1Cademy - Solving a Two-Number Word Problem Using a System of Equations by Substitution

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Problem-Solving Strategy for Systems of Linear Equations

Example

Solving a Two-Number Word Problem Using a System of Equations by Substitution

Apply the seven-step problem-solving strategy for systems of linear equations to an abstract number problem, using substitution to solve the resulting system.

Problem: The sum of two numbers is zero. One number is nine less than the other. 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 zero" gives $n + m = 0$ . "One number is nine less than the other" gives $n = m - 9$ . The system is:

$\left\{\begin{array}{l} n + m = 0 \\ n = m - 9 \end{array}\right.$

Solve using substitution. Because the second equation is already solved for $n$ , substitute $m - 9$ for $n$ in the first equation:

$m - 9 + m = 0$

Combine like terms: $2m - 9 = 0$ . Add $9$ to both sides: $2m = 9$ . Divide both sides by $2$ : $m = \frac{9}{2}$ .

Substitute $m = \frac{9}{2}$ into the second equation to find $n$ :

$n = \frac{9}{2} - 9 = \frac{9}{2} - \frac{18}{2} = -\frac{9}{2}$

Check: Does $\frac{9}{2} + \left(-\frac{9}{2}\right) = 0$ ? Yes ✓. Is $-\frac{9}{2}$ nine less than $\frac{9}{2}$ ? $\frac{9}{2} - 9 = -\frac{9}{2}$ ✓.
Answer: The numbers are $\frac{9}{2}$ and $-\frac{9}{2}$ .

This example demonstrates how using two variables and a system of equations can simplify the setup of a word problem. Each verbal statement translates directly into one equation — one for the sum constraint and one for the relationship between the numbers — without needing to express both unknowns through a single variable. The substitution method is ideal here because the second equation already expresses $n$ in terms of $m$ , so substitution can begin immediately. The fractional solution reinforces the practical advantage of substitution over graphing for non-integer answers.

Updated 2026-04-21

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