1Cademy - Solving a Two-Number Sum and Difference Word Problem Using a System of Equations by Elimination

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Problem-Solving Strategy for Systems of Linear Equations

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Solving a Two-Number Sum and Difference Word Problem Using a System of Equations by Elimination

Apply the seven-step problem-solving strategy for systems of linear equations to an abstract number problem where both the sum and the difference of the two unknowns are given, using elimination to solve the resulting system.

Problem: The sum of two numbers is 39. Their difference is 9. Find the numbers.

Read the problem.
Identify what to find: two numbers.
Name the unknowns: Let $n$ = the first number and $m$ = the second number.
Translate into a system of equations. "The sum of two numbers is 39" gives $n + m = 39$ . "Their difference is 9" gives $n - m = 9$ . The system is:

$\left\{\begin{array}{l} n + m = 39 \\ n - m = 9 \end{array}\right.$

Solve using elimination. Both equations are already in standard form, and the coefficients of $m$ are already opposites ( $+1$ and $-1$ ). Adding the two equations directly eliminates $m$ :

$n + m + n - m = 39 + 9$

$2n = 48$

Divide both sides by $2$ : $n = 24$ .

Substitute $n = 24$ into the first equation:

$24 + m = 39$

$m = 15$

Check: $24 + 15 = 39$ ✓ and $24 - 15 = 9$ ✓.
Answer: The numbers are 24 and 15.

This example illustrates why the elimination method is a natural fit for certain word problems. Because "sum" and "difference" translate directly into equations in standard form ( $n + m = 39$ and $n - m = 9$ ), the coefficients of $m$ are already opposites, so no multiplication step is needed — the equations can be added immediately. Compare this with the substitution approach to similar number problems, where an additional step of isolating a variable would be required before substituting.

Updated 2026-04-21

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