Equations: scales in balance · Martinとコード

What an equation is, what it means to solve it, and why both sides must change the same way.

Scales you must not rock

This is the lesson the whole level is named after. An equation is a writing with an equals sign, in which an unknown number is hiding: x + 3 = 8.

Picture old kitchen scales with two pans. On the left pan lies x + 3, on the right 8. The scales are in balance — both sides weigh the same.

Solving an equation means finding what x is. Here you can see it even without calculating: x = 5, because 5 + 3 = 8.

But how do you go about it when you cannot see it? The only rule: what you do to one pan, you must do to the other. You add a kilo on the left? Add it on the right too, otherwise the scales rock and the equality dies. You take a three off the left? Take it off the right too.

This one rule will last you until the end of secondary school. Everything else around equations is just consequences.

Filip is holding a bag of sweets and three extra sweets. Ema has eight sweets and claims they both have the same. “How many are in the bag?” Jonáš asks. Filip shrugs. Ema: “So we will play it. We both put three sweets away — we still have the same, right?” Filip is left with only the bag, Ema with five sweets. “So there are five in the bag,” says Jonáš. Ema nods: “That is exactly how an equation works. The bag is x, and when you take the same off both sides, the truth stays the truth.”

💡

Write every rearrangement on a new line and mark after the equation with a slash what you are doing: x + 3 = 8 / −3. Then the notebook can be read by a teacher — and most of all by you in a week.

How to solve an equation by rearranging

The goal: get x alone on one side. Clear everything else — but always from both sides at once.

Equation with plus: x + 3 = 8. The three is in the way. I subtract it from both sides:

x + 3 − 3 = 8 − 3, so x = 5.

Equation with minus: x − 4 = 10. I add four to both sides: x = 14.

Equation with times: 3x = 12. The left side is three times x. I divide both sides by three: x = 4.

Equation with divide: x/5 = 6. I multiply both sides by five: x = 30.

Notice the pattern: an obstacle is removed by the opposite operation. Plus cancels minus, times cancels divide.

At the end always a check: substitute the result into the original equation. x = 5 in the equation x + 3 = 8: left side 5 + 3 = 8, right side 8. They are equal — solved correctly. The check is your private marker; with it you never hand in a wrong result.

Example 1: x + 7 = 15

The seven is in the way of x. I subtract it from both sides:

x + 7 = 15 / −7

x = 15 − 7

x = 8.

Check: left side 8 + 7 = 15, right side 15. It fits.

The scales in action: seven disappeared from both pans — the balance stayed, only what x weighs was revealed.

Example 2: 4x = 28

The left side is four times x. I divide both sides by four:

4x = 28 / :4

x = 28 : 4

x = 7.

Check: 4 · 7 = 28. It fits.

Watch the difference: with x + 4 = 28 I would subtract, with 4x = 28 I divide. Look at how the number is stuck to x: with plus, or with multiplying?

Example 3: x − 2.5 = 4

It works with decimals too. I add 2.5 to both sides:

x − 2.5 = 4 / +2.5

x = 4 + 2.5

x = 6.5.

Check: 6.5 − 2.5 = 4. It fits.

Equations are not afraid of decimal points or negative numbers. The scales rule works for all numbers the same.

⚠️

Never rearrange only one side. x + 3 = 8, “cross out the three” and leave 8 — that is a rocked scale and a wrong result. Every rearrangement = both sides. No exception.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. スタート with ten. When it goes well, add more.

Solve the equation and write only x. On paper write the rearrangements with a slash — you are training the writing, not just the result.

Practise the questions

On paper

Equations are solved on lines. One line = one rearrangement.

Solve with rearrangements line by line: x + 9 = 14, x − 6 = 11, x + 3.5 = 10.
Solve: 5x = 35, x/4 = 8. For each one write which opposite operation you used.
For every equation from steps 1 and 2 write a check: substitute the result and compare both sides.
Invent your own equation whose solution is x = 6, and let someone in the family solve it.

Now you 💪

I know an equation is a balance and I do rearrangements on both sides.
I can pick the opposite operation: plus cancels minus, times cancels divide.
Every solution ends with a check.

✅

Done when: You have five solved equations with rearrangements written, checks for all of them, and a run of five correct in practice.

What to take from this lesson

An equation = scales in balance. Solving it = finding what x is.
What you do to one side, do to the other too.
Obstacles are removed by the opposite operation.
Write rearrangements with a slash after the equation: / −3, / :4.
A check by substituting is a required full stop after every equation.