Checking Solutions

Finding an answer is not always the end of a mathematics problem. A strong mathematical solution also shows that the answer is correct and reasonable.

For a system of equations, an ordered pair is a solution only if it satisfies both original equations.

For example:

x + y = 9

2x − y = 3

Suppose we find:

(x, y) = (4, 5)

Check the first equation:

4 + 5 = 9 ✓

Check the second equation:

2(4) − 5 = 3

8 − 5 = 3 ✓

Since both equations are true:

(4, 5) is verified as the solution.

A useful checking process is:

Substitute → Calculate → Compare → Evaluate → Communicate

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What Does It Mean to Check a Solution?

To check or verify a solution means to test whether it satisfies the original mathematical conditions.

For a system:

Equation 1

Equation 2

the proposed values must make:

Equation 1 true

AND

Equation 2 true

Satisfying only one equation is not enough.

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Why Check Both Equations?

Consider:

x + y = 10

2x − y = 5

Suppose someone claims:

(4, 6)

Check the first:

4 + 6 = 10 ✓

But check the second:

2(4) − 6 = 2

The equation requires:

5

So:

2 ≠ 5 ✗

Therefore:

(4, 6) is not a solution to the system.

It lies on one line but not the other.

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The Graphical Meaning

Each equation represents a line.

A solution to the system must lie on both lines simultaneously.

Therefore, the solution is normally the point where the lines intersect.

A point lying on only one line is not a solution to the system.

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The Basic Verification Process

Suppose the proposed solution is:

(3, 2)

for:

2x + y = 8

x + 3y = 9

Check Equation 1

Substitute:

x = 3

y = 2

into:

2x + y = 8

2(3) + 2 = 8

6 + 2 = 8

8 = 8 ✓

Check Equation 2

3 + 3(2) = 9

3 + 6 = 9

9 = 9 ✓

Therefore:

(3, 2) is the solution.

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Use the Original Equations

Whenever possible, check using the original equations, not equations that were rearranged or multiplied during the solution.

Suppose the original system is:

2x + 3y = 12

5x + 2y = 19

During elimination you might multiply these equations.

But once you find:

(3, 2)

check:

2(3) + 3(2) = 12

and:

5(3) + 2(2) = 19

This checks the actual problem you were asked to solve.

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Checking a Substitution Solution

Solve:

y = 2x + 1

3x + y = 11

Substitute:

3x + (2x + 1) = 11

5x + 1 = 11

5x = 10

x = 2

Then:

y = 2(2) + 1

y = 5

Proposed solution:

(2, 5)

Now verify it.

First equation:

5 = 2(2) + 1

5 = 5 ✓

Second equation:

3(2) + 5 = 11

11 = 11 ✓

Therefore:

(2, 5) is correct.

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Checking an Elimination Solution

Solve:

x + y = 7

x − y = 1

Add:

2x = 8

x = 4

Substitute:

4 + y = 7

y = 3

Proposed solution:

(4, 3)

Check:

4 + 3 = 7 ✓

4 − 3 = 1 ✓

Therefore:

(4, 3) is verified.

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Checking Fractional Solutions

Fractions are not automatically a sign that something went wrong.

Suppose:

x + y = 4

2x − y = 1

Add:

3x = 5

Therefore:

x = 5/3

Then:

y = 4 − 5/3

y = 12/3 − 5/3

y = 7/3

Check:

5/3 + 7/3 = 12/3 = 4 ✓

Second equation:

2(5/3) − 7/3

= 10/3 − 7/3

= 3/3

= 1 ✓

Therefore:

(5/3, 7/3)

is correct.

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Checking Decimal Solutions

Suppose:

x = 2.5

y = 1.5

for:

x + y = 4

2x − y = 3.5

Check:

2.5 + 1.5 = 4 ✓

Second equation:

2(2.5) − 1.5

= 5 − 1.5

= 3.5 ✓

Therefore:

(2.5, 1.5)

is valid.

