LEARNER JOURNEY Solve 6x = 42
Name the goalSix equal copies of x total 42.
Choose the next moveDivide both sides by 6.
Carry out the mathematicsx = 42/6 = 7.
Check and interpretCheck: 6(7) = 42.
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
Solve equations in which the variable is multiplied or divided by a nonzero number.
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
- ax = b
- Divide by a
- Coefficient → 1
- x = b/a
- Check
Read the complete visual relationship as text
- ax = b
- Divide by a
- Coefficient → 1
- x = b/a
- Check
Definition in plain language
For ax = b with a nonzero, divide both sides by a. For x/a = b, multiply both sides by a.
Why this matters
This isolates the variable by turning its coefficient into 1 while preserving equality.
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 the signed coefficient or divisor.
Use division or multiplication as the inverse.
Apply it to both complete sides.
Check the result, including its sign.
The same method in words
- Identify the signed coefficient or divisor.
- Use division or multiplication as the inverse.
- Apply it to both complete sides.
- Check the result, including its sign.
Interactive decision model
Build the reasoning, one decision at a time
Solve −3x = 24.
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 −4x = 28
Name the goalThe coefficient is negative 4.
Choose the next moveDivide both sides by −4.
Carry out the mathematicsx = 28/(−4) = −7.
Check and interpretCheck: −4(−7) = 28.
Solve x/5 = −6
Name the goalx is divided by 5.
Choose the next moveMultiply both sides by 5.
Carry out the mathematicsx = −30.
Check and interpretCheck: −30/5 = −6.
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 = 18.
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 = 20.
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 5x = -35.
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 = -42.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
2. Solve 8x = 56.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
3. Solve 7x = -21.
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 = 72.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
5. Solve 12x = 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 -5x = 45.
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 11x = 0.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
2. Solve 2x = -16.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
3. Solve -3x = -27.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
4. Solve 10x = 90.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
5. Solve -8x = 24.
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 the solution of −5x = 30 is negative.
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
Dividing both sides of −5x = 30 by −5 gives x = −6. The quotient is negative because a positive divided by a negative is negative. Checking gives −5(−6) = 30.
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.