Checking an equation · Martinとコード

L = R: how to write a check properly, why it is 完了 in the original equation, and how to find your own mistake.

Your built-in marker

You have been doing a check in the last lessons. Today we promote it from a quick look to a proper tool — because the check is the only part of maths where you are the teacher for yourself.

The idea: take the calculated x and substitute it into the original equation. Calculate the left side (marked L) separately and the right side (R) separately. When L = R, the solution is correct. When not, there is a mistake somewhere — and the equation told you before the test did.

Why into the original equation? Because a mistake could have been born in any rearrangement step. Substituting into a rearranged equation only checks the steps after it — it lets a mistake from the first line through.

A check also has its official writing, which teachers want to see: L = …, R = …, L = R. Three lines. You will learn it today and you will write it with every equation until your final exams.

And you will also learn the most valuable thing: what to do when the check does not work.

Adam handed in the test first and was sure. It came back with two mistakes: in both equations a wrongly flipped sign. Ema shows him hers: with every equation three extra lines — L, R, a tick. “Doesn’t that eat time?” Adam asks. “A minute per question. But on the third equation I found a mistake myself before the teacher found it. L came out 14, R came out 10, so I knew I had to look.” Since that test Adam writes checks too. Certainty cannot be talked into — it can be calculated.

💡

When L ≠ R, do not look for the mistake from the start. Go through the rearrangements FROM THE BACK — last step, second last… The mistake is usually where signs changed or something was multiplied.

How to write a check like a pro

You solved the equation 3x + 4 = 19 and got x = 5. Check:

1. Left side. Write L = and substitute: L = 3 · 5 + 4 = 15 + 4 = 19.

2. Right side. Write R = and substitute (here there is nothing to substitute, you copy the number): R = 19.

3. Compare. L = R. Done, the solution holds.

When x is on both sides, you substitute into both: with 7x − 4 = 2x + 26 and x = 6, L = 38 and R = 38.

When L ≠ R:

First check the check itself — you can slip in that too.
Then go through the rearrangements from the back. Look for: flipped signs, a forgotten term when multiplying, a wrong division.
Fix it and do the check again.

The check reveals THAT there is a mistake, not WHERE it is. But that is enough — you know you have to look, and you do not look needlessly where everything is fine. Without a check you hand in blind.

Example 1: a full write-up with a check

Equation: 5x − 7 = 18.

Solution: / +7 → 5x = 25, / :5 → x = 5.

Check:

L = 5 · 5 − 7 = 25 − 7 = 18

R = 18

L = R — the solution x = 5 holds.

That is what the whole official write-up looks like. Five lines from the question to the tick. Photograph this pattern into your head.

Example 2: a check reveals a mistake

Filip solved 4x + 6 = 30 and got x = 9 (he divided wrongly: 36 : 4 came out 9 for him).

Check: L = 4 · 9 + 6 = 42. R = 30. L ≠ R — there is a mistake somewhere.

He goes from the back: the division 24 : 4… wait, did he subtract 6 wrongly? Check: 30 − 6 = 24, not 36. He found it. Correctly: 4x = 24, x = 6.

New check: L = 4 · 6 + 6 = 30 = R. Now it holds. Mistake found and fixed at home, not with a red pen.

Example 3: a check with x on both sides

Equation 3x + 8 = 5x − 4, it came out x = 6.

L = 3 · 6 + 8 = 18 + 8 = 26

R = 5 · 6 − 4 = 30 − 4 = 26

L = R. It holds.

You substitute into BOTH sides — each is calculated separately and nothing may be carried between them. If L came out 26 and R say 28, you would look for the mistake in the rearrangements.

⚠️

A check into a rearranged equation is not a check. A mistake from the first step goes through unnoticed, because the rearranged equation is already infected by the mistake. Always substitute into the equation from the question.

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, write a check with L and R on paper, and only then write x into the box.

Practise the questions

On paper

Today the write-up is marked. L, R and a conclusion with every equation.

Solve 6x − 5 = 19 and write the check exactly in the form L = …, R = …, L = R.
Solve 2x + 3 = 4x − 7 and write the check with substituting into both sides.
A classmate claims that the solution of 3(x − 2) = 15 is x = 5. Check whether they are right. If not, find the correct solution.
Solve any equation from the last lesson again and deliberately make a mistake in one step. Then show how the check reveals it.

Now you 💪

I write the check in the form L = …, R = …, L = R.
I always substitute into the original equation.
When L ≠ R, I look for the mistake from the back through the rearrangements.

✅

Done when: You have three equations with an official check write-up, one revealed someone else’s mistake, and a run of five correct in practice.

What to take from this lesson

A check = substituting the result into the original equation.
L and R are calculated separately; the solution holds when L = R.
The check says THAT there is a mistake — then look from the back.
Teachers want to see the writing L = …, R = …, L = R.
A minute of checking saves marks you would otherwise lose blindly.