Most equations have exactly one solution. Two other things can happen.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
No solution
2x + 3 = 2x + 5 collapses to 3 = 5, which is false. No value of x can fix that.
Infinitely many
2x + 6 = 2(x + 3) collapses to 6 = 6, which is always true. Every value of x works.
Read what is left
If the variable vanishes, look at what remains. False means none; true means all.
Three possible outcomes
Solving a linear equation ends in one of three ways: x equals a number, giving one solution; a false statement like 3 = 5, giving no solutions; or a true statement like 5 = 5, giving infinitely many.
What no solution looks like
2x + 3 = 2x + 7 reduces to 3 = 7, which is false for every x. Graphically, the two sides are parallel lines that never meet. No value works, so the solution set is empty.
What infinitely many looks like
2(x + 3) = 2x + 6 reduces to 6 = 6, true whatever x is. The two sides are the same line drawn twice. Every number is a solution, which is a genuine answer and not a failure to finish.
Do not panic when the letter disappears
When the variable cancels out, that is information rather than an error. Read what is left: if it is true, all values work; if false, none do. Students often assume they went wrong, and usually they did not.
Step 2: Try It Yourself
Tap and try it out.
The slope is 2: for every 1 across, the line goes 2 up.
Step 3: Watch an Example
One step at a time.
Watch Leo Hit a Contradiction
Leo solves 4x + 1 = 4x + 7.
- Step 1
He subtracts 4x from both sides.
Step 4: Your Turn
Practice makes it stick.
The Collapse
Problem 1 of 2
3x + 2 = 3x + 9 leaves 2 = 9. How many solutions?
The Identity
Problem 2 of 2
5(x + 2) = 5x + 10 leaves 10 = 10. Does every value of x work?
How Many Solutions?
1 of 4
2x + 4 = 2x + 4. How many solutions?
2 of 4
6x + 1 = 6x + 8. How many solutions?
3 of 4
Solve 3x + 4 = x + 10.
4 of 4
4(x + 1) = 4x + 4. How many solutions?
Step 5: Quick Check
Show what you know.
Question 1 of 2
7x + 2 = 7x + 9. How many solutions?
Question 2 of 2
The variable vanishes and leaves a true statement. What does that mean?
What You Learned
- Most equations have one solution.
- A false leftover statement means no solution.
- A true leftover statement means infinitely many.