Substitution: rewriting one unknown so only the other survives

0:00Here is a pair of equations. Seven x minus fifteen y equals two, and x plus two y equals three. Look at the second one. It says something quite specific: whatever x is, it is three minus two y. That is not a fact about some x. It is a licence. Anywhere in this problem where the letter x appears, you are entitled to write three minus two y instead, and nothing changes.

0:31So do it. In the first equation, x becomes three minus two y. And now there is only one letter left. That is the whole method. Everything else in this video is either the arithmetic, or the reason you are allowed to do it. First, the choice. You have two equations and four letters to pick from, and the choice is worth a moment. x plus two y equals three gives x equals three minus two y. Whole numbers throughout.

1:05Seven x minus fifteen y equals two would give x equals two sevenths plus fifteen sevenths y. Both work. Both reach the same answer. But count what you actually write down along each route: the expression, the equation it leaves, and the two answers. Open the second equation and four of those six quantities are whole numbers. Open the first and none of them are. You are in sevenths from the very first line.

1:36So the rule is not a rule. Take the letter with a coefficient of one, or nearest to it, and you have saved yourself two lines of fractions. Now carry it through. x equals three minus two y. Put that into seven x minus fifteen y equals two. Seven times the bracket three minus two y, minus fifteen y, equals two. Expand: twenty-one minus fourteen y, minus fifteen y, equals two.

2:10Collect: twenty-one minus twenty-nine y equals two. So twenty-nine y equals nineteen, and y is nineteen twenty-ninths. Notice what has happened. Two unknowns became one, and one unknown is a thing you already know how to handle. You are not finished. The question asked for a pair. Go back to the rearranged equation — x equals three minus two y — and put the value in. x equals three minus two times nineteen twenty-ninths.

2:45Three is eighty-seven twenty-ninths, and thirty-eight twenty-ninths taken from it leaves forty-nine twenty-ninths. So the answer is x equals forty-nine twenty-ninths, y equals nineteen twenty-ninths. And go back to the REARRANGED equation, not to one of the two you started with. That one already has x on its own; the others would make you solve for x all over again. Now the part that is usually skipped, which is why the replacement is allowed at all.

3:20You did not solve the pair you were given. You solved a different pair — the rearranged equation, together with the one-letter equation the substitution left behind. Why should the answer to that be the answer to this? Because the replacement is reversible. It swaps a letter for something equal to it, and it can be swapped straight back. So the two systems have exactly the same solutions. Not roughly. Exactly the same ones.

3:52That was measured rather than assumed. Five hundred and sixty-five pairs of equations, and for each of them a grid of points, one point at a time. A million one hundred thousand points, each asked two questions: does this point satisfy both of the original equations, and does it satisfy the substituted pair? The answers agreed every single time. Three hundred and fifty-eight points said yes to both. Not one point said yes to one and no to the other.

4:30There is one way to get this badly wrong, and it is the commonest mistake in the method. You rearranged the second equation. The expression has to go into the FIRST one — the other one. Put it back into the equation it came from and watch what happens. x equals three minus two y, substituted into x plus two y equals three, gives three minus two y plus two y equals three.

4:59Nought equals nought. Every letter has gone. Which is true, and completely useless. You have asked a question you had already answered. And it is worse than useless, because nought equals nought is also what you get when a pair really does have endlessly many answers. The slip looks exactly like a genuine result. Of the same five hundred and sixty-five pairs, four hundred and sixty-two have a letter that comes free at all. Substituting back into the source equation gave nought equals nought on every one of those four hundred and sixty-two, whatever the pair was.

5:41Doing it properly settled a value on three hundred and ninety-eight of them. Then check. Put the pair back into both of the equations you were given, not into your own working. Seven times forty-nine twenty-ninths, minus fifteen times nineteen twenty-ninths, is three hundred and forty-three minus two hundred and eighty-five, over twenty-nine. Fifty-eight over twenty-nine. Two. And forty-nine twenty-ninths plus thirty-eight twenty-ninths is eighty-seven over twenty-nine. Three. Both hold, so the pair is right.

