LEARNER JOURNEY Solve x/3 + 2 = 7
Name the goalThe denominator is 3.
Choose the next moveMultiply every term by 3.
Carry out the mathematicsx + 6 = 21, so x = 15.
Check and interpretCheck: 15/3 + 2 = 7.
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
Clear fractions or decimals safely, then solve the resulting equivalent linear equation.
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.
- inverse operation
- An operation that reverses another operation, such as subtraction reversing addition.
- coefficient
- A numerical factor multiplying a variable.
Look first, then name the mathematics
Build the visual meaning
- Common multiplier
- Every term
- Denominators cancel
- Then solve
- Check
Read the complete visual relationship as text
- Common multiplier
- Every term
- Cancel denominators
- Equivalent equation
- Solve
Definition in plain language
Multiplying every term on both sides by a common denominator or power of ten removes fractional or decimal coefficients without changing the solution.
Why this matters
Clearing denominators reduces arithmetic clutter while preserving the exact equation.
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.
Identify every denominator or decimal place.
Choose a common multiplier.
Multiply every term on both sides.
Solve the resulting linear equation and check in the original.
The same method in words
- Identify every denominator or decimal place.
- Choose a common multiplier.
- Multiply every term on both sides.
- Solve the resulting linear equation and check in the original.
Interactive decision model
Build the reasoning, one decision at a time
Solve x/6 + 1/3 = 2.
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 0.4x + 1 = 5
Name the goalOne decimal place suggests multiplying by 10.
Choose the next move10(0.4x + 1) = 10(5).
Carry out the mathematics4x + 10 = 50, so 4x = 40 and x = 10.
Check and interpretCheck: 4 + 1 = 5.
Solve (x − 2)/4 = 3
Name the goalThe whole numerator is divided by 4.
Choose the next moveMultiply both sides by 4.
Carry out the mathematicsx − 2 = 12, so x = 14.
Check and interpretCheck: (14 − 2)/4 = 3.
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 x/3 + 2 = 7.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
2. Solve x/4 − 1 = 5.
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 x/5 + 3 = −1.
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 x/2 − 6 = 3.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
2. Solve 0.5x + 2 = 7.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
3. Solve 0.25x − 1 = 2.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
4. Solve 1.2x = 9.6.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
5. Solve 0.4x + 1.6 = 4.8.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
6. Solve (x − 2)/3 = 4.
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 (x + 5)/2 = −3.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
2. Solve x/6 + 1/3 = 2.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
3. Solve x/8 − 1/4 = 1.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
4. Solve 0.75x = 6.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
5. Solve 2.5x − 5 = 10.
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
Solve 0.25x + 1.5 = 4 and explain why multiplying by 100 is valid even though 10 would also work.
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
Multiplying by 100 gives 25x + 150 = 400, so 25x = 250 and x = 10. Multiplying both sides by the same nonzero number preserves equality; 10 also clears the decimals and leads to the same result. Check: 2.5 + 1.5 = 4.
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.