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Be Careful with Negative Values

Suppose the proposed solution is:

(−3, 5)

for:

2x + y = −1

x − y = −8

Check:

2(−3) + 5

= −6 + 5

= −1 ✓

Second equation:

−3 − 5 = −8 ✓

Therefore:

(−3, 5)

is correct.

Using parentheses around negative substituted values helps prevent sign errors.

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Parentheses Make Checking Safer

If:

x = −4

then:

3x

should be written:

3(−4)

If:

x = −4

then:

x²

should be checked as:

(−4)²

Writing substituted values in parentheses makes the calculation clearer and reduces mistakes.

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Identifying Calculation Errors

Checking is especially useful for finding arithmetic mistakes.

Suppose a student solves:

x + y = 8

2x − y = 1

and claims:

(3, 5)

Check the first:

3 + 5 = 8 ✓

Check the second:

2(3) − 5

= 6 − 5

= 1 ✓

The solution is correct.

Now suppose the student claims:

(4, 4)

First equation:

4 + 4 = 8 ✓

Second:

2(4) − 4 = 4

but the equation requires:

1

Therefore an error occurred somewhere in the original solution.

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Checking Helps Locate Errors

If both equations fail, there may be a major error such as:

• incorrect algebra
• incorrect ordered pair
• wrong system copied
• incorrect variable values

If one equation works and one fails, there may be:

• a substitution error
• an arithmetic error
• an incorrect second variable
• a sign error

Checking does more than say "wrong." It gives clues about where the mistake occurred.

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Sign Errors

Suppose:

3x − 2y = 11

and:

x = 5, y = 2

Check:

3(5) − 2(2)

= 15 − 4

= 11 ✓

A student might incorrectly calculate:

15 + 4 = 19

because they ignored the subtraction sign.

Writing every step clearly makes these mistakes easier to identify.

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Distribution Errors

Suppose an equation contains:

3(x − y) = 6

and the proposed solution is:

x = 4, y = 2

Substitute:

3(4 − 2) = 6

3(2) = 6

6 = 6 ✓

Do not incorrectly write:

3(4) − 2

because the 3 multiplies the entire expression inside the parentheses.

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Errors from Elimination

Suppose:

2x + 3y = 12

5x + 2y = 19

A student multiplies the first equation by 2 and accidentally writes:

4x + 6y = 12

The correct multiplication is:

4x + 6y = 24

Checking the final solution in the original equations can expose this type of error.

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Errors from Substitution

Suppose:

y = 3x − 4

and:

2x − y = 6

When substituting, write:

2x − (3x − 4) = 6

not:

2x − 3x − 4 = 6

The correct simplification is:

2x − 3x + 4 = 6

because the negative sign applies to the entire substituted expression.

Verification can reveal errors caused by incorrect substitution.

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Checking the Ordered Pair

Remember:

(x, y)

means:

• first coordinate = x
• second coordinate = y

If:

x = 2

and:

y = 7

then:

(2, 7)

not:

(7, 2)

A correct calculation can still produce an incorrect final answer if the ordered pair is written backward.

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Reasonableness

An answer can satisfy algebraic calculations but still need to be interpreted in context.

Ask:

• Is the sign reasonable?
• Is the value too large or too small?
• Are the units correct?
• Can this quantity be fractional?
• Can this quantity be negative?
• Does the result fit the situation?

This is called checking for reasonableness.

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Mathematical Versus Practical Reasonableness

Suppose a system models:

x = number of buses

and the calculation gives:

x = 4.3

Mathematically, 4.3 is a valid number.

Practically, you cannot normally hire 0.3 of a bus.

If 4 buses are insufficient, the situation may require:

5 buses

The context determines how the mathematical answer should be interpreted.

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Negative Answers

Suppose a word problem gives:

x = −6

If x represents temperature, −6°C may be perfectly reasonable.

If x represents the number of students attending an event, −6 students is impossible.

Therefore, negative answers are not automatically wrong.

They must be interpreted according to the variable's meaning.

