A Plan Is Not Yet a Solution
Suppose we have already decided to solve
\[ 3x+5=20 \]using inverse operations.
The strategy is correct, but execution still matters.
A clear solution is:
\[ 3x+5=20, \] \[ 3x=15, \] \[ x=5. \]Each line follows from the previous line by a valid operation.
A careless version might jump directly from the original equation to
\[ x=5 \]without showing how the relationship was preserved.
The answer may be right, but the reasoning is much harder to inspect.
\[ \boxed{ \text{execution} = \text{valid steps} + \text{visible reasoning} + \text{controlled calculation} } \]What Does It Mean to Carry Out a Solution?
To carry out a solution means to execute the selected method until the requested result has been obtained.
Execution may involve:
- substituting values,
- performing operations,
- transforming equations,
- applying laws and identities,
- following an algorithm,
- building a table,
- refining an approximation,
- recording intermediate states.
The purpose is not merely to reach an answer. It is to preserve the logic of the chosen strategy while moving from known information to the unknown.
Start from the Chosen Plan
Before calculating, state the relationship or procedure you intend to use.
Example
A rectangle has length \(9\text{ m}\) and width \(4\text{ m}\). Find its area.
Plan:
\[ A=\ell w. \]Then execute:
\[ A=(9\text{ m})(4\text{ m}) = 36\text{ m}^2. \]Writing the relationship first makes the calculation easier to interpret and verify.
Substitute Carefully
Substitution replaces a variable with a known value or expression.
Suppose
\[ f(x)=x^2-3x+2 \]and
\[ x=-2. \]Substitute with grouping:
\[ f(-2)=(-2)^2-3(-2)+2. \]Then:
\[ f(-2)=4+6+2=\boxed{12}. \]The parentheses around \(-2\) protect the sign during exponentiation.
Use Valid Transformations
Every algebraic step should follow from a recognized law or operation.
For example:
\[ 2(x+3)=14. \]Distribute:
\[ 2x+6=14. \]Subtract \(6\) from both sides:
\[ 2x=8. \]Divide both sides by \(2\):
\[ x=4. \]Each transformation preserves the solution set.
Preserve Equivalence When Required
When solving equations, many steps should produce an equivalent equation.
\[ E_1\iff E_2 \]means the two equations have the same solutions.
For example:
\[ x+7=12 \]is equivalent to
\[ x=5. \]But some operations can introduce or remove candidate solutions if their conditions are not handled carefully.
Example: Squaring
From
\[ x=3, \]we may square both sides to obtain
\[ x^2=9. \]But the reverse implication is not equivalent because
\[ x^2=9 \]has two solutions:
\[ x=\pm3. \]Execution should therefore distinguish between transformations that preserve full equivalence and those that require later checking.
Record Intermediate Results
Intermediate values help us:
- trace the reasoning,
- detect arithmetic mistakes,
- reuse derived quantities,
- verify later steps.
Example
A car travels \(180\text{ miles}\) in \(3\text{ hours}\), then continues for \(2\text{ hours}\) at the same rate.
First derive:
\[ r=\frac{180}{3}=60\text{ mph}. \]Then use the intermediate result:
\[ d=rt=(60)(2)=120\text{ miles}. \]The rate is not the final answer, but it is an important intermediate known.
Carry Units through the Work
Units should remain attached whenever they help interpret the operations.
For example:
\[ v= \frac{150\text{ km}}{3\text{ h}} = 50\frac{\text{km}}{\text{h}}. \]This shows that the operation is dimensionally sensible.
Likewise:
\[ A=(7\text{ cm})(5\text{ cm}) = 35\text{ cm}^2. \]The squared unit appears naturally from multiplication.
Keep Restrictions Visible
Restrictions should not disappear while the algebra is being performed.
Suppose:
\[ \frac{x+1}{x-4}=2. \]Record:
\[ x\neq4. \]Then solve:
\[ x+1=2(x-4), \] \[ x+1=2x-8, \] \[ x=9. \]Finally confirm:
\[ 9\neq4. \]The solution is admissible.
