Unit 3: Finding x

An equation is a promise

An equation like x + 4 = 11 promises you something: whatever x is, adding 4 to it gives 11.

Think of it as a balance with the two sides weighing exactly the same. Your job is to get x alone on one side without ever tipping it.

The rule is short: whatever you do to one side, do to the other.

Undo what was done

In x + 4 = 11, something was added to x. To undo adding, you subtract — from both sides.

x + 4 - 4 = 11 - 4, which tidies up to x = 7.

Always check by putting it back: 7 + 4 = 11. It fits.

Times and divide undo each other too

5x means 5 times x. To undo multiplying by 5, divide both sides by 5.

So 5x = 40 becomes x = 8.

And the other way round: x/3 = 7 becomes x = 21, because multiplying undoes dividing.

Undo in the opposite order

3x + 5 = 20 did two things to x: multiplied by 3, then added 5.

Undo them backwards, like taking off your shoes before your socks. Subtract the 5 first, giving 3x = 15. Then divide by 3, giving x = 5.

Getting the order wrong is the single most common mistake here.

When x is on both sides

If x appears on both sides, gather it up first.

5x + 2 = 3x + 12. Take 3x off both sides, and the right side has none left: 2x + 2 = 12.

Now it is an ordinary two-stepper. 2x = 10, so x = 5.

Brackets first

A bracket means everything inside gets multiplied.

3(x + 2) = 21 is really 3x + 6 = 21.

Or — quicker — divide both sides by 3 straight away and get x + 2 = 7. Both routes reach x = 5, and choosing the shorter one is a skill worth having.

Multiply the whole equation

Fractions in an equation are a nuisance, so get rid of them first.

For x/3 = 4, multiply both sides by 3 and you have x = 12.

For x/2 + x/3 = 5, multiply everything by 6 — the lowest common multiple from unit 5. That gives 3x + 2x = 30, so x = 6.

Cross-multiplying is the same trick

When two fractions are equal — x/3 = 8/12 — multiplying both sides by both bottoms gives 12x = 24.

People call that cross-multiplying, and draw a little X. It is not a new rule. It is multiplying both sides by the same thing, twice.

Solving with letters instead of numbers

A = l × w gives the area of a rectangle. But what if you know the area and the length and want the width?

Divide both sides by l:

w = A/l

Exactly the same moves as solving for x. The only difference is that the answer is a formula rather than a number.

Undo in the opposite order, again

To make x the subject of y = 3x + 7, undo the +7 first and then the × 3:

x = (y - 7)/3

That bracket matters enormously. y - 7/3 would take a third of the 7 only, which is a different formula and a different answer.

Add the equations together

Two unknowns need two equations. The trick is to combine them so one letter disappears.

x + y = 12
x - y = 4

Add them and the y terms cancel: 2x = 16, so x = 8. Put that back into the first: y = 4.

One letter gone, and what is left is a lesson-1 equation.

Sometimes you subtract instead

If the letter you want to lose has the same sign in both, subtract rather than add.

3x + y = 17
x + y = 7

Subtracting kills the y: 2x = 10, so x = 5 and then y = 2.

Same sign — subtract. Opposite signs — add. That is the whole decision.

Substitution: put one inside the other

When one equation already tells you what a letter *is*, put it straight into the other.

y = 2x and x + y = 9.

Replace y with 2x: x + 2x = 9, so 3x = 9 and x = 3. Then y = 6.

That is substitution, and it beats adding equations whenever one letter is already alone.

Which method when?

Both methods always work, so pick the one that means less writing.

Substitution when a letter is already on its own, like y = 3x - 1.

Adding or subtracting when the equations are lined up neatly, like 2x + 3y = 12 and 2x - 3y = 0.

Choosing well is a real skill, and it is worth a few seconds before you start.

Give the unknown a name

Most real problems arrive as a story, not an equation. The trick is always the same: name the thing you do not know, then write down what the story says about it.

"I think of a number, double it, add 5, and get 23."

Call the number x. The story says 2x + 5 = 23. Now it is just algebra.

Two unknowns need two facts

One equation cannot pin down two mystery numbers. x + y = 10 has loads of answers.

But two facts can. If you also know x - y = 4, then adding the two equations makes y vanish: 2x = 14, so x = 7 and y = 3.

Making one unknown disappear is the whole idea.