How to Solve Systems of Equations Word Problems? (+FREE Worksheet!)
Systems of equations word problems are a practical application of algebra: you translate a real-world scenario into two equations and solve the system to find the two unknown quantities. Mastering this skill connects abstract algebra to everyday situations like pricing, distance-rate-time, and mixture problems.
What Are Systems of Equations Word Problems?
These problems describe a real situation with two unknown quantities and give you two pieces of information — one for each unknown. Your job is to define variables, write two equations from the given information, and solve the system to find the values of the unknowns.
How to Solve Systems of Equations Word Problems
Step 1: Define Your Variables
Assign a letter to each unknown. Be specific: for example, let \(\color{blue}{a}\) = number of adult tickets and \(\color{blue}{c}\) = number of child tickets.
Step 2: Write Two Equations
Translate each piece of given information into an equation. Look for a total (sum) equation and a value (cost, distance, or rate) equation.
Step 3: Solve the System
Use substitution or elimination — whichever fits the equations better.
Step 4: Interpret and Check
Make sure your answer makes sense in context (e.g., no negative tickets) and verify it satisfies both original conditions.
Step-by-Step Summary
- Read carefully and identify the two unknowns.
- Define variables with clear labels.
- Write one equation for each piece of numerical information.
- Solve by substitution or elimination.
- Answer in a complete sentence and check both conditions.
Watch: Systems of Equations Word Problems
Khan Academy sets up and solves a real-world system from scratch:
Systems Word Problems – Worked Examples
Example 1: Ticket Sales
A school sold 100 tickets for a concert. Adult tickets cost $10 and child tickets cost $6. The total revenue was $840. How many of each type were sold?
Let \(\color{blue}{a}\) = adult tickets and \(\color{blue}{c}\) = child tickets.
Equation 1 (total tickets): \(\color{blue}{a + c = 100}\)
Equation 2 (total revenue): \(\color{blue}{10a + 6c = 840}\)
From Eq. 1: \(\color{blue}{c = 100 – a}\). Substitute into Eq. 2:
\(\color{blue}{10a + 6(100 – a) = 840}\) → \(\color{blue}{4a = 240}\) → \(\color{blue}{a = 60}\), \(\color{blue}{c = 40}\).
60 adult tickets and 40 child tickets.
Example 2: Speed Problem
Juan runs 2 mph faster than Maria. Together, each running for 2 hours, they cover a combined 26 miles. Find each person’s speed.
Let \(\color{blue}{m}\) = Maria’s speed and \(\color{blue}{j}\) = Juan’s speed.
Eq. 1: \(\color{blue}{j = m + 2}\) (Juan is 2 mph faster)
Eq. 2: \(\color{blue}{2j + 2m = 26}\) (combined distance)
Substitute: \(\color{blue}{2(m + 2) + 2m = 26}\) → \(\color{blue}{4m = 22}\) → \(\color{blue}{m = 5.5}\), \(\color{blue}{j = 7.5}\).
Maria: 5.5 mph; Juan: 7.5 mph.
Example 3: Coin Problem
A piggy bank holds 30 coins, all dimes and quarters. The total value is $5.55. How many dimes and how many quarters are there?
Let \(\color{blue}{d}\) = dimes and \(\color{blue}{q}\) = quarters.
Eq. 1: \(\color{blue}{d + q = 30}\)
Eq. 2: \(\color{blue}{0.10d + 0.25q = 5.55}\)
Multiply Eq. 2 by 100: \(\color{blue}{10d + 25q = 555}\). Multiply Eq. 1 by 10: \(\color{blue}{10d + 10q = 300}\). Subtract: \(\color{blue}{15q = 255}\) → \(\color{blue}{q = 17}\), \(\color{blue}{d = 13}\).
13 dimes and 17 quarters.
More Practice: Walk & Ride Word Problem
Khan Academy solves a distance-rate-time word problem using a system of equations:
Exercises: Systems of Equations Word Problems
- Two numbers have a sum of 48 and a difference of 12. Find the two numbers.
- A store sells pens for $1.50 and notebooks for $3.00. Maya bought 8 items for $18.00. How many of each did she buy?
- A boat travels 60 miles downstream in 3 hours and 60 miles upstream in 5 hours. Find the speed of the boat in still water and the speed of the current.
- The perimeter of a rectangle is 56 cm. The length is 4 cm more than twice the width. Find the length and width.
- Two friends start 100 miles apart and drive toward each other. One drives at 40 mph and the other at 60 mph. How long until they meet?
Answers
- 30 and 18
- 4 pens and 4 notebooks
- Boat speed in still water: 16 mph; current speed: 4 mph
- \(\color{blue}{\text{ Width } = 8}\) cm; \(\color{blue}{\text{ length } = 20}\) cm
- 1 hour
Frequently Asked Questions
How do I identify what the variables represent?
Look for the two quantities the problem asks you to find — those become your variables. Give each a descriptive label so you don’t confuse them mid-solution.
What are the most common types of systems word problems?
The four most common types are: (1) total \(\color{blue}{\text{ count } + \text{ total }}\) value (tickets, coins), (2) distance-rate-time, (3) mixture problems, and (4) age problems. Each type has a standard setup that becomes familiar with practice.
My answer is correct algebraically, but it doesn’t make sense. What happened?
Check whether you set up the equations correctly. A negative number of tickets, for instance, signals that the equations were written incorrectly. Re-read the problem and verify each equation matches what the problem states.
Related Topics
Related to This Article
More math articles
- Division Facts Review for 4th Grade
- Top 10 Tips to Overcome TSI Math Anxiety
- How to Graph the Secant Function?
- Free Grade 7 English Worksheets for New Jersey Students
- How is the SSAT Test Scored?
- The Best Grade 4 ELA Practice Tests for Washington Students
- How to Write and Solve Direct Variation Equations
- Top 10 Tips You MUST Know to Retake the ACCUPLACER Math
- 3rd Grade FSA Math Worksheets: FREE & Printable
- The Ultimate NYSTCE Grades 1-6 Math (222) Course (+FREE Worksheets & Tests)
What people say about "How to Solve Systems of Equations Word Problems? (+FREE Worksheet!) - Effortless Math: We Help Students Learn to LOVE Mathematics"?
No one replied yet.