Elimination: scaling the equations so a variable cancels on addition

0:00Two quantities are equal. Two more quantities are equal. Add them up, and the totals are equal too. That is not a trick. It is what being equal means, and it is the first of the two permissions this whole method rests on. Two balances. On one, seven x minus fifteen y sits level against two; on the other, x plus two y sits level against three. Tip both onto a third, and it still balances.

0:31Eight x minus thirteen y equals five. So any pair satisfying the two you started with satisfies the sum as well. Nothing has been lost. Now the useful version. Arrange for one of the two letters to carry opposite coefficients before you add, and that letter goes. That is the entire method. Everything else is choosing what to multiply by. Because you almost never get opposite coefficients for free. You have to make them.

1:06So here is the second permission. Multiply an equation right through by a number, and the pairs that satisfy it do not change. Two x plus three y equals nine, multiplied by four, is eight x plus twelve y equals thirty-six. Every pair fitting the first fits the second, and every pair fitting the second fits the first. Every part of it gets multiplied, and that includes the number on the right. Leaving the right-hand side alone is the commonest arithmetic slip in the step.

1:44One number is forbidden, and it is nought. Multiply an equation by nought and you get nought equals nought. True everywhere, satisfied by every point in the plane, and the information that was in it is gone for good. Multiplying by four can be undone by dividing by four. Multiplying by nought cannot be undone by anything, and that is exactly why it is the one number forbidden here. Here is the problem the method was built for.

2:20Two people earn in the ratio nine to seven, and spend in the ratio four to three. Each of them saves two thousand euros a month. What does each earn? Earning in the ratio nine to seven does not tell you either income. It tells you they are nine of something and seven of the same something. So call that something x. The incomes are nine x and seven x, and whatever x turns out to be, the ratio comes out right.

2:53Do the same with the spending. Call it y, and the expenditures are four y and three y. Now saving is income minus expenditure, and both savings are two thousand. Nine x minus four y equals two thousand. Seven x minus three y equals two thousand. Two equations, two letters, and neither letter comes free without fractions. Look at the y column: minus four and minus three. They do not match, so subtracting the equations would not remove y. But four and three both go into twelve.

3:34Twelve is the lowest common multiple of four and three, and that is the whole of the choice. Multiply the first equation by three and the second by four. Twenty-seven x minus twelve y equals six thousand. Twenty-eight x minus twelve y equals eight thousand. Now the y columns are identical. Both are minus twelve y. And because they match rather than oppose, it is subtraction that removes them. Take the first from the second.

4:09Twenty-eight x minus twenty-seven x is x. Minus twelve y minus, minus twelve y is nothing at all. Eight thousand minus six thousand is two thousand. x equals two thousand. Note what decided the sign: matched coefficients get subtracted, opposite ones get added. Backwards, and you double the letter you meant to remove — the commonest arithmetic failure in the method. Put x back into either original equation and y equals four thousand.

4:48Now do it again, and remove the other letter instead. The x column is nine and seven, whose lowest common multiple is sixty-three. Heavier numbers, but no harder in principle. Multiply the first by seven and the second by nine. Sixty-three x minus twenty-eight y equals fourteen thousand. Sixty-three x minus twenty-seven y equals eighteen thousand. Subtract the first from the second and the x terms go. Minus twenty-seven y minus, minus twenty-eight y is plus y. Eighteen thousand minus fourteen thousand is four thousand.

5:33y equals four thousand, straight out, with no back-substitution at all. And then x comes back as two thousand. Two routes, sharing nothing but the equations they started from, arriving at the same pair. That is the most convincing demonstration this topic has, and it is worth doing every time you are unsure. Now the step that is skipped more often than any other. The question did not ask for x. It asked what each person earns.

6:07x is two thousand, and two thousand is not an income. It is the size of one share of the earning ratio. The incomes are nine x and seven x. Nine times two thousand is eighteen thousand. Seven times two thousand is fourteen thousand. And the expenditures are four y and three y. Four times four thousand is sixteen thousand. Three times four thousand is twelve thousand. Reporting two thousand as somebody's income is the standard error on any problem modelled with a ratio: it comes of stopping when the algebra stops rather than when the question is answered.

6:53So: eighteen thousand and fourteen thousand euros a month. Check it, and check it against the sentences rather than against the equations you wrote. Eighteen thousand minus sixteen thousand is two thousand. Fourteen thousand minus twelve thousand is two thousand. Both savings are right. The incomes, eighteen thousand to fourteen thousand, divide by two thousand to give nine to seven. That ratio is right. And here is the one people skip. The expenditures, sixteen thousand to twelve thousand, divide by four thousand to give four to three.