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Fractional Answers

Suppose:

x = 2.5

If x represents:

2.5 hours

this may be reasonable.

If x represents:

2.5 people

it is normally unreasonable.

Always connect the mathematical value to the real quantity.

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Checking Units

Suppose a system models ticket sales and produces:

x = 25

y = 40

If x and y represent numbers of tickets, the answer should be communicated as:

25 adult tickets and 40 student tickets

not simply:

(25, 40)

In practical mathematics, labels and units are part of a complete answer.

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Estimation as a Check

Before or after solving, estimate what kind of answer seems reasonable.

Suppose:

5 notebooks cost approximately $20.

A calculated notebook price of:

$400

is probably unreasonable.

Even before substituting back into the equations, the size of the answer suggests that an error occurred.

Estimation is a useful first check.

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Checking by a Different Algebraic Method

One powerful way to verify a solution is to solve the same system using another method.

Suppose:

x + y = 7

2x − y = 5

Elimination

Add:

3x = 12

x = 4

Then:

y = 3

Solution:

(4, 3)

Now verify using substitution.

From:

x + y = 7

write:

y = 7 − x

Substitute:

2x − (7 − x) = 5

3x − 7 = 5

3x = 12

x = 4

y = 3

Both methods agree.

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Checking Algebraically and Graphically

Suppose algebra gives:

(2, 4)

If the graphs of the equations intersect at approximately:

(2, 4)

this provides additional evidence that the solution is correct.

Graphing can therefore be used as an independent check.

However, if the graph is hand drawn, remember that it may provide only an approximate answer.

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Multiple Methods Increase Confidence

A solution can be checked in several ways:

Direct substitution

Put the values into both equations.

Alternative algebraic method

Solve again using substitution or elimination.

Graphing

Check whether the lines intersect at the calculated point.

Estimation

Determine whether the values are reasonable.

Context check

Determine whether the answer makes sense in the real situation.

Using several forms of verification makes a solution more reliable.

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Exact and Approximate Checks

Suppose the exact solution is:

(7/3, 5/3)

A graph might show an intersection near:

(2.33, 1.67)

These are consistent because:

7/3 ≈ 2.33

and:

5/3 ≈ 1.67

The algebraic solution is exact.

The graphical solution is approximate.

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Checking a Graphical Solution

Suppose a graph suggests:

(3, 4)

Do not assume it is automatically exact.

Substitute:

x = 3, y = 4

into both equations.

If both equations become true, the graphical reading is correct.

If not, the graph may have been read inaccurately.

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No-Solution Systems

Consider:

y = 2x + 3

y = 2x − 4

If we try substitution:

2x + 3 = 2x − 4

Subtract 2x:

3 = −4

This is impossible.

Therefore:

No solution

Graphically, the lines have the same slope but different intercepts, so they are parallel.

This is another example of verifying an algebraic conclusion using a different method.

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Infinitely Many Solutions

Consider:

y = 2x + 3

2y = 4x + 6

Substitute:

2(2x + 3) = 4x + 6

4x + 6 = 4x + 6

This is always true.

Therefore:

Infinitely many solutions

We can verify this by simplifying the second equation:

2y = 4x + 6

Divide by 2:

y = 2x + 3

The equations are identical.

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Checking Special Systems

For a system with no solution, there is no ordered pair to substitute into both equations.

Instead, verification might involve:

• showing that elimination produces a contradiction
• comparing slopes and intercepts
• graphing the parallel lines

For a system with infinitely many solutions, verification might involve:

• showing that one equation simplifies to the other
• obtaining an identity such as 0 = 0
• graphing and observing that the lines coincide

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Communicating a Complete Solution

A complete mathematical solution should be easy for another person to follow.

It should normally include:

1. the method used
2. clear algebraic steps
3. values of both variables
4. the ordered pair
5. verification in both original equations
6. interpretation when the problem has a real-world context

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Example of a Complete Solution

Solve:

2x + y = 11

x − y = 1

Use elimination.

