Setting Up and Solving an Equation from a Context

Say a plumber comes out to fix your boiler. Three hours later the job is done, and the bill reads ninety five pounds. Twenty of that is a call-out fee, charged before he even opened the toolbox. The rest is his hourly rate, three times over. Nowhere on that bill does it say what he charges an hour. You would quite like to know, because the next quote you get should be checked against it. You could guess your way there. Thirty pounds an hour would make the bill a hundred and ten, too much. Twenty an hour would make it eighty, too little. Keep narrowing and you would land on it eventually. There is a way to pull the exact number straight out of the sentence, first time, and it works on every question built like this one.

Start with the difference between an expression and an equation, because it decides everything that follows. Naming quantities with a letter gives you an expression. The width is w, so the length is w plus three. Building those is the whole job of the video called Writing an Expression or Formula from a Context. An expression has nothing to solve. There is no equals sign in it, so there is nothing to balance. The equals sign has to come from somewhere, and it comes from one specific place. It comes from the fact in the question that is already a number. A perimeter of twenty six metres. A bill of ninety five pounds. A total age of forty. Hunting down that number is the move that makes an equation appear. Four words carry the whole method, and they are short enough to walk into an exam hall with. Let, build, solve, say. Let names the letter, build writes the equation, solve does the algebra, and say turns the number back into an answer about the actual thing.

Time to run those four words on a garden with a fence round it. A rectangular garden is three metres longer than it is wide. Its perimeter is twenty six metres. Form and solve an equation to find the width. Let comes first. Let w be the width of the garden, in metres. Writing the unit into that sentence is not decoration, it is what makes the final step possible. Three metres longer than the width means the length is w plus three, and that is the naming done. Now build. Perimeter means all four sides added together: w, plus w plus three, plus w, plus w plus three. So which of these three is the equation? Four w plus three equals twenty six. Four w plus six equals twenty six. Or two w plus three equals twenty six. Take your pick. I'll wait. The answer is four w plus six equals twenty six. There are two long sides, not one, so the extra three metres gets counted twice. Label all four sides of the rectangle before you add them up. The side you forget to draw is the side that goes missing. Now solve, and briskly, because the balancing itself belongs to the video on solving linear equations by balancing. Take six off both sides and four w equals twenty. Divide both sides by four and w equals five. And now say, which is the step that quietly gets dropped. On its own, w equals five is not an answer about a garden. The width is five metres. The letter turns back into a width, the number picks up its unit, and the sentence answers what was asked. Check it in your head. Width five, length eight, and five plus eight plus five plus eight comes to twenty six. Now watch what changes if the question asks for the length instead. The solving is identical, w still equals five, but the answer is eight metres. Same equation, same working, different final sentence. That is why say is a step of its own and not an afterthought.

Your turn now, on the boiler bill from the start. A plumber charges a twenty pound call-out fee plus an hourly rate. A three hour job cost ninety five pounds in total. Form and solve an equation to find the hourly rate. Run all four words, and remember that the last one wants a unit on the end of it. Pause it there and work it out. I'll wait. Let h be the hourly rate, in pounds. The call-out is a flat twenty, the work is three lots of h, and the number already given to you is ninety five. So twenty plus three times h equals ninety five. Solve it. Take twenty off both sides, so three times h equals seventy five. Divide both sides by three, and h equals twenty five. Then say it properly: the plumber charges twenty five pounds an hour. Check it against the bill. Twenty pounds call-out, plus three lots of twenty five, comes to ninety five exactly.

Examiners have written about the shortcut round all of this, and their words are worth reading. Here is one line from an examiner report, on a question where students had to write ages in terms of x. The most successful method was Trial and improvement, with even the more able students writing the ages in terms of x and then not being able to form an equation when attempting an algebraic approach. Two separate things are sitting in that sentence. Trial and improvement found the right number, and the algebra never got as far as an equation. The second half is the one to look at. Naming the ages in terms of x was done. It was the equals sign that nobody could find. Which is exactly what build is for. Go hunting for the fact that is already a number, because that is the side your equation balances against. And when a question says form and solve an equation, that wording is an instruction rather than a suggestion. An answer with no equation written down has not done what it was asked to do. Trial and improvement is still worth having in your hands. Use it to check the answer your algebra gave you, never to replace it.

One more shape to recognise, because it needs something all of this does not. Two coffees and a cake cost eight pounds. One coffee and one cake cost five pounds. Find the price of a cake. Let works as normal here. Let c be the price of a coffee and k the price of a cake, both in pounds. Build hands you two equations instead of one. Two c plus k equals eight, and c plus k equals five. A single equation carrying two unknown letters has endless answers. It takes a second fact, and a second equation, to pin both prices down. The garden had two quantities as well, a width and a length. It stayed one equation only because the length could be written in terms of the width. Coffee and cake have no link like that, so both letters have to survive into two separate equations. Solving a pair together is its own method, taught in the video on solving simultaneous linear equations by elimination. Recognising that you need a pair is the part that belongs here.

Before the round-up, here is a small one to test yourself on. I think of a number, double it, then add seven. The result is twenty five. What was my number? Have a think. I'll wait. Let n be the number I thought of. Doubling it and adding seven gives two times n plus seven, and the number already handed to you is twenty five. So two n plus seven equals twenty five, which gives n equals nine. Then say it: the number was nine. Nine doubled is eighteen, and eighteen plus seven is twenty five.

Right then, back to those four words, and what each one is protecting you from. Let means naming the letter and saying what it stands for, unit included. Skip that and the final step has nothing to come home to. Build means finding the fact that is already a number, and balancing the description against it. Solve is ordinary balancing, done briskly, exactly as in the video on solving linear equations by balancing. Say turns the letter back into the thing it stood for, with its unit, answering the question that was actually asked. And if two letters survive with no link between them, that is a pair of equations, not one.

Next in the chapter: Recalling Circle, Pythagoras and Trig Formulae.

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