Class 7 Mathematics Chapter 15: Finding the Unknown

A scale that balances

Picture a weighing scale that is perfectly level. On the left pan sit three identical bags and four loose blocks. On the right pan sit nineteen loose blocks.

The scale balances, so the two pans weigh the same. How heavy is one bag?

Writing the balance down

Call the weight of one bag b, counted in blocks. Then the left pan weighs 3b + 4 and the right pan weighs 19.

Balanced means equal. So 3b + 4 = 19.

That is an equation

A statement that two expressions have the same value is called an equation. The line 3b + 4 = 19 is one.

The part on the left of the equals sign is the left side. The part on the right is the right side.

An equation is a claim

Notice what the equation does and does not say. It does not tell you what b is.

It makes a claim: whatever b stands for, these two amounts match. Finding that value is called solving the equation.

What a solution is

A solution is a value for the letter that makes the two sides agree.

Try b as 5. The left side becomes 3 times 5 plus 4, which is 19. The right side is 19. They agree, so 5 is the solution.

A value that is not a solution

Now try b as 6. The left side becomes 3 times 6 plus 4, which is 22, while the right side is still 19.

Twenty-two is not 19, so 6 is not a solution. Any value can be tested this way, which is one of the best features of an equation.

Forming an equation from a situation

Equations usually arrive as words rather than as symbols. A pattern of matchsticks might need 2n + 1 sticks at position n.

Asking which position uses 41 sticks is asking for the n that makes 2n + 1 equal 41. So the equation is 2n + 1 = 41.

Trying values

One way to find n is simply to try numbers. Start somewhere easy.

At n as 10 the left side is 21, far below 41. At n as 15 it is 31, closer but still short.

Closing in

At n as 19 the left side reaches 39. That is near, and only 2 away.

At n as 20 the left side is 41 exactly. So 20 is the solution, and the position wanted is the twentieth.

This method has a name

Trying values until one works is called the trial and error method. It is an honest method, and it always confirms its own answer.

It is also the first thing most people invent for themselves, which is a point in its favour.

But it is slow

Trial and error is slow, and it gets worse as the numbers grow. Solving 2n + 1 = 4001 by trying values would take a long while.

It is slower still when the answer is not a whole number. Nothing tells you which value to try next.

So something better is needed

Look again at the balance of bags and blocks. Nobody solves that by guessing bag weights.

You take four blocks off each pan, so the pans still match, and the problem gets smaller. That idea is the subject of the next topic.

One letter, one value

A last point about notation. In one equation the same letter always stands for the same number.

So in 5x - 4 = 26 both appearances of the number 5 and the single x behave consistently, and x turns out to be 6, since 5 times 6 is 30 and 30 less 4 is 26.

Two sides, two expressions

One more habit worth forming. Read each side of an equation as an expression in its own right.

The left side of 3b + 4 = 19 is an expression with a letter in it. The right side is an expression too, just a very short one.

An equation is not a formula

A formula tells you how to work something out, and every value is allowed. An equation makes a claim that only some values satisfy.

The difference matters. Asking to solve a formula makes no sense, and checking an equation always does.

What carries forward

An equation says two expressions have the same value, and the parts either side of the equals sign are its two sides. A solution is a value for the letter that makes those two sides agree, and you can always test a value by substituting it.

Trial and error finds solutions by trying values, as 20 solves 2n + 1 = 41. It is reliable but slow, and it offers no hint about what to try next. The balance suggests a better way: change both sides in the same way, and keep them equal.

Scenario:

A class is given two puzzles. In the first, a scale balances three identical bags and four loose blocks against nineteen loose blocks. One pupil guesses that each bag weighs 6 blocks. In the second, a pattern of matchsticks needs 2n + 1 sticks at position n, and the class must find the position using 41 sticks. A pupil tries n as 19, gets 39, and says that is close enough. A third pupil complains that trial and error is not real mathematics at all.

Analysis:

Three claims, and each one can be settled by substituting.

Activity:

Make up three equations of your own, swap them with a partner, and solve them by trying values.

Title:

A balance made of coins

Safety:

Balance the ruler low down so nothing falls far.

What You Notice:

The loose coins on the far side should come to 19 if each envelope holds 5, giving 3b + 4 = 19. Trying values should reach b as 5, and opening the envelope confirms it.

The equals sign in an equation states a fact, unlike the one in a sum, where it means work this out.

Trial and error is the one method that proves its own answer at the moment it finds it.

Solving unknowns with letters was called bijaganita in ancient India, from the word for a seed.

Balanced means equal, both sides the same.

Substitute, and let the two sides argue.

Back to the scale

Nobody solves a balance puzzle by guessing. If both pans hold four loose blocks, you take those four off each pan.

The scale is still level, and the puzzle is simpler. That single move is the whole method of this topic.