Add:

3x = 12

Therefore:

x = 4

Substitute into:

x − y = 1

4 − y = 1

−y = −3

y = 3

Solution:

(4, 3)

Check Equation 1:

2(4) + 3 = 11

11 = 11 ✓

Check Equation 2:

4 − 3 = 1

1 = 1 ✓

Therefore:

The solution is (4, 3).

This is complete, organized, and verifiable.

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Real-World Example: Ticket Sales

A school sells adult tickets for $12 and student tickets for $8.

A total of 50 tickets are sold for $480.

Let:

a = adult tickets

s = student tickets

System:

a + s = 50

12a + 8s = 480

Suppose solving gives:

a = 20

s = 30

Now check.

Number of tickets:

20 + 30 = 50 ✓

Revenue:

12(20) + 8(30)

= 240 + 240

= 480 ✓

The values also make sense:

• neither is negative
• both are whole numbers
• together they total 50 tickets

Therefore:

20 adult tickets and 30 student tickets were sold.

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Real-World Example: Ages

Suppose:

a + b = 30

a = b + 6

Solving gives:

a = 18

b = 12

Check:

18 + 12 = 30 ✓

and:

18 = 12 + 6 ✓

The values are positive and reasonable for ages.

Therefore the solution is valid in both the algebra and the context.

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Real-World Example: Measurement

Suppose a rectangle has:

l + w = 17

l − w = 5

Solving gives:

l = 11

w = 6

Check:

11 + 6 = 17 ✓

11 − 6 = 5 ✓

If the measurements are in centimetres:

length = 11 cm

width = 6 cm

Both dimensions are positive, so the result is physically reasonable.

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Catching an Unreasonable Answer

Suppose a rectangle problem produces:

length = 11 cm

width = −6 cm

Even before checking the equations, a negative physical width is a warning.

This suggests:

• an algebraic error
• an incorrect equation
• an incorrect interpretation

Reasonableness is an important error-detection tool.

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Checking Before Submitting

A useful final checklist is:

Equation Check

Does the solution satisfy Equation 1?

Does it satisfy Equation 2?

Calculation Check

Are arithmetic and signs correct?

Method Check

Could another method confirm the answer?

Reasonableness Check

Does the answer make sense?

Communication Check

Are variables, ordered pairs, labels, and units clearly stated?

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A Systematic Verification Strategy

Use:

Substitute → Calculate → Compare → Cross-Check → Interpret → Communicate

Substitute

Put the proposed values into both original equations.

Calculate

Evaluate each side carefully.

Compare

Confirm that the left and right sides are equal.

Cross-Check

Use another method when appropriate.

Interpret

Determine whether the answer makes sense in context.

Communicate

State the final solution clearly with appropriate notation and units.

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Common Mistakes

Mistake: Checking only one equation.

A system solution must satisfy both equations.

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Mistake: Checking transformed equations instead of the originals.

Use the original equations whenever possible.

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Mistake: Substituting x and y in the wrong positions.

Remember that (x, y) gives x first and y second.

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Mistake: Forgetting parentheses around negative numbers.

Write:

3(−4)

rather than relying on mental substitution.

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Mistake: Assuming a fraction means the answer is wrong.

Fractional solutions are common and can be completely correct.

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Mistake: Assuming negative values are always wrong.

Their reasonableness depends on what the variable represents.

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Mistake: Accepting an algebraic answer that makes no practical sense.

Interpret the answer in context.

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Mistake: Ignoring units.

A complete practical answer should identify what the numbers measure.

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Mistake: Using an approximate graph to reject an exact algebraic solution.

A hand-drawn graph may not be precise enough.

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Mistake: Repeating the same calculation as a "different check."

A stronger verification uses independent evidence, such as substitution after elimination or graphing after algebra.

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Mistake: Writing only the final ordered pair.

Show enough work that another person can understand and verify the solution.

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Key Terms

Check — test whether a mathematical result satisfies the required conditions.