Preserve Exactness When Useful
If an exact form is manageable, keep it until approximation is actually needed.
For example:
\[ x=\frac{2+\sqrt5}{3} \]is exact.
Converting immediately to
\[ x\approx1.412 \]throws away information and may introduce rounding error into later steps.
A useful habit is:
\[ \boxed{ \text{exact first} \to \text{approximate later if needed} } \]Control Approximation and Rounding
When approximation is required, record enough digits to protect the final accuracy.
Suppose:
\[ \sqrt7\approx2.64575131. \]If that value will be used in several later operations, rounding immediately to \(2.65\) may create avoidable accumulated error.
A common approach is:
- retain extra working digits,
- perform the full calculation,
- round the final result to the required precision.
Track Changing State
Algorithms and iterative computations often require explicit state tracking.
For example:
total = 0
FOR i = 1 TO 4
total = total + i
END FOR| Iteration | \(i\) | total |
|---|---|---|
| start | — | 0 |
| 1 | 1 | 1 |
| 2 | 2 | 3 |
| 3 | 3 | 6 |
| 4 | 4 | 10 |
A trace table makes the changing state visible.
Organize Multi-Step Work
For longer solutions, divide the work into meaningful stages.
For example:
- derive a rate,
- convert units,
- apply the model,
- simplify,
- interpret the result.
This is easier to review than one uninterrupted chain of arithmetic.
Labels can help when the solution naturally has different roles:
Step 1: Convert minutes to hours.
Step 2: Apply \(d=rt\).
Step 3: Check units.
Know When to Reassess
Do not keep executing a strategy merely because it has already consumed time.
Reassess when:
- the algebra becomes much more complicated than expected,
- a denominator becomes zero,
- the units become inconsistent,
- the result violates a known restriction,
- an iterative process fails to converge,
- a simpler representation becomes obvious.
Execution should be disciplined, not stubborn.
Show Enough Work
How much work should be shown?
Enough that another reader can:
- identify the method,
- see the important transformations,
- locate intermediate quantities,
- check the arithmetic,
- understand the final result.
Routine arithmetic need not be expanded into excessive detail. But important logical steps should not disappear.
The goal is clarity, not maximum line count.
Worked Examples
Example 1: Linear Equation
Solve:
\[ 4x-7=21. \]Add \(7\) to both sides:
\[ 4x=28. \]Divide both sides by \(4\):
\[ \boxed{x=7}. \]Example 2: Formula with Units
A car travels at \(55\text{ mph}\) for \(2.5\text{ h}\). Find the distance.
\[ d=rt, \] \[ d= \left(55\frac{\text{mi}}{\text{h}}\right)(2.5\text{ h}), \] \[ d=137.5\text{ mi}. \] \[ \boxed{d=137.5\text{ miles}}. \]Example 3: Preserve an Exact Value
Find the diagonal of a square with side length \(5\).
\[ d^2=5^2+5^2, \] \[ d^2=50, \] \[ d=\sqrt{50}=5\sqrt2. \]Exact:
\[ \boxed{d=5\sqrt2}. \]Approximate only if needed:
\[ d\approx7.071. \]Example 4: Keep a Restriction
Solve:
\[ \frac{3}{x-1}=2. \]Restriction:
\[ x\neq1. \]Then:
\[ 3=2(x-1), \] \[ 3=2x-2, \] \[ 5=2x, \] \[ x=\frac52. \]Since
\[ \frac52\neq1, \]the solution is valid:
\[ \boxed{x=\frac52}. \]Example 5: Multi-Step Percentage Problem
An \$80 item is discounted \(25\%\), then taxed \(8\%\).
Discounted price:
\[ 80(0.75)=60. \]After tax:
\[ 60(1.08)=64.8. \] \[ \boxed{\$64.80}. \]Separating the stages makes the order of operations clear.
