Word problems with two variables ask you to turn a real situation into two equations and solve them as a system. This cheat sheet helps students identify unknowns, write equations, choose a solving method, and check whether the answer makes sense. It is especially useful for mixture, ticket, age, distance, and comparison problems.
Clear setup is often the hardest part, so the focus is on translating words into algebra.
Key Facts
- A system of two linear equations can often be written as , where and represent the two unknown quantities.
- Define variables clearly before writing equations, such as and .
- A total amount often translates to an addition equation such as .
- A value or cost relationship often translates to a weighted equation such as .
- In substitution, solve one equation for one variable, such as , then replace in the other equation.
- In elimination, add or subtract equations so one variable cancels, such as giving .
- The solution to a system is the ordered pair that makes both equations true.
- Always check answers in the original word problem because negative, fractional, or unrealistic values may not fit the situation.
Vocabulary
- System of equations
- A set of two or more equations that use the same variables and must be solved together.
- Variable
- A letter or symbol that represents an unknown quantity in a problem.
- Substitution
- A method for solving a system by replacing one variable expression with an equivalent expression from another equation.
- Elimination
- A method for solving a system by adding or subtracting equations to cancel one variable.
- Ordered pair
- A solution written as where the first value gives and the second value gives .
- Constraint
- A condition from the real problem that limits possible answers, such as values needing to be whole numbers or nonnegative.
Common Mistakes to Avoid
- Not defining the variables first, which makes it unclear what and represent and often leads to reversed equations.
- Mixing up totals and values, which can turn a count equation like into an incorrect money equation.
- Using the same units incorrectly, because combining hours with minutes or dollars with cents without converting gives wrong equations.
- Forgetting to check both original equations, which can hide arithmetic errors from substitution or elimination.
- Accepting an unrealistic answer, because a solution like tickets may satisfy an equation but not the real situation.
Practice Questions
- 1 A school sells adult tickets for \10\. If tickets are sold for a total of \640$, write and solve a system to find the number of each ticket.
- 2 Two numbers have a sum of and a difference of . Let be the larger number and be the smaller number. Write and solve the system.
- 3 A boat travels miles downstream in hours and miles upstream in hours. Let be the boat speed in still water and be the current speed. Use and to find both speeds.
- 4 A system from a word problem has the solution . Explain why you must know what and represent before writing the final answer.