Word Problems with Systems — SAT Math Explained
Real-world scenarios that require setting up and solving a system of two equations because two unknown quantities are described by two separate conditions.
The Core Idea
The art of word problems is translation — converting English sentences into math equations. Two unknowns means you need two equations. Each constraint in the problem becomes one equation.
Translation Guide
x + y = 20
x = y + 3
x + y = 500
x = 2y
x + y = 15
Problem Types
Two unknown numbers with given sum/difference/relationship
Combining two substances with different concentrations or values
Two objects moving at different speeds
Two items at different prices with given total quantities and costs
Money split between two investments at different interest rates
Process For Solving Word Problems
1. Read the entire problem before writing anything
2. Identify what the two unknowns are — define variables clearly (let x = ... , let y = ...)
3. Identify the two conditions given — each becomes one equation
4. Solve the system using substitution or elimination
5. Interpret the solution — does it answer the actual question asked?
6. Check: do both original conditions hold with your answer?
Common Errors to Avoid
Defining variables vaguely (e.g., 'let x = tickets' instead of 'let x = number of adult tickets')
Writing only one equation when two are needed
Answering the wrong question (solving for x when the problem asks for x + y)
Practice: Word Problems with Systems
5 SAT-style questions. Select your answer and get an instant explanation.
Two numbers add up to 20. The larger number is 4 more than the smaller. What are the two numbers?
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