Example 6: Reassess a Candidate
Suppose a physical length calculation gives:
\[ x^2=16. \]Algebraically:
\[ x=\pm4. \]But if \(x\) is a positive length, the restriction gives:
\[ \boxed{x=4}. \]Execution includes applying the problem's conditions to the candidate results.
Common Execution Errors
1. Skipping Transformations That Matter
A correct answer with invisible reasoning is difficult to verify.
2. Losing a Negative Sign during Substitution
Use parentheses around substituted negative values.
3. Applying an Operation to Only One Side of an Equation
Equivalent transformations must preserve the relationship.
4. Dropping Units Midway
Units are useful throughout the solution, not only on the final answer.
5. Forgetting Restrictions
A candidate result must still satisfy the original domain and context.
6. Rounding Too Early
Premature rounding can accumulate error.
7. Combining Too Many Steps Mentally
Long invisible calculations increase the chance of mistakes.
8. Forcing a Failing Strategy
Execution should stop and reassess when the method becomes invalid or clearly inefficient.
Practice
Substitution
Evaluate
\[ f(x)=x^2+2x \]at \(x=-3\).
Check Answer
\[ f(-3)=(-3)^2+2(-3) =9-6 =\boxed{3}. \]Equation Step
Solve:
\[ 5x+2=17. \]Check Answer
\[ 5x=15, \] \[ \boxed{x=3}. \]Units
Compute the area of a rectangle measuring \(8\text{ cm}\) by \(3\text{ cm}\).
Check Answer
\[ A=(8\text{ cm})(3\text{ cm}) = \boxed{24\text{ cm}^2}. \]Restriction
For
\[ \frac{1}{x+2}, \]what restriction should remain visible during the work?
Check Answer
\[ \boxed{x\neq-2}. \]Exact or Decimal?
If a problem accepts an exact answer, which is usually preferable during the work:
\[ \sqrt3 \]or
\[ 1.732? \]Check Answer
\[ \boxed{\sqrt3} \]because it preserves exactness. Approximate later if the application requires a decimal.
The Deeper Idea
Carrying out a solution is a controlled sequence of mathematical state changes.
\[ S_0 \to S_1 \to S_2 \to \cdots \to S_n. \]Each state should preserve the information needed for the next step.
This connects the previous problem-solving lessons:
- known and unknown information determines the starting state,
- objects and restrictions define what is allowed,
- representation determines how the state is expressed,
- the selected operation, law, model, or algorithm determines the update,
- execution carries out those updates,
- verification checks whether the final state actually solves the original problem.
Execution is therefore not merely calculation. It is the disciplined movement of a mathematical model from givens to a justified result.
Summary
- Carrying out a solution means executing the chosen strategy through valid, visible mathematical steps.
- State the relationship or procedure before beginning detailed calculation.
- Use parentheses and careful notation when substituting values.
- Each algebraic transformation should be justified by a valid operation or law.
- Distinguish transformations that preserve full equivalence from those that may introduce candidate solutions requiring later checks.
- Intermediate results help organize, trace, and verify multi-step work.
- Carry units through calculations whenever they clarify dimensional structure.
- Keep domain and contextual restrictions visible.
- Preserve exact values until approximation is useful or required.
- Delay final rounding when additional calculations remain.
- Trace changing state explicitly in iterative or algorithmic work.
- Break long solutions into meaningful stages.
- Reassess when the chosen strategy becomes invalid, inefficient, or inconsistent with known conditions.
- Show enough work that another reader can reconstruct the method and check the important steps.
A strong solution is not a mysterious jump from question to answer. It is a sequence of valid moves that keeps the mathematical structure intact. When those moves are clear, the solution becomes easier to trust, easier to debug, and easier to learn from.
Problem Solving and Verification
- Previous: Choosing an Operation, Law, Model, or Algorithm
- Problem Solving and Verification — return to the lesson directory.