The expression (x² - 9) / (x² + 5x + 6) is given, where x ≠ -2 and x ≠ -3.
Which of the following is equivalent to the expression?
Pick an answer to get instant coaching from Nova.
Straight-up "Factor first, cancel second": fully factor the top and bottom before you touch anything, because only whole factors cancel. The moment you spot the shared (x + 3), it drops out and the answer falls out. Crossing off the x² terms early is exactly the cancel-too-early trap.
Factor both parts: x² - 9 = (x + 3)(x - 3), and x² + 5x + 6 = (x + 2)(x + 3). The (x + 3) factors cancel, leaving (x - 3)/(x + 2), choice A. Check with x = 1: the original gives -8/12 = -2/3, and choice A gives -2/3. Choice C is the term-canceling trap: crossing out the x² on top and bottom is not a legal move, and plugging in x = 1 gives -9/11, which does not match.
In the app, drills like this adapt to your level, and your AI coach can walk you through any step you get stuck on.
Nova
Straight-up "Factor first, cancel second": fully factor the top and bottom before you touch anything, because only whole factors cancel. The moment you spot the shared (x + 3), it drops out and the answer falls out. Crossing off the x² terms early is exactly the cancel-too-early trap.
Set it up yourself in the blank calculator, then read the answer straight off the graph. Muscle memory beats watching.
The Desmos trap: when to close the calculator
- A two-variable expression graphs as a surface, not a 2D curve, so this graph-both-and-compare check does not apply to it.
- A removable hole can make two curves look identical even when they differ at one point, so plug in a quick value to sanity-check if the match looks too clean.
Nova
One heads-up: this is a single-variable move. If the expression has two variables it graphs as a surface, not a 2D curve, so this check does not apply. And watch for a removable hole, a spot where one curve has a gap can still make two graphs look identical, so sanity-check a point if you are unsure.
Put this move into real practice
"Check if Two Expressions Are Equivalent" shows up all over the real SAT. SAT Prep Quest drills the skill at your level, so reaching for it becomes automatic by test day.
Practice freeSAT is a registered trademark of the College Board, which was not involved in producing, and does not endorse, this product. Score estimates are Learner Labs projections, not official College Board scores.