Two steps to x
Last time one rearrangement was enough. Today equations grow: 2x + 3 = 11. Next to x a number is stuck by multiplying and by adding. Two obstacles, two steps.
In what order to clear them? Remember: plus and minus first, then times and divide. First you subtract the three (2x = 8 is left), then you divide by two (x = 4).
Why this direction? Because the three is “furthest from x” — it stuck on last, so it comes off first. It is like undressing: the jacket went on last, you take it off first.
Today you will also get used to x coming out negative or decimal. The equation 2x + 10 = 4 has the solution x = −3 and that is completely fine. The result does not have to be pretty — it has to pass the check.
And we add equations where x is on both sides right away. You will see that the scales rule can handle that too.
Sofie is saving for headphones at 1100 Kč. She already has 300 Kč and each week she puts aside 200 Kč. “In how many weeks will I have them?” Adam writes: 200t + 300 = 1100. “What is t?” — “The number of weeks. Two hundred times weeks plus what you already have.” They subtract 300 from both sides: 200t = 800. They divide by two hundred: t = 4. “Four weeks,” says Sofie and writes it in the calendar. The equation planned her saving — no guessing, just a calculation.
Tidy order: first numbers with plus and minus, then the coefficient of x. A memory trick: what stuck on last comes off first.
Method for ax + b = c
Step 1: clear the constant term. 2x + 3 = 11 / −3 → 2x = 8.
Step 2: clear the coefficient. 2x = 8 / :2 → x = 4.
Step 3: check. 2 · 4 + 3 = 11. It fits.
With a negative result: 3x + 12 = 3 / −12 → 3x = −9 / :3 → x = −3. Check: 3 · (−3) + 12 = −9 + 12 = 3. It fits. A negative x is a normal solution, not a mistake.
With x on both sides: 5x + 2 = 3x + 10. First move all the x to one side — subtract 3x from both sides:
5x − 3x + 2 = 10 → 2x + 2 = 10.
And from there you know it: / −2 → 2x = 8 → x = 4.
Agreement: move x to the side where there is more of it — you avoid needless minuses.
Write every line under the previous one, mark the rearrangement with a slash. An equation in three lines checks itself; an equation calculated in your head is a lottery.
Example 1: 4x − 5 = 19
Step 1 — I clear −5, so I add 5 to both sides:
4x − 5 = 19 / +5 → 4x = 24.
Step 2 — I divide by four:
4x = 24 / :4 → x = 6.
Check: 4 · 6 − 5 = 24 − 5 = 19. It fits.
Two steps, two lines, a check. This rhythm is the whole of solving equations.
Example 2: 2x + 15 = 7
Step 1: 2x + 15 = 7 / −15 → 2x = −8.
Step 2: 2x = −8 / :2 → x = −4.
Check: 2 · (−4) + 15 = −8 + 15 = 7. It fits.
On the right a smaller number came out than on the left, so x had to be negative. Do not cross it out, do not panic — a negative solution is a solution. The check decides, not a feeling.
Example 3: 7x − 4 = 2x + 26
x is on both sides. There is more of it on the left, so I move left — I subtract 2x from both sides:
7x − 2x − 4 = 26 → 5x − 4 = 26.
Now the classic: / +4 → 5x = 30, / :5 → x = 6.
Check into the ORIGINAL equation: left 7 · 6 − 4 = 38, right 2 · 6 + 26 = 38. It fits.
The check is always 완료 in the original equation — in the rearranged one it would not catch a mistake from the first step.
When you do the step “/ −3”, subtract three from the WHOLE right side, not just from somewhere. And do not forget: the check belongs in the original equation. A check in an already rearranged equation is like checking copied text against your own copy.
Practice: now you do it 🎲
Now try it for real. You will see all the questions at once, like on paper. 시작 with ten. When it goes well, add more.
Solve the equation and write only x. Write both steps on paper — the box wants the result, a test wants the working.
On paper
Rhythm: rearrangement, line, rearrangement, line, check.
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Solve: 3x + 7 = 25, 5x − 8 = 17, 2x + 9 = 3. All with rearrangements line by line and a check.
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Solve an equation with x on both sides: 6x + 5 = 4x + 17.
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Sofie is saving: she has 500 Kč and each week she adds 150 Kč. Write and solve the equation for how many weeks until she has 1400 Kč.
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Solve 10x + 3 = 3 and write in a sentence what the result x = 0 means.
Now you 💪
- I tidy in the right order: plus and minus first, then times.
- I move x on both sides to where there is more of it.
- I do the check in the original equation.
Done when: You have six solved equations including one with a negative and one with a zero solution, all with a check, and a run of five correct in practice.
What to take from this lesson
- ax + b = c you solve in two steps: first −b, then :a.
- What stuck on last comes off first.
- A negative or decimal result is normal — the check decides.
- x on both sides? First gather it onto one.
- The check always in the original equation.
© 2026 Ing. Martin Polak / AlgoRhino · 콘텐츠 이용 약관