Systems of Equations Word Problems
Learn to translate real-world situations into systems of equations and solve them step by step.
Definition
| Word/Phrase | Mathematical Operation |
|---|---|
| sum, total, combined | (addition) |
| difference, less than | (subtraction) |
| times, product, each | (multiplication) |
| per, for each, rate | typically multiplication |
| is, equals, was | (equals sign) |
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Worked Examples
Adult tickets cost 12 dollars and child tickets cost 8 dollars. A group bought 15 tickets for a total of 148 dollars. How many of each type of ticket did they buy?
Define variables
Let = number of adult tickets Let = number of child tickets → Two unknowns: and
Write the first equation (total tickets)
The group bought 15 tickets total: →
Write the second equation (total cost)
Adults: dollars Children: dollars Total: →
Solve using substitution
From equation 1: Substitute into equation 2: →
Simplify and solve for c
→ child tickets
Find a
→ adult tickets
Check the answer
Tickets: ✓ Cost: ✓ → Both equations satisfied!
Answer: The group bought 7 adult tickets and 8 child tickets.
Common Mistakes
Using only one variable when two are needed
Why it's wrong: If you have two unknown quantities, you need two variables and two equations. Using one variable often leads to contradictions or overly complex expressions.
Correct: Always identify ALL unknown quantities first, then assign a different variable to each.
Mixing up which equation represents which condition
Why it's wrong: Writing 'cost equation' when you meant 'quantity equation' gives wrong answers even with correct algebra.
Correct: Label your equations clearly: 'Equation 1 (total quantity)' and 'Equation 2 (total value)'
Forgetting to check the answer in the original problem
Why it's wrong: Calculation errors happen. The answer might solve your equations but not match the word problem conditions.
Correct: Always substitute back into BOTH original conditions and verify in context (not just in your equations).
Setting up rate problems incorrectly
Why it's wrong: Downstream, the boat and current speeds ADD. Upstream, you SUBTRACT the current from boat speed.
Correct: Think about it: going with the current makes you faster (add), going against makes you slower (subtract).
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Practice Problems
15 problemsThe sum of two numbers is 20 and their difference is 6. Which system of equations represents this situation?
Why It Matters
- Business: Finding the right mix of products to maximize profit
- Chemistry: Mixing solutions to achieve a desired concentration
- Finance: Calculating how much to invest in different accounts
- Travel: Determining speeds when two objects are moving
- Shopping: Finding individual prices when you know totals
Real World Applications
Running a Business
Business owners use systems of equations to make pricing decisions and manage inventory.
Example:
A coffee shop sells small coffees for 3 euros and large coffees for 5 euros. Yesterday they sold 120 coffees totaling 480 euros. How many of each size did they sell?
A bakery sells cookies for 2 euros and brownies for 3 euros. One day they sold 85 items for a total of 195 euros.
How many cookies and brownies did they sell?
Step 1: Write the mathematical expression
Set up the system: Let = cookies, = brownies
Travel Planning
Planning trips involves calculating distances, speeds, and times that often require systems of equations.
Example:
A cyclist traveled 50 km. Going uphill took 3 hours, and downhill took 2 hours. What were the uphill and downhill speeds?
A train travels from City A to City B (300 km). The first half of the journey takes 2 hours, and the second half takes 3 hours due to track conditions.
What are the speeds for each half of the journey?
Step 1: Write the mathematical expression
Use distance = rate × time for each section
Chemistry and Mixing
Chemists and pharmacists use systems to calculate how much of each substance to mix.
Example:
A chemist needs 100 mL of a 25% acid solution. She has 10% and 40% solutions available. How much of each should she mix?
A juice company wants to create 500 mL of a 30% fruit juice blend using 20% and 50% juice concentrations.
How many mL of each concentration is needed?
Step 1: Write the mathematical expression
Let = mL of 20% juice, = mL of 50% juice
Key Takeaways
- 1Define a variable for EACH unknown quantity in the problem
- 2Identify TWO conditions that relate the variables and write an equation for each
- 3Solve using substitution or elimination (choose the method that seems easier)
- 4Always check your answer in the ORIGINAL word problem, not just the equations
- 5Common problem types: sum/difference, cost/quantity, rate problems, mixture problems
Frequently Asked Questions
Glossary
- System of equations
- Two or more equations with the same variables that must all be true simultaneously
- Variable
- A letter (like or ) that represents an unknown quantity
- Substitution
- Solving one equation for a variable and plugging that expression into the other equation
- Elimination
- Adding or subtracting equations to cancel out one variable
- Rate
- A quantity measured per unit of something else (speed = distance per time, price = cost per item)
Formula Card
Setting Up Variables
Assign a variable to each unknown quantity
Distance Formula
Distance equals rate times time
Total Value Formula
Sum of price times quantity for each item
Mixture Formula
Concentration times amount equals final mixture