7:37Skipping the second ratio is how a wrong y survives a check. The savings alone do not pin it down; you need the ratio it came from. A check worth running has to be able to reject. Swap the two coordinates, flip a sign, move either by one, halve both — every one of those fails this check, which is why the pair that passes passes for a reason. So the shape of the method, in four steps.

8:09One. Scale both equations by numbers that are not nought, until one letter carries the same coefficient in both. Two. Add or subtract, whichever makes that letter go: subtract when the coefficients match, add when they oppose. Three. Solve what is left, which has one letter in it. Four. Put that value back into either of the original equations for the other letter. And a branch. Sometimes step two leaves a statement with no letter in it at all.

8:43If that statement is true, the pair has endlessly many answers. If it is false, the pair has none. Step one is the only place judgement enters. Everything after it is forced. Here is a pair that runs into the branch. Two x plus three y equals eight, together with four x plus six y equals seven. Match the x column: multiply the first by two. Four x plus six y equals sixteen, alongside four x plus six y equals seven.

9:20Something has happened that did not happen before. Both columns match, not just the one you were aiming at. Subtract, and both letters go together. Nought equals nine. Which is false. So there is no pair of numbers satisfying both, and the pair is called inconsistent. And the coefficient test from earlier says the same thing. Two over four is a half, three over six is a half, and eight over seven is not. Coefficient ratios agreeing while the constant ratio differs is the parallel pattern.

9:59Two methods, one answer, as it must be. The two odd endings are worth separating, because they look alike on the page and mean opposite things. Change that seven to sixteen and everything changes. Two x plus three y equals eight, with four x plus six y equals sixteen. Scale the first by two, subtract, and you get nought equals nought. True. Which means the second equation was the first one doubled — one condition written twice, and endlessly many pairs satisfy it.

10:38One line, drawn on top of itself. Whereas nought equals nine is false, and false by nine wherever you look. Two lines with the same slope and different intercepts, never meeting. So the test is simple. No letter left, and a true statement: endlessly many answers. No letter left, and a false one: no answer at all. And notice that both readings come out of the arithmetic without anybody drawing a line.

11:10One last problem, because it hides something worth naming. A two-digit number added to the number you get by reversing its digits comes to sixty-six. The two digits differ by two. Find the number. Call the tens digit x and the units digit y. The number is ten x plus y, and the reverse is ten y plus x. Add them and you get eleven x plus eleven y — eleven times the digit sum, and that is true of every two-digit number and its reverse. So the digits add to six.

11:49Now the second condition. The digits differ by two. And that is not a linear equation at all until you decide which digit is the larger one. A difference of size two is two separate statements: x minus y equals two, or y minus x equals two — both of them perfectly good linear equations, which is why the split is allowed. Solve each with x plus y equals six. One branch gives four and two, so the number is forty-two; the other gives two and four, so the number is twenty-four. Both survive, so there are two answers and reporting only one is wrong.

12:33But two branches do not by themselves mean two answers. Had the digits differed by six, one branch gives sixty and the other wants a leading nought, which is no two-digit number at all. What produces two answers is not the split. It is what survives it. A word on the checking, because a claim about what a step PRESERVES is confirmed for free by a check that tests nothing.

13:03Scaling by three and scaling by one both leave the answers where they were, so a check that quietly scales by one has confirmed the permission for free. Four hundred and five pairs were taken, and each transformed three ways: scaled, combined, and — deliberately — multiplied by nought. Then five hundred and fifty-four thousand points were put to the original pair and to all three of those, one point at a time.

13:35The two legitimate operations lost nothing and invented nothing. The one that multiplies by nought invented plenty: eight thousand seven hundred and sixty-eight points satisfy it and do not satisfy the pair it came from. That occupied cell is what makes the empty ones worth having. A check that had stopped checking would leave every cell empty and look exactly the same. Every transformation was also recorded as having genuinely rewritten the pair it was handed, on all four hundred and five.

14:14The answers were then found three ways — removing x, removing y, and determinants, which removes nothing — and all three agreed on every pair. Beside them ran a fourth that matches the coefficients and then adds where it should subtract. It agreed with the others on none of the four hundred and four pairs it was handed. Two permissions and one choice. You may multiply an equation through by any number except nought, and the answers do not move.

14:50You may add two true equations, and the answers do not move. The choice is the multipliers: match one letter's coefficients on their lowest common multiple, and that letter leaves by itself. Subtract when the matched coefficients agree in sign; add when they oppose. No letter left and a true statement means endlessly many answers. No letter left and a false statement means none. Then go back to what was asked. If the letters were shares of a ratio, the answer is nine x and seven x, not x.

15:28And when a stated condition is a size rather than a signed quantity, split it into the linear cases it covers, solve every one of them, and report every answer that survives.