Free maths tool · Level: Ages 12 to 14
Solving equations
An equation is a balance that is level. Step through the moves and watch what happens to both pans.
With the balance method you do exactly the same thing to both sides, so the balance stays level. For 2x + 3 = 11 you take 3 off both sides to get 2x = 8, then divide both sides by 2 to get x = 4. Put 4 back into the original equation and it holds, which is your check.
Two pans that stay equal
The equals sign says the left and the right weigh the same. Take something off one side and the balance tips. Take the same off both sides and it stays level, leaving a simpler equation with the same solution.
That is where the rule comes from. Not because a number "moves across and changes sign", but because you did the same thing to both sides. Anyone who can picture the