Linear Equation Word Problems

Introduction to Word Problems

Word problems translate real-world situations into mathematical equations. The key is understanding the language and identifying the mathematical relationships.

Why Word Problems Matter:

• They connect math to real life
• They develop critical thinking skills
• They appear on standardized tests
• They show practical applications of algebra

The Problem-Solving Process

Step 1: Read and Understand

• Read the entire problem carefully
• Identify what you're looking for
• Note any given information

Step 2: Define Variables

• Choose a letter to represent the unknown
• Write what the variable represents in words

Step 3: Write an Equation

• Translate words into math symbols
• Use the relationships described

Step 4: Solve the Equation

• Use algebraic techniques
• Show all work step-by-step

Step 5: Check and Interpret

• Does the answer make sense?
• Answer the question in a complete sentence
• Check in the original problem (not just the equation)

Key Words and Phrases

Addition (+):

• sum, total, combined, more than, increased by, plus, added to

Subtraction (-):

• difference, less than, decreased by, minus, fewer, reduced by, subtracted from

Multiplication (×):

• product, times, of, multiplied by, at (rate), per, twice, double, triple

Division (÷):

• quotient, divided by, ratio, per, split, shared equally

Equals (=):

• is, equals, is equal to, amounts to, results in, the same as

Number Problems

These involve relationships between numbers.

Example 1: Consecutive Integers Problem: The sum of three consecutive integers is 72. Find the integers.

Let x = first integer Then x + 1 = second integer And x + 2 = third integer

Equation: x + (x + 1) + (x + 2) = 72 3x + 3 = 72 3x = 69 x = 23

Answer: The integers are 23, 24, and 25.

Check: 23 + 24 + 25 = 72 ✓

Example 2: Number Relationships Problem: One number is 5 more than twice another number. Their sum is 38. Find the numbers.

Let x = smaller number Then 2x + 5 = larger number

Equation: x + (2x + 5) = 38 3x + 5 = 38 3x = 33 x = 11

Answer: The numbers are 11 and 27.

Check: 11 + 27 = 38 ✓ and 27 = 2(11) + 5 ✓

Example 3: Even Consecutive Integers Problem: Find three consecutive even integers whose sum is 126.

Let x = first even integer Then x + 2 = second even integer And x + 4 = third even integer

Equation: x + (x + 2) + (x + 4) = 126 3x + 6 = 126 3x = 120 x = 40

Answer: The integers are 40, 42, and 44.

Age Problems

These involve relationships between people's ages.

Example 1: Current Ages Problem: Maria is 3 years older than her brother. The sum of their ages is 27. How old is each?

Let x = brother's age Then x + 3 = Maria's age

Equation: x + (x + 3) = 27 2x + 3 = 27 2x = 24 x = 12

Answer: Brother is 12, Maria is 15.

Example 2: Ages in the Future Problem: Sarah is 8 years old. In how many years will she be 20?

Let x = number of years Her future age: 8 + x

Equation: 8 + x = 20 x = 12

Answer: In 12 years.

Example 3: Ages in the Past Problem: Tom is currently 24. His age is 6 more than twice his age 5 years ago. Is this true?

Let x = Tom's age 5 years ago Then x + 5 = Tom's current age (24)

So x = 19 (Tom was 19 five years ago) Check: Is 24 six more than twice 19? 2(19) + 6 = 38 + 6 = 44 ≠ 24

This reveals an inconsistency in the problem setup.

Money and Cost Problems

Example 1: Shopping Problem: You buy 3 notebooks and 2 pens for $11. If each pen costs$1.50, what is the cost of each notebook?

Let x = cost of one notebook Total: 3x + 2(1.50) = 11

Equation: 3x + 3 = 11 3x = 8 x = 2.67 (approximately)

Answer: Each notebook costs about $2.67.

Example 2: Ticket Sales Problem: Student tickets cost $3 and adult tickets cost$5. If 200 tickets were sold for $840, how many were student tickets?

Let x = number of student tickets Then 200 - x = number of adult tickets

Equation: 3x + 5(200 - x) = 840 3x + 1000 - 5x = 840 -2x = -160 x = 80

Answer: 80 student tickets were sold.

Example 3: Budget Problem: You have $50 to buy supplies. Binders cost$4 each and you need to buy 3 folders at $2 each. How many binders can you buy?

Let x = number of binders Cost equation: 4x + 3(2) ≤ 50

4x + 6 ≤ 50 4x ≤ 44 x ≤ 11

Answer: You can buy at most 11 binders.

Geometry Problems

Example 1: Perimeter Problem: The perimeter of a rectangle is 50 cm. The length is 5 cm more than the width. Find the dimensions.

