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: Picture It
Start with something you already know.
Two runners at one speed
- Two runners run at exactly the same speed.
- One starts 4 feet ahead of the other.
- A minute later the gap is still 4 feet, and it never closes.
- If they start side by side instead, they are level the whole race.
The same speed is the same x term on both sides, and taking it away leaves only the head start.
A head start leaves a false line and no solution, while starting level makes every number work.
Runners at different speeds are level at just one moment, perhaps before the start, as in the last lesson.
Step 2: 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 3: 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 4: 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 5: 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 6: Your Turn
Practice makes it stick.
Two Streaming Plans
Problem 1 of 15
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 15
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.
Problem 3 of 15
Solve 5x + 4 = 5x + 9. What does it have?
Problem 4 of 15
Solve 3x + 8 = 5x + 2. What is x?
Problem 5 of 15
Solve 4(x − 2) = 4x − 8. What does it have?
Problem 6 of 15
Solve 6x − 3 = 3(x + 1). What is x?
Problem 7 of 15
Solve 3x + 7 = 5x + 7. What does it have?
Whose Step?
One line in each solve is a slip. Find it, then fix it.
Problem 8 of 15
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
Problem 9 of 15
What should Priya's last line say?
Problem 10 of 15
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
Problem 11 of 15
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.
Problem 12 of 15
To decide 5(x + 2) = 5x + 3 without solving it out, what is the first move, and why?
Problem 13 of 15
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?
Problem 14 of 15
Select every equation that has every number as a solution.
Problem 15 of 15
Select every equation that, like 2x + 5 = 2x + 9, has no solution.
Step 7: 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.