Verify — confirm that a solution is correct.

Substitution — replacing a variable with a known value or equivalent expression.

System of equations — two or more equations involving the same variables.

Solution to a system — values that satisfy every equation simultaneously.

Ordered pair — pair written (x, y) representing values of two variables.

Original equation — equation as it appeared before rearranging, multiplying, or otherwise transforming it.

Calculation error — mistake made during arithmetic or algebraic manipulation.

Sign error — mistake involving positive or negative signs.

Reasonableness — whether an answer makes sense mathematically and in its context.

Exact solution — precise value such as 7/3.

Approximate solution — estimated value such as 2.33.

Cross-check — verify a result using another method or source of evidence.

Graphical verification — using a graph to confirm the approximate location or nature of a solution.

Interpretation — explaining what a mathematical result means in its context.

Complete solution — solution showing sufficient working, verification, and interpretation to communicate the reasoning clearly.

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Key Takeaways

• Solving and checking are both important parts of mathematics.
• A solution to a system must satisfy both original equations.
• Checking only one equation is not enough.
• Substitute both variable values carefully into each original equation.
• Parentheses are especially useful when substituting negative values.
• Verification can reveal arithmetic, sign, substitution, and multiplication errors.
• Fractional and decimal solutions can be completely correct.
• Negative solutions may or may not be reasonable depending on the context.
• Mathematical correctness and practical reasonableness are different checks.
• Real-world answers should include appropriate labels and units.
• Estimation can help identify obviously unreasonable answers.
• A second algebraic method can provide independent verification.
• Graphing can be used to check an algebraic solution visually.
• Graphical solutions may be approximate while algebraic solutions can be exact.
• Different correct methods must agree on the same system solution.
• A contradiction such as 0 = 5 indicates no solution.
• An identity such as 0 = 0 may indicate infinitely many solutions.
• For special systems, verification may involve comparing equations or graphs rather than checking one ordered pair.
• Verification should normally use the original equations.
• A complete solution should show enough working for another person to follow.
• In contextual problems, explain what each variable value represents.
• Always ask whether the final answer makes sense.
• Checking is not merely an extra step—it is part of producing a reliable mathematical solution.

Check Your Understanding

1. What must an ordered pair do to be a solution to a system of equations?
2. Explain why checking only one equation is not enough.
3. Verify whether (3, 2) solves 2x + y = 8 and x + 3y = 9.
4. Verify whether (4, 6) solves x + y = 10 and 2x − y = 5.
5. Verify whether (−3, 5) solves 2x + y = −1 and x − y = −8.
6. Check whether (5/3, 7/3) solves x + y = 4 and 2x − y = 1.
7. Explain why parentheses are useful when checking negative values.
8. A student multiplies 2x + 3y = 12 by 2 and writes 4x + 6y = 12. Identify the error.
9. A student substitutes y = 3x − 4 into 2x − y = 6 and writes 2x − 3x − 4 = 6. Identify the error.
10. Explain why it is best to verify a solution using the original equations.
11. What is the difference between an exact solution and an approximate solution?
12. Explain how graphing can be used to check an algebraic solution.
13. Explain how elimination could be used to check a solution originally found using substitution.
14. A problem about the number of students produces x = −12. Explain why this answer should be questioned.
15. A travel problem produces t = 2.5 hours. Explain why the decimal value is not necessarily unreasonable.
16. A bus problem produces 4.2 buses. Explain how the context affects the interpretation of this answer.
17. Explain what 0 = 9 tells you about a system and describe one other way to verify this conclusion.
18. Explain what 0 = 0 may tell you about a system and describe how you could verify it.
19. Adult tickets cost $12 and student tickets cost $8. A school sells 50 tickets for $480. Verify that 20 adult tickets and 30 student tickets satisfy the system, then explain why the answer is reasonable.
20. Solve 2x + y = 11 and x − y = 1 using a method of your choice. Then verify the solution in both original equations, check it using a second method, and communicate the complete solution clearly.