Doing the same to both sides

Here is the rule. Whatever you do to one side of an equation, do exactly the same to the other.

Then the two sides still have the same value, so the equation still says something true. The letter has not changed, so its solution has not changed either.

Why it is allowed

The reason is worth stating plainly. An equation says its two sides name one and the same value.

So one operation applied to that value gives one answer, however it is written. Both sides must still match.

A first equation

Take 4a + 9 = 37. The 9 is in the way, so subtract 9 from both sides.

The left side becomes 4a and the right becomes 28. So 4a = 28.

Finishing it

Now 4a is four copies of a. Divide both sides by 4.

The left side becomes a and the right becomes 7. So a = 7.

Always check

The check costs one line. Put 7 back into the original equation.

Four sevens are 28, and 28 plus 9 is 37, which is the right side. So 7 is correct, and nothing was taken on trust.

All four operations are allowed

Adding, subtracting, multiplying and dividing all work, as long as you do them to both sides. Dividing by zero is the one thing never allowed.

Usually you choose the operation that removes something. That is why subtracting 9 was the sensible first move above.

Inverse operations

Subtracting undoes adding, and dividing undoes multiplying. Pairs like that are called inverse operations.

So you remove an added 9 by subtracting 9, and remove a multiplier of 4 by dividing by 4. The inverse is how you choose the step.

A useful shortcut

Inverses also save work on plain arithmetic. Suppose you are told that 582 - 47 + 19 - 8 + 64 = 610.

To find 582 - 47 + 19 - 8, subtract 64 from both sides. The answer is 546, and nothing had to be worked out from scratch.

And with multiplication

The same trick works the other way round. Suppose 13 times 7 times 5 times 4 comes to 1820.

Then 13 times 7 times 5 is found by dividing both sides by 4, which gives 455.

When the letter is subtracted

Now 20 - 3b = 5. Here the letter is being taken away, which looks harder.

Subtract 20 from both sides, giving -3b = -15. Then divide both sides by -3, which gives b = 5. Checking, 20 less 15 is 5.

When the letter has a divisor

Try c/4 + 6 = 10. Subtract 6 from both sides, giving c/4 = 4.

Now multiply both sides by 4, since multiplying undoes dividing. That gives c = 16, and 16 divided by 4 plus 6 is indeed 10.

Letters on both sides

Sometimes the letter appears twice. Take 7d - 2 = 5d + 12.

The first job is to gather the letters on one side. Subtract 5d from both sides, which leaves 2d - 2 = 12.

Finishing that one

Now the equation looks like the first one. Add 2 to both sides, giving 2d = 14.

Divide both sides by 2, so d = 7. Checking, the left side is 47 and so is the right.

Nothing actually moves

One habit to drop. People often say a number moves across the equals sign and changes its sign on the way.

Nothing moves anywhere. You subtracted 9 from both sides, and that is the only reason the 9 turns up as a subtraction on the right.

A routine, not a trick

Notice the shape of all four solutions. Gather the letters, remove what is added or subtracted, then undo the multiplier or divisor.

Every step is the same change to both sides, and every answer is checked by substituting. There is nothing to remember beyond that.

What carries forward

To solve an equation, do the same thing to both sides, and keep doing it until the letter stands alone. This is allowed because the two sides name one value, so one operation on both leaves them equal.

Choose each step by its inverse: subtract an added number, divide by a multiplier, multiply by a divisor. Gather letters on one side first when the letter appears twice. And always finish by substituting the answer back into the original equation.

Scenario:

A teacher writes four equations for the class. They are 4a + 9 = 37, then 20 - 3b = 5, then c/4 + 6 = 10, and finally 7d - 2 = 5d + 12. One pupil begins the first by dividing both sides by 4. Another begins the second by writing 3b = 5 + 20. A third looks at the fourth equation and says it cannot be done. Her reason is that the letter sits on both sides at once. So there is nothing left to work towards, and no side for the answer to end up on. The rest of the class is unsure whether she is right.

Analysis:

Three pupils, three different habits, and one rule that settles all of them.

Activity:

Write four equations of your own, one of each kind, and solve them with the steps and checks written out.

Title:

Undo it with a partner

Safety:

No hazards here.

What You Notice:

Your partner should recover the secret number every time, because each step undoes exactly one operation. The subtracted-letter version is where signs usually go wrong.

Nothing ever travels across the equals sign; the sign appears to flip only because the same operation was done to both sides.

Dividing by zero is the single operation barred from both sides, since no number results from it.

An inverse operation tells you which step to take next, which is exactly what trial and error could not do.

Same to both sides, every time.

Added, so subtract; times, so divide.

The cheapest step in algebra

Solving an equation takes several lines. Checking the answer takes one.

That one line is the only part of the work that proves anything. It is also the part most often skipped.

