L5 · Equation problem-solving clinic

Learning goal

“I can build an equation from a real situation - choosing the variable, translating the story into symbols - and solve any equation this unit covers (rational coefficients, conversions, parentheses, variables on both sides), verifying every solution by substitution and interpreting the answer back in the situation.”

Model — (verb) to translate a situation into an equation: deciding what the variable stands for and what equality the situation claims.

Consecutive integers — integers that follow one another: n, n + 1, n + 2, ...

Interpret — to state the solution back in the situation's language, with units: not "x = 5" but "the job takes 5 hours."

Worked example

A plumber charges a $45 call-out fee plus $60 per hour. An electrician charges a $25 call-out fee plus $70 per hour. For what job length do the two tradespeople cost exactly the same? Step 1: Model: let h be the job length in hours. Plumber's cost: 45 + 60h. Electrician's cost: 25 + 70h. The situation claims these are equal: 45 + 60h = 25 + 70h. Step 2: Collect: subtract 60h from both sides -> 45 = 25 + 10h. Subtract 25: 20 = 10h. Step 3: h = 2. Step 4: Verify AND interpret: plumber 45 + 60(2) = $165; electrician 25 + 70(2) = $165. Interpretation: for a 2-hour job, both charge $165.

Check yourself — explain it

Why does the model put 45 + 60h on one side and 25 + 70h on the other, rather than adding all four numbers together?

Paired problem — example still visible

Two water tanks: Tank A starts with 30 L and fills at 4 L/min; Tank B starts with 10 L and fills at 6 L/min. When do they hold the same amount? -> 30 + 4t = 10 + 6t -> 20 = 2t -> t = 10 min (check: 30 + 40 = 70 L and 10 + 60 = 70 L)

Faded completion — last step is yours

"45 + 60h = 25 + 70h: subtract 60h and then 25 from both sides to reach 20 = 10h, so h = ___" -> 2

Ready to practise?

10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

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