Put the Answer Back into the Problem
Suppose we solve
\[ 3x+5=20 \]and obtain
\[ x=5. \]A substitution check places the candidate back into the original equation:
\[ 3(5)+5=15+5=20. \]The original condition is satisfied, so the solution passes the check.
Now consider a process check. If we start from \(5\), multiply by \(3\), and then add \(5\), we get:
\[ 5 \xrightarrow{\times3} 15 \xrightarrow{+5} 20. \]Undoing the process should reverse those steps:
\[ 20 \xrightarrow{-5} 15 \xrightarrow{\div3} 5. \]Both checks ask the same deeper question:
\[ \boxed{ \text{Does the proposed answer actually reproduce the original relationship?} } \]What Is a Substitution Check?
A substitution check replaces the unknown in the original equation, formula, or condition with the proposed answer.
If the original relationship becomes true, the candidate passes that check.
Example
Candidate:
\[ x=4. \]Original equation:
\[ 2x+3=11. \]Substitute:
\[ 2(4)+3=8+3=11. \]So
\[ \boxed{x=4} \]satisfies the equation.
Check the Original Condition
The strongest substitution check usually uses the original equation or condition, not merely the last transformed line.
Suppose a long solution ends with
\[ x=6. \]Checking only a nearby intermediate equation may repeat an earlier algebraic mistake.
Instead, return to the starting condition and test:
\[ \text{original left side} \stackrel{?}{=} \text{original right side}. \]This reaches back across the entire solution chain.
What Is an Inverse-Operation Check?
An inverse-operation check verifies a process by undoing the operations in reverse order.
For example, suppose the forward process is:
\[ x \xrightarrow{\times4} 4x \xrightarrow{-7} 4x-7. \]To reverse it:
\[ y \xrightarrow{+7} y+7 \xrightarrow{\div4} \frac{y+7}{4}. \]The inverse process reverses both:
- the operation,
- the order of the operations.
Reverse the Process
If the forward process is
\[ x \xrightarrow{+3} x+3 \xrightarrow{\times2} 2(x+3), \]then the inverse process must begin by undoing multiplication:
\[ y \xrightarrow{\div2} \frac y2 \xrightarrow{-3} \frac y2-3. \]A common mistake is to undo the operations in the same order they were performed.
\[ \boxed{ \text{inverse process} = \text{inverse operations in reverse order} } \]Forward and Reverse Verification
Checking Equation Solutions
For an equation such as
\[ 5x-2=18, \]suppose we obtain
\[ x=4. \]Check:
\[ 5(4)-2=20-2=18. \]The candidate satisfies the original equation.
If the left and right sides do not match, either:
- the candidate is wrong,
- the substitution was performed incorrectly,
- the original equation was copied incorrectly.
Checking Formula Rearrangements
Suppose
\[ d=rt \]is rearranged to solve for time:
\[ t=\frac dr. \]We can check the rearrangement by substituting the new expression back into the original formula:
\[ d=r\left(\frac dr\right). \]For \(r\neq0\):
\[ d=d. \]The identity confirms the rearrangement.
Checking Function Values
Suppose
\[ f(x)=x^2-4x+1 \]and we calculate
\[ f(3)=-2. \]Check directly:
\[ f(3) = 3^2-4(3)+1 = 9-12+1 = -2. \]The check repeats the definition with the stated input.
Checking Roots and Radical Solutions
Suppose we solve
\[ x^2=25 \]and obtain
\[ x=\pm5. \]Check each candidate:
\[ 5^2=25, \] \[ (-5)^2=25. \]Both satisfy the original equation.
For radical equations, substitution is especially important because squaring can introduce extra candidates.
Detecting Extraneous Solutions
An extraneous solution is a candidate produced during the algebra that does not satisfy the original equation.
