Systems of equations word problems are among the most common question types on the Digital SAT. You'll be given a real-world scenario involving two unknowns and enough information to write two equations. The challenge is translating English into algebra — once you set up the system correctly, solving it is straightforward.
These problems cover contexts like pricing and purchasing, mixtures, speed and distance, ages, and work rates. The key skill is defining your variables clearly and identifying the two relationships described in the problem.
Core Concepts
Setting Up the System
- Define variables. Let and represent the two unknowns.
- Write equation 1 from the first relationship.
- Write equation 2 from the second relationship.
- Solve using substitution or elimination.
- Answer the question — which may ask for , , or an expression like .
Common Problem Types
Pricing/Purchasing:
- Equation 1: total number of items
- Equation 2: total cost
Mixtures:
- Equation 1: total volume
- Equation 2: total concentration or value
Speed/Distance:
- Equation 1: total time or distance
- Equation 2: relationship between speeds/distances
Ages:
- Equation 1: relationship between ages now
- Equation 2: relationship between ages at a different time
Example: Pricing Problem
A store sells shirts for $15 and hats for $10. A customer buys 8 items and spends $95. How many shirts were bought?
Let = shirts, = hats.
From equation 1:
Substitute:
→ →
3 shirts were bought.
Example: Mixture Problem
A chemist mixes a 20% acid solution with a 50% acid solution to make 30 litres of a 30% acid solution. How much of each solution is needed?
Let = litres of 20% solution, = litres of 50% solution.
From equation 1:
→ →
litres of 20%, litres of 50%.
Strategy Tips
Tip 1: Define Variables First
Always state what your variables represent before writing equations. This prevents confusion.
Tip 2: Look for Two Distinct Relationships
Every system word problem gives you two pieces of information. One is usually about quantity/number, the other about value/cost/amount.
Tip 3: Check That Your Answer Makes Sense
If you get shirts, something went wrong. Answers should be realistic in context.
Tip 4: Read the Question Carefully
The question might ask for one variable, the other, their sum, or their difference. Don't stop too early.
Tip 5: Units Must Be Consistent
If one equation uses dollars and the other uses cents, convert to the same unit first.
Worked Example: Example 1
A movie theatre sold 200 tickets. Adult tickets cost $12 and child tickets cost $7. Total revenue was $1,900. How many adult tickets were sold?
and
→
→ →
100 adult tickets.
Worked Example: Example 2
Two friends drive toward each other from cities 300 miles apart. One drives 50 mph and the other 70 mph. After how many hours do they meet?
Let = time (same for both). Distance 1 + Distance 2 = 300.
→ → hours.
Worked Example: Example 3
The sum of two numbers is 48. One number is 3 times the other. Find the numbers.
and
→ → ,
Worked Example: SAT-Style
A catering company charges $25 per adult meal and $15 per child meal. For an event with 80 guests, the total food cost is $1,700. How many children attended?
and
→
→ →
30 children.
Worked Example: Example 5
A piggy bank contains quarters and dimes. There are 40 coins worth $7.00 total. How many quarters are there?
and
→
→ →
20 quarters.
Practice Problems
Problem 1
A bakery sells muffins for $3 and croissants for $4. 50 items were sold for $170 total. How many of each?
Problem 2
The perimeter of a rectangle is 56 cm. The length is 4 cm more than the width. Find the dimensions.
Problem 3
A boat travels 36 miles downstream in 2 hours and 24 miles upstream in 2 hours. Find the boat's speed in still water and the current's speed.
Problem 4
Tom is twice as old as Sam. In 5 years, Tom will be 1.5 times Sam's age. How old is Sam now?
Problem 5
A shop sells 120 items — some at $5 and some at $8. Revenue is $780. How many $8 items were sold?
Problem 6
A phone plan charges $0.10/text and $0.05/email. Last month, a user sent 200 messages total and was charged $14. How many texts were sent?
Want to check your answers and get step-by-step solutions?
Common Mistakes
- Not defining variables clearly. Ambiguity leads to setting up wrong equations.
- Using only one equation. You need two equations for two unknowns.
- Mixing up which equation goes with which condition. The "total number" condition and "total value" condition must be separate equations.
- Forgetting units/conversions. Quarters are $0.25, not $25.
- Not checking that the answer makes sense in context. Negative quantities or fractional people signal an error.
Frequently Asked Questions
How do I know which method to use — substitution or elimination?
If one equation easily isolates a variable (like → ), use substitution. Otherwise, elimination may be faster.
What if the problem gives me three or more pieces of information?
On the SAT, you'll only need two equations for two unknowns. Extra information might be used to verify your answer.
Can I use a calculator to solve systems?
Yes — on Desmos, graph both equations and find the intersection point.
What if the numbers don't come out nicely?
That's okay. The SAT can have fractional answers, especially in student-produced response questions.
How often do word problems with systems appear on the SAT?
Very frequently — typically 2–3 per test.
Key Takeaways
Define variables first, then write two equations from two different relationships.
Common patterns: total items + total value, total distance + speed relationships, sum + ratio.
Solve by substitution or elimination — choose whichever is more efficient.
Answer what's asked — the question may want one variable, a sum, or a difference.
Check your answer against both equations and against the context.
Practise translating English into algebra — this is the most important skill.