6:23That step is not a formality. It is the only step that tests your answer against the problem instead of against your own arithmetic. One warning about it. Do not round first. Rounded to two places, the answer is one point six nine and nought point six six, and put into the equations that leaves minus nought point nought seven and plus nought point nought one. Neither is zero, and the check would fail for a reason that has nothing to do with the method.

6:59So the method has a shape, and it is worth holding the shape rather than the steps. Free one letter from whichever equation gives it up most cheaply. Put that expression into the other equation, and solve what is left. Return the value to the rearranged equation to recover the letter you freed. And there is a branch. Sometimes step two leaves no letter at all. That is not a failure. It is an answer in a different form, and which answer depends only on whether the statement you are left with is true or false.

7:36Both of those happen, and both are worth seeing. Before the odd endings, one problem that arrives as sentences rather than as equations. A man and his daughter. Seven years ago he was seven times her age. In three years' time he will be three times her age. Call his age s and hers t, both in years. Seven years ago: s minus seven equals seven times, bracket, t minus seven.

8:07In three years: s plus three equals three times, bracket, t plus three. Tidy the second one and it says s equals three t plus six. That is already a letter set free, so put it into the first. Three t plus six minus seven equals seven t minus forty-nine, which gives four t equals forty-eight, so t is twelve and s is forty-two. Check it against the sentences, not against the equations. Seven years ago they were thirty-five and five, and thirty-five is seven fives. In three years they are forty-five and fifteen, and forty-five is three fifteens.

8:54Now the first odd ending. Two pencils and three erasers cost nine euros. Four pencils and six erasers cost eighteen. Two x plus three y equals nine, and four x plus six y equals eighteen. Free x from the first: x equals nine minus three y, all over two. Put that into the second: four times that bracket is eighteen minus six y, plus six y, equals eighteen. Eighteen equals eighteen. The letters are gone and what is left is true.

9:31That does not mean x and y are nought. It means nothing further has been pinned down. And you can see why. The second equation is exactly the first, doubled. It was never a second condition at all. So any price fitting the first sentence fits both. A pencil at three with an eraser at one. A pencil at one fifty with an eraser at two. Endlessly many, all along one line, and no single price can be named for either item.

10:06And the other ending. Two straight rails: x plus two y minus four equals nought, and two x plus four y minus twelve equals nought. Will they ever cross? Free x from the first: x equals four minus two y. Put it into the second: two times four minus two y, plus four y, minus twelve. Eight minus four y plus four y minus twelve. The y terms cancel and you are left with minus four equals nought.

10:40Which is false. Not unsolved — false. And it is false by the same amount for every y you could ever choose. Minus four, always. So there is no pair of numbers satisfying both. The rails never meet. Two endings, two pictures. A true statement is one line written twice. A false statement is two lines that never touch. The algebra found the geometry without drawing anything. A word on the checking, because this topic is unusually easy to confirm without testing anything.

11:19The claim that matters is not the answer. It is that the replacement neither creates solutions nor destroys them — and that is a statement about two sets, which no amount of looking at one point can establish. So it was tested point by point. Five hundred and sixty-five pairs, a grid of sixths, and every point asked both questions. But a table that agrees with itself would come out looking the same, so the very same comparison was run a second time against the system you get from the slip — substituting back into the equation you rearranged.

12:01That one does NOT agree, and it disagrees in twelve thousand six hundred and seven places: points on the first line that satisfy nothing else. That disagreement is what makes the first table worth having. A comparison that had stopped comparing would have been diagonal in both. The answers themselves were also found a second way, by determinants, which never replaces anything, and the two routes were asked to agree pair by pair rather than in total.

12:35And the check on the answer was put to nine points, not one: the answer, which had to pass, and eight near misses — a twenty-ninth out, a sign flipped, the two coordinates swapped, the rounded version — every one of which had to fail. Every one did. An equation is permission to replace, and the permission is what makes the method honest. Free the letter that comes free most cheaply.

13:04Put the expression into the OTHER equation, and never into the one it came from. Solve, then return the value to the rearranged equation for the second unknown. Check both original equations, without rounding first. No letter left and a true statement: one line, endlessly many answers. No letter left and a false statement: no answer at all. And when neither letter comes free without fractions — nine x minus four y equals two thousand beside seven x minus three y equals two thousand — substitution is the wrong tool, and there is a lighter one coming.