applications

Word Problems with Linear Equations

Algebra I30 minDifficulty

You should know: linear equation, systems of linear equations

Overview

Word problems translate real-world situations into algebraic equations. For linear problems: identify the unknown, assign a variable, write an equation from the problem's constraints, solve, then check the answer makes sense in context. Common types include age problems, distance-rate-time, mixture problems, and number problems.

Intuition

The key is translating English into math: 'is' means =, 'more than' means +, 'times as much' means *, 'less than' means -. Define one variable clearly, express everything else in terms of it, write the equation from the problem's main condition, and solve. Always check your answer makes sense (negative ages don't!)

Formal Definition

Definition

Problem-solving strategy: (1) Read carefully, identify what's unknown. (2) Let x = unknown. (3) Express other quantities in terms of x. (4) Write equation from the constraint. (5) Solve. (6) Check. Distance formula: d = r*t. Mixture: amount_1 * conc_1 + amount_2 * conc_2 = total * conc_total.

d = r \cdot t

Distance = rate * time

c_1 v_1 + c_2 v_2 = c_f(v_1 + v_2)

Mixture: concentration balance

Notation

Notation	Meaning
$d = rt$	Distance-rate-time relationship

Theorems

Theorem 1: Linear Model

Any situation where one quantity changes at a constant rate with respect to another is modeled by a linear equation. The slope is the rate of change; the y-intercept is the initial value.

Worked Examples

1
Let n = the number. Write the equation.
$n + 7 = 3n - 5$
2
Solve: 7 + 5 = 3n - n => 12 = 2n.
$n = 6$

✓ Answer

n = 6. Check: 6 + 7 = 13 = 3(6) - 5 = 13. Correct.

Practice Problems

Easyapplication

Emma is 3 times as old as her brother. In 6 years, she'll be twice as old. How old are they now?

Common Mistakes

Common Mistake

Not checking the answer in the original problem after solving.

Always substitute back. A non-integer or negative answer that doesn't fit the context (negative age, fractional number of people) signals an error.

Quiz

If a train travels at 80 mph for t hours and covers 240 miles, then t equals:

234160

Historical Background

Word problems appear in the oldest mathematical texts. The Rhind Papyrus (c. 1650 BCE) contains practical arithmetic problems. Al-Khwarizmi's 'Al-Kitab al-Mukhtasar fi Hisab al-Jabr wal-Muqabala' (c. 830 CE), which gave us the word 'algebra,' is largely a collection of practical problem types. Problem-solving has always been the primary motivation for developing algebraic methods.

830 CE
Al-Khwarizmi's algebra text presents systematic problem types
Muhammad ibn Musa al-Khwarizmi

Summary

Strategy: define variable, express quantities, write equation, solve, check.
d = rt for distance problems.
Age problems: set up equations for current ages, then for future/past ages.
Mixture: amount * concentration = total solute; set up for before and after.

References

BookHall, B. and Fabricant, M. Algebra 1. Prentice Hall, 2001.