Let w = width Then w + 5 = length

Perimeter formula: 2w + 2(w + 5) = 50 2w + 2w + 10 = 50 4w = 40 w = 10

Answer: Width is 10 cm, length is 15 cm.

Check: 2(10) + 2(15) = 20 + 30 = 50 ✓

Example 2: Angles Problem: Two angles are supplementary (sum to 180°). One angle is 30° more than twice the other. Find both angles.

Let x = smaller angle Then 2x + 30 = larger angle

Equation: x + (2x + 30) = 180 3x + 30 = 180 3x = 150 x = 50

Answer: The angles are 50° and 130°.

Example 3: Triangle Problem: The second angle of a triangle is twice the first. The third angle is 20° more than the first. Find all three angles.

Let x = first angle Then 2x = second angle And x + 20 = third angle

Sum of angles: x + 2x + (x + 20) = 180 4x + 20 = 180 4x = 160 x = 40

Answer: The angles are 40°, 80°, and 60°.

Distance, Rate, and Time Problems

Use the formula: Distance = Rate × Time (d = rt)

Example 1: Travel Time Problem: You drive 240 miles at 60 mph. How long does it take?

Using d = rt: 240 = 60t t = 4

Answer: 4 hours.

Example 2: Meeting Problem: Two cars leave the same point driving in opposite directions. One travels at 50 mph, the other at 60 mph. After how many hours are they 330 miles apart?

Let t = time in hours Distance apart = distance car 1 + distance car 2

Equation: 50t + 60t = 330 110t = 330 t = 3

Answer: After 3 hours.

Example 3: Catch Up Problem: A car leaves at noon traveling 55 mph. Another car leaves at 1 PM traveling 65 mph on the same route. When does the second car catch up?

Let t = time the second car travels (hours) Then t + 1 = time the first car travels

When they meet, distances are equal: 65t = 55(t + 1) 65t = 55t + 55 10t = 55 t = 5.5

Answer: The second car catches up at 6:30 PM.

Percent Problems

Example 1: Discount Problem: A shirt is on sale for 25% off. If the sale price is $30, what was the original price?

Let x = original price Sale price = original - discount

Equation: x - 0.25x = 30 0.75x = 30 x = 40

Answer: The original price was $40.

Example 2: Tax Problem: The total cost of an item including 8% sales tax is $54. What was the pre-tax price?

Let x = pre-tax price Total = price + tax

Equation: x + 0.08x = 54 1.08x = 54 x = 50

Answer: The pre-tax price was $50.

Mixture Problems (Preview)

Example: Combining Solutions Problem: How many liters of 20% acid solution should be mixed with 5 liters of 50% acid solution to get a 30% solution?

Let x = liters of 20% solution

Amount of pure acid from each: 0.20x + 0.50(5) = 0.30(x + 5) 0.20x + 2.5 = 0.30x + 1.5 1 = 0.10x x = 10

Answer: 10 liters of 20% solution.

Common Mistakes to Avoid

1. Not defining the variable clearly Always write what your variable represents!

2. Translating words incorrectly "5 less than x" is x - 5, NOT 5 - x

3. Forgetting units Keep track of dollars, hours, miles, etc.

4. Not checking the answer in context Does a negative age make sense? Can you buy 3.7 tickets?

5. Answering the wrong question If asked "How old is Maria?" don't just solve for x and stop!

Problem-Solving Tips

• Draw a picture or diagram when possible
• Make a table to organize information
• Use the variable for the quantity you know least about
• Write formulas you might need (d = rt, P = 2l + 2w, etc.)
• Work backwards from the answer choices if given
• Check reasonableness: Is the answer logical?

Practice Strategy

1. Start with easier problems to build confidence
2. Identify the problem type (age, money, geometry, etc.)
3. Look for keywords that indicate operations
4. Write the equation before solving
5. Show all steps clearly
6. Always check your work
7. Practice translating English to algebra

Quick Reference - Common Setups

| Problem Type | Variable Setup | Common Formula | |-------------|----------------|----------------| | Consecutive integers | x, x+1, x+2 | Sum or product | | Age | x = age now | Current, past, future | | Perimeter rectangle | w, l = w+k | P = 2l + 2w | | Supplementary angles | x, 180-x | Sum = 180° | | Distance | d = rt | Distance = rate × time | | Discount | x = original | Sale = x - discount |

Word Problem Checklist

Before submitting your answer:

• ☐ Did I define my variable?
• ☐ Did I write an equation?
• ☐ Did I solve correctly?
• ☐ Did I check my solution?
• ☐ Does my answer make sense?
• ☐ Did I answer in a complete sentence?
• ☐ Did I include units if needed?