Sometimes solving makes every x disappear. What is left decides the answer: a false statement means no solution, a true statement means every number works.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Try it first
Try x = 1, then x = 2, then x = 3 in 2(x + 3) = 2x + 6. What happens?
Stuck is fine. Let’s Learn shows the way next. Got it first time? You can jump ahead to Watch an Example.
A false statement
2x + 3 = 2x + 7: subtract 2x from both sides and 3 = 7 remains. No x was involved in the falsehood, so no x can repair it. The equation has no solution.
A true statement
2x + 6 = 2(x + 3): expand the right, 2x + 6 = 2x + 6. Subtract 2x: 6 = 6, true whatever x is. Every number is a solution. An equation like this is called an identity.
Why the x vanished
Both sides had the same x term. Taking it from both leaves only numbers, and numbers do not depend on x. Same coefficient of x on both sides means no solution or every solution, never exactly one.
What it means graphically
y = 2x + 3 and y = 2x + 7 are parallel lines. They never meet, so there is no x where the two sides agree. 2x + 6 and 2(x + 3) are one line drawn twice, so every x is a meeting point.
Spot it early
Before solving, compare the x terms. 4x + 1 = 4x + 9 will collapse to 1 = 9. 3x + 1 = 4x + 9 will not: its x terms differ, so it has exactly one solution.
Step 2: Try It Yourself
Tap and try it out.
Solve 4x + 1 = 4x + 9, then 3(x + 2) = 3x + 6
- 14x + 1 = 4x + 9the first equation
Step 0 of 4
The slope is 2: for every 1 across, the line goes 2 up.
Step 3: In Real Life
Two identical price lists
Two stores both charge 5 plus 2 per item, so 5 + 2x = 5 + 2x: every quantity costs the same, and x vanishes leaving 0 = 0. Add a 3-dollar fee to one and 0 = 3: no quantity ever matches.
Step 4: Watch an Example
One step at a time.
Watch Noor Solve 3(2x − 1) + 4 = 6x + 5
A bracket to clear, two numbers to collect, and then something no earlier lesson prepared her for.
- Step 1
Distributing the 3 across the bracket gives 6x − 3 + 4 = 6x + 5. Both terms inside were multiplied: 3 × 2x = 6x, and 3 × (−1) = −3, so the minus travels with the 1 rather than being added back afterwards.
Watch Ravi Solve 4(x + 2) − 3 = 2(2x + 1) + 3
A bracket on each side. Ravi tidies both, and you write the last two lines.
- Step 1
Distributing on the left: 4(x + 2) is 4x + 8, so the left side reads 4x + 8 − 3.
Read the Same Two Equations as Lines
Noor's and Ravi's equations again, each side drawn as a line.
- Step 1
Noor's equation tidied to 6x + 1 = 6x + 5. Reading each side as a line, y = 6x + 1 and y = 6x + 5, both have slope 6, and their intercepts are 1 and 5.
Step 5: Your Turn
Practice makes it stick.
Two Streaming Plans
Problem 1 of 2
Plan A costs $8 a month plus $3 a rental. Plan B costs $12 a month plus $3 a rental. For how many rentals r in a month do the two plans cost the same?
Two Car Washes
Problem 2 of 2
A company fleet is c cars and 2 vans. Wash A charges $6 for every vehicle: 6(c + 2). Wash B charges $6 a car plus a flat $12 for the two vans: 6c + 12. For how many cars c do the two washes cost the same?
Solve or Decide
Some have one answer, some have none, some have every number. Read what is left.
1 of 5
Solve 5x + 4 = 5x + 9. What does it have?
2 of 5
Solve 3x + 8 = 5x + 2. What is x?
3 of 5
Solve 4(x − 2) = 4x − 8. What does it have?
4 of 5
Solve 6x − 3 = 3(x + 1). What is x?
5 of 5
Solve 3x + 7 = 5x + 7. What does it have?
Whose Step?
One line in each solve is a slip. Find it, then fix it.
1 of 4
Priya solved 3(x + 2) − 6 = 3x and wrote "no solution". One line is a slip. Which?
Priya's solve of 3(x + 2) − 6 = 3x
- 13(x + 2) − 6 = 3x
- 23x + 6 − 6 = 3x
- 33x = 3x
- 40 = 0− 3x
- Answerno solution
2 of 4
What should Priya's last line say?
3 of 4
Dev solved 2(5 + 3x) = 3x + 16 and wrote "no solution". One line is a slip. Which?
Dev's solve of 2(5 + 3x) = 3x + 16
- 12(5 + 3x) = 3x + 16
- 210 + 3x = 3x + 16
- 310 = 16− 3x
- Answerno solution
4 of 4
Fix Dev's line 2 and finish the solve. What is x?
Pick the Move
No solving needed. Choose the first move, and the reason for it.
1 of 2
To decide 5(x + 2) = 5x + 3 without solving it out, what is the first move, and why?
2 of 2
4x + 9 = 6x + 9. Which x term do you take from both sides first, and why?
Same Solution?
Without solving all the way, which of these share an answer?
1 of 2
Select every equation that has every number as a solution.
2 of 2
Select every equation that, like 2x + 5 = 2x + 9, has no solution.
Step 6: Quick Check
Show what you know.
Question 1 of 1
8x + 2 = 8x + 11. How many solutions?
Ready for more?Optional. Past the lesson, for anyone who wants it.
Stretch 1 of 2
3x + k = 3x + 5, where k is a number you choose. For which values of k does the equation have no solution?
Stretch 2 of 2
Find the number a that makes a(x + 2) = 3x + 6 true for every x.
What You Learned
- If every x cancels, read the statement that remains.
- A false statement means no solution; a true one means every number works.
- Graphically these are parallel lines, or one line drawn twice.
- Equal x terms on both sides are the warning sign.
For the grown-up
The misconception this lesson targets is reading the leftover statement as an answer about x: "0 = 0, so nothing is left, so no solution" and "3 = 3, so x = 3" are its two common forms. Once the x is gone, the only question is whether what remains is true (every number) or false (none); and x = 0, when it arrives, is one ordinary solution, not the absence of one.