Example
Solve:
\[ \sqrt{x+6}=x. \]Square both sides:
\[ x+6=x^2. \] \[ x^2-x-6=0. \] \[ (x-3)(x+2)=0. \]Candidates:
\[ x=3, \qquad x=-2. \]Check \(x=3\):
\[ \sqrt{9}=3. \]Valid.
Check \(x=-2\):
\[ \sqrt4=2\neq-2. \]So:
\[ \boxed{x=3} \]and \(x=-2\) is extraneous.
Detecting Sign Errors
Substitution is excellent for catching lost negative signs.
Suppose someone solves
\[ 2x-7=-15 \]and claims
\[ x=11. \]Check:
\[ 2(11)-7=15, \]but the right side should be \(-15\).
The mismatch immediately exposes an error.
Checking Fractional Results
Suppose
\[ 3x+1=5 \]gives
\[ x=\frac43. \]Check:
\[ 3\left(\frac43\right)+1 = 4+1 = 5. \]A fractional answer is not suspicious merely because it is not an integer. The original condition determines validity.
Checking Systems of Equations
A solution to a system must satisfy every equation in the system.
Suppose:
\[ x+y=7, \] \[ x-y=1. \]Candidate:
\[ (x,y)=(4,3). \]Check first equation:
\[ 4+3=7. \]Check second equation:
\[ 4-3=1. \]Both are true, so:
\[ \boxed{(4,3)} \]is a solution to the system.
Checking Inverse Functions
If \(f^{-1}\) is truly the inverse of \(f\), then composition should recover the original input:
\[ f^{-1}(f(x))=x \]for appropriate \(x\) in the domain.
Example
\[ f(x)=3x+2. \]Candidate inverse:
\[ f^{-1}(x)=\frac{x-2}{3}. \]Check:
\[ f^{-1}(f(x)) = \frac{(3x+2)-2}{3} = x. \]And:
\[ f(f^{-1}(x)) = 3\left(\frac{x-2}{3}\right)+2 = x. \]Both compositions confirm the inverse relationship.
Verification Is Not Always Re-Solving
A good check is often shorter than the original solution.
If a quadratic equation took several steps to solve, substitution may verify each root in one line.
If a formula was rearranged, substituting the rearranged form into the original identity may be faster than deriving it again.
Verification should be efficient enough that it is worth doing routinely.
What These Checks Can and Cannot Do
Substitution is powerful, but it has limits.
- It verifies that a candidate satisfies a stated condition.
- It does not always prove that all possible solutions have been found.
- It may repeat the same arithmetic if performed carelessly.
- It depends on substituting into the correct original relationship.
For example, checking one root of a polynomial proves that root works, but it does not prove no other roots exist.
Verification and completeness are related but different questions.
Worked Examples
Example 1: Linear Equation
Solve:
\[ 7x+1=29. \] \[ 7x=28, \] \[ x=4. \]Check:
\[ 7(4)+1=29. \] \[ \boxed{x=4} \]passes.
Example 2: Inverse-Operation Check
A process multiplies a number by \(5\) and then subtracts \(8\), producing \(27\).
Reverse:
\[ 27+8=35, \] \[ 35\div5=7. \]Forward check:
\[ 7(5)-8=27. \] \[ \boxed{7} \]is the original number.
Example 3: Formula Rearrangement
From
\[ A=\pi r^2, \]solve for \(r\):
\[ r=\sqrt{\frac A\pi} \]for a nonnegative radius.
Check:
\[ \pi\left(\sqrt{\frac A\pi}\right)^2 = \pi\left(\frac A\pi\right) = A. \]The rearrangement reproduces the original area formula.
Example 4: Extraneous Root
Solve:
\[ \sqrt{x+2}=x. \]Squaring:
\[ x+2=x^2, \] \[ x^2-x-2=0, \] \[ (x-2)(x+1)=0. \]Candidates:
\[ x=2, \qquad x=-1. \]Check:
\[ \sqrt4=2 \]so \(x=2\) works.
\[ \sqrt1=1\neq-1 \]so \(x=-1\) is extraneous.
\[ \boxed{x=2} \]Example 5: System Check
Candidate:
\[ (x,y)=(2,5). \]System:
\[ x+y=7, \] \[ 2x+y=9. \]Check:
\[ 2+5=7, \] \[ 2(2)+5=9. \]Both equations are satisfied.