How to check

Take your answer and put it into the original equation, not into any of your own later lines. Work out the left side and the right side separately.

If the two agree, the answer is right. If they do not, something earlier was wrong.

A check that passes

Solve 6x + 5 = 29. Subtracting 5 from both sides gives 6x = 24, and dividing by 6 gives x = 4.

Now check. Six fours are 24, and 24 plus 5 is 29, which is the right side. So 4 is correct.

A check that fails

Suppose someone solving the same equation wrote 6x = 29 + 5 instead. That gives 6x = 34, and then x = 17/3.

Check it. Six times 17/3 is 34, and 34 plus 5 is 39, not 29. The check exposes the slip at once.

Mistake one: the sign

Look at what went wrong there. The 5 was added on the left, so it must be subtracted from both sides.

Writing 29 + 5 treats the 5 as though it hopped over the equals sign and kept its sign. It should have become a subtraction.

Mistake two: a subtracted letter

Now 12 - 5y = 2. A common line here is 5y = 12 + 2, giving 5y = 14 and then y = 14/5.

Check it: 12 less 14 is -2, not 2. Done properly, subtract 12 from both sides for -5y = -10, so y = 2. And 12 less 10 is 2.

Mistake three: gathering with the wrong sign

Try 3z + 4 = z + 16. A tempting line is 3z + z = 16 - 4, giving 4z = 12 and z = 3.

But the z on the right must be SUBTRACTED from both sides, not added. The right line is 3z - z = 16 - 4, so 2z = 12 and z = 6.

Checking that one

Test the wrong answer first. At z as 3 the left side is 13 and the right side is 19, which disagree.

Now the right answer. At z as 6 the left side is 22 and so is the right. The check settles which is which.

Mistake four: a letter term read as a number

Here is 10w - 4w = 30. Someone writes 10w = 30 + 4w and then 10w = 30, giving w = 3.

The 4w cannot be dropped halfway through a line. There is no need to move it at all: 10w - 4w is just 6w, so 6w = 30 and w = 5. And 50 less 20 is 30.

Mistake five: dividing only one term

Last, 2v + 3 = -9. Someone divides the left side by 2 and writes v + 3 = -9/2.

Dividing a side by 2 means dividing all of it, so the 3 must be halved too. Better still, subtract 3 from both sides first: 2v = -12, so v = -6.

What the five have in common

Not one of these is careless arithmetic. Each is a rule applied in the wrong place, which is why the working looks so tidy.

That is exactly why a check is needed. Tidy working is not evidence of a right answer.

The check catches all five

Run the substitution on each wrong answer and all five fail. The checks give 39 against 29, then -2 against 2, then 13 against 19, then 18 against 30, and then -12 against -9.

One method catches every kind of slip. You do not have to remember the list.

When the check fails

A failed check does not tell you where the mistake is. It tells you that there is one, which is already worth a great deal.

Then read your working back line by line, asking of each step what was done to both sides. The line where the answer stops being honest is the line to mend.

Where this all comes from

Solving for unknowns with letters has a long history in India. The subject was called bijaganita, from the word bija, meaning a seed.

The idea is that an answer lies hidden inside a problem as a tree lies hidden in a seed. Brahmagupta set out rules for working with unknown numbers in his book of 628 CE.

What carries forward

Check every answer by substituting it into the original equation and comparing the two sides. It is one line, it is the only part that proves anything, and it catches every kind of mistake.

The five slips in this topic all come from applying a rule in the wrong place: keeping a sign when a term should change, adding a letter term that should be subtracted, dropping a letter term, or dividing only part of a side. Ask of every line what was done to both sides, and these cannot survive.

Scenario:

A teacher collects four answers and marks none of them yet. The first pupil solved 6x + 5 = 29 and wrote 6x = 29 + 5, reaching 17/3. The second solved 3z + 4 = z + 16 by writing 3z + z = 16 - 4, reaching 3. The third solved 10w - 4w = 30 by writing 10w = 30 + 4w and then 10w = 30, reaching 3. The fourth solved 2v + 3 = -9 by dividing the left side by 2, reaching -9/2 less 3. Every page looks neat, and every pupil says the steps felt right.

Analysis:

Four tidy pages, four wrong answers, and one method that finds them all.

Activity:

Swap your own solutions with a partner and check each other by substituting into the original equations only.

Title:

Plant a mistake

Safety:

No hazards here.

What You Notice:

The check should find the planted mistake every time, even though both pages look equally neat. Naming the exact line is harder, and needs the working read back.

None of the five common mistakes is careless arithmetic; each is a real rule applied in the wrong place.

A failed check is still useful, because knowing a mistake exists is most of the work of finding it.

Solving for unknowns with letters was called bijaganita in India, from bija, meaning a seed.

One line proves it: substitute and compare.

Neat is not right.