Solving Equations (Special Cases)
Most equations you solve end with the variable equal to one number. But two special cases break that pattern. Sometimes the variable terms cancel completely and leave a statement that is flat-out false, like — that equation has no solution. And sometimes they cancel and leave a statement that is always true, like — that equation has infinitely many solutions, because every value of the variable works.
The skill in this lesson isn't new solving moves. It's recognizing what it means when the variable disappears on you.
The two special cases
Solve normally: distribute, combine like terms, gather variables on one side. If the variable terms cancel and what's left is a false statement, there is no solution — no number can make the equation true. If what's left is a true statement, there are infinitely many solutions — the two sides were the same expression all along, so every number works.
In , subtracting from both sides leaves . False, so no solution. In , distributing gives ; subtracting leaves . True, so infinitely many solutions.
What it looks like on a graph
Think of each side of the equation as its own line. A solution is an -value where the two lines meet. When an equation has no solution, its two sides are parallel lines — same slope, different starting points — so they never touch.
When an equation has infinitely many solutions, both sides graph as the exact same line, so they "meet" at every single point. And the ordinary case, one solution, is two lines with different slopes crossing exactly once.
Spotting the cases early
You can often see a special case before finishing the algebra. If both sides have the same variable term — like on the left and on the right — compare the constants. Different constants mean no solution; identical sides mean infinitely many.
But be careful: the coefficients have to match after distributing and combining, not before. doesn't look like a special case until you distribute and see .
Worked examples
Example 1: no solution
Solve .
Answer: No solution
Example 2: infinitely many solutions
Solve .
Answer: Infinitely many solutions
Example 3: the ordinary case, for contrast
Solve .
Answer: — one solution
Try one yourself
Common questions
How do I tell no solution and infinitely many solutions apart?
Look at the statement left after the variable cancels. False statement, like : no solution. True statement, like : infinitely many solutions.
Is the same as no solution?
No — this trips up a lot of students. is one perfectly good solution; the number zero works when you plug it in. No solution means the variable vanished and left a false statement, so nothing works.
Did I make a mistake if the variable disappears?
Not necessarily. If your algebra is clean and the variable terms cancel, the equation really is a special case. Re-check your distributing and signs once, then trust the result.
Does "infinitely many solutions" mean any number I pick will work?
Yes. Both sides are the same expression in disguise, so any value of the variable makes them equal. Try one — plug into and both sides give .
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