Example 6: Function Inverse
\[ f(x)=2x-3, \] \[ g(x)=\frac{x+3}{2}. \]Check:
\[ g(f(x)) = \frac{(2x-3)+3}{2} = x. \]So \(g\) undoes \(f\) on the relevant domain.
Common Errors
1. Checking Only the Last Intermediate Equation
Return to the original condition whenever possible.
2. Repeating the Same Mistake in the Check
Substitute carefully and simplify independently.
3. Undoing Operations in the Wrong Order
Inverse operations must be applied in reverse order.
4. Forgetting to Check Every Candidate
Radical equations and squared equations may produce multiple candidates.
5. Checking Only One Equation in a System
A system solution must satisfy every equation.
6. Treating a Successful Check as Proof of Completeness
A candidate can be valid even if additional solutions remain undiscovered.
7. Ignoring Domain Restrictions
A value may satisfy a transformed equation but still violate the original domain.
Practice
Substitution Check
Does \(x=6\) solve
\[ 2x+1=13? \]Check Answer
\[ 2(6)+1 = 12+1 = 13. \]Yes.
Inverse Check
A process adds \(4\) and then multiplies by \(3\), producing \(30\). What is the original number?
Check Answer
Undo multiplication first:
\[ 30\div3=10. \]Then undo addition:
\[ 10-4=\boxed6. \]Extraneous Candidate
If a transformed equation gives candidates \(x=4\) and \(x=-4\), what should you do before accepting both?
Check Answer
Substitute both candidates into the original equation or condition.
System Check
Does \((3,2)\) satisfy
\[ x+y=5, \] \[ x-y=1? \]Check Answer
\[ 3+2=5, \] \[ 3-2=1. \]Yes.
Inverse Function Check
If
\[ f(x)=x+7, \]what function should undo \(f\)?
Check Answer
\[ \boxed{f^{-1}(x)=x-7}. \]The Deeper Idea
Substitution and inverse-operation checks test whether the path from input to output can be connected back to the original mathematical relationship.
\[ \boxed{ \text{candidate} \to \text{original condition} \to \text{true or false} } \]and
\[ \boxed{ \text{forward process} \leftrightarrow \text{inverse process} } \]These checks connect several foundational ideas:
- substitution evaluates a mathematical expression,
- inverse operations undo earlier operations,
- equivalence preserves solution sets,
- functions and inverse functions reverse mappings,
- verification tests whether a candidate belongs to the original problem.
The practical habit is simple: when a solution is available, send it back through the mathematics and see whether the original structure is restored.
Summary
- A substitution check places a candidate solution into the original equation, formula, or condition.
- The original relationship is usually the best place to verify a result.
- An inverse-operation check undoes a process using inverse operations in reverse order.
- Equation solutions can be checked by comparing the original left and right sides after substitution.
- Formula rearrangements can be checked by substituting the rearranged form back into the original formula.
- Function values can be verified directly from the function definition.
- Radical and squared equations should be checked carefully because extraneous candidates may appear.
- Substitution can expose sign, arithmetic, and transcription errors.
- A solution to a system must satisfy every equation in the system.
- Inverse functions can be checked through composition.
- Verification is often shorter than re-solving the entire problem.
- A successful substitution check proves that a candidate satisfies the condition, but not necessarily that every possible solution has been found.
A candidate answer is only a proposal until it has been tested against the problem. Substitution asks whether the answer fits. Inverse operations ask whether the process can be undone. Together, they provide two of the simplest and most reusable verification tools in mathematics.
Problem Solving and Verification
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