Two unknowns need two conditions
A pair of linear equations is a mathematical model of two conditions involving the same two unknown quantities.
Learning goals
- Define two variables consistently with units.
- Translate verbal conditions into standard linear equations.
- Check whether the pair represents every original condition.
Concept lesson
Translate comparison words
If x is 4 more than y, write x=y+4. If x is twice y, write x=2y. Read the subject of the sentence first.
Define before translating
Write one fixed meaning and unit for each variable, such as x=number of girls and y=number of boys. Use those meanings in both equations and in the final answer.
Standard form and independence
Each condition becomes ax+by+c=0. Two useful conditions must constrain the same variables; merely rewriting one equation as a multiple does not add new information.
Translate a past-age statement
Five years ago Noori's age x was three times Sonu's age y. Write standard form.
Form a pair from total and difference
Ten students joined a quiz. There were four more girls than boys.
Fix variable meanings
Let x and y be the prices in rupees of one pencil and one pen. Form equations for 5 pencils and 7 pens costing 50 rupees, and 7 pencils and 5 pens costing 46 rupees. Keep the meanings fixed.
Check every phrase
For each formed equation, underline the phrase represented by every term. Substitute a plausible pair and restate what the equality says in words.
Detect a redundant condition
A student uses x+y=12 and 2x+2y=24 as two conditions. Explain why the second gives no new information and why a unique answer cannot follow.
One-mark modelling answer
Define variables if the question does not already do so, translate time and comparison words before simplifying, then write ax+by+c=0 cleanly.
Exit ticket
Two numbers total 32 and the larger is 8 more than the smaller. Define variables, form the pair, and verify that the proposed values 20 and 12 satisfy both verbal conditions.