LEARNER JOURNEY Solve 3x + 2x = 20
Name the goalBoth terms contain x to the first power.
Choose the next moveAdd the coefficients 3 and 2.
Carry out the mathematics5x = 20, so x = 4.
Check and interpretCheck: 12 + 8 = 20.
Start here
Build the concept from the beginning
Starting point
No prior equation-solving skill is assumed. Arithmetic operations, variables, equality, and every new algebra decision are explained on this page.
Learning target
Combine like terms correctly before isolating the variable.
Evidence of mastery
Construct the reasoning path, solve at least 4 of 5 mastery problems, and revise an unfamiliar explanation.
Essential terms
- equation
- A statement that two mathematical expressions have the same value.
- solution
- A value that makes an equation true when substituted for the variable.
- coefficient
- A numerical factor multiplying a variable.
- constant
- A number without a variable.
Look first, then name the mathematics
Build the visual meaning
- Like x-terms
- Like constants
- Add coefficients
- Keep x
- Then solve
Read the complete visual relationship as text
- Like x-terms
- Like constants
- Add coefficients
- Keep x
- Then solve
Definition in plain language
Like terms have the same variable part and exponent. Their coefficients can be added or subtracted; unlike terms cannot be merged.
Why this matters
Combining like terms reduces an equation to a simpler equivalent form without changing its solutions.
Reasoning path
See the method as a sequence
Move from left to right. Each decision prepares the next one; on a phone, follow the numbered cards from top to bottom.
Group x-terms with x-terms.
Group constants with constants.
Add signed coefficients carefully.
Solve the simplified equation and check.
The same method in words
- Group x-terms with x-terms.
- Group constants with constants.
- Add signed coefficients carefully.
- Solve the simplified equation and check.
Interactive decision model
Build the reasoning, one decision at a time
Solve 5x + 3x − 6 = 18.
Choose this step, then check the construction.
Choose this step, then check the construction.
Choose this step, then check the construction.
The reasoning path changes as each decision is selected.
Choose every step, then check the construction.
Model before practice
Three fully explained examples
Every example identifies the goal, explains the next move, carries out the algebra, and checks or interprets the result.
Solve 7x − 4x + 5 = 14
Name the goalCombine the variable terms first.
Choose the next move7x − 4x = 3x.
Carry out the mathematics3x + 5 = 14, so 3x = 9 and x = 3.
Check and interpretCheck: 21 − 12 + 5 = 14.
Solve −2x − 5x − 6 = 15
Name the goalBoth x-coefficients are negative.
Choose the next move−2x − 5x = −7x.
Carry out the mathematics−7x − 6 = 15, so −7x = 21 and x = −3.
Check and interpretCheck: 6 + 15 − 6 = 15.
Practice with decreasing support
Guided practice
Practice immediately after the model
Predict first. Then use the visual relationship and reasoned feedback to check two decisions.
1. Solve 3x + 2x + (4) = 29.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
2. Solve 5x + -2x + (7) = 16.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
Not completed yet.
Partially completed problem
Finish the missing step
A prompt supplies part of the reasoning. Predict the missing mathematical move before choices appear.
1. Solve -3x + -4x + (5) = -30.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
Not completed yet.
Independent practice
Six problems for repetition
Solve all six. Use the specific feedback to revise a method or sign error.
1. Solve 6x + 1x + (-3) = 32.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
2. Solve 4x + 3x + (2) = 23.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
3. Solve -5x + 2x + (8) = -13.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
4. Solve 7x + -3x + (-4) = 12.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
5. Solve 2x + 5x + (1) = 36.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
6. Solve -6x + 4x + (9) = -11.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
Not completed yet.
Mastery and transfer
Mastery check
Demonstrate understanding
Attempt all five. At least four correct answers are required for mastery.
1. Solve 8x + -5x + (-2) = 19.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
2. Solve 3x + 6x + (-7) = 38.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
3. Solve -4x + -2x + (3) = -15.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
4. Solve 9x + -7x + (4) = 12.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
5. Solve 5x + 4x + (-6) = 48.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
Not completed yet.
Unfamiliar transfer
Explain, compare, and revise
Explain why 4x + 3 cannot be simplified to 7x, then solve 4x + 3 = 19.
Guided self-check
Judge the locked first attempt against every criterion. Each judgment gives revision guidance.
Choose Met or Needs revision.
Choose Met or Needs revision.
Choose Met or Needs revision.
Model reasoning
4x and 3 are unlike terms, so 4x + 3 does not become 7x. Subtract 3 from both sides to get 4x = 16, then divide by 4 to get x = 4. Checking gives 19 = 19.
Write a first attempt before opening the self-check.
The record reveals exact answers and model reasoning only after the corresponding attempt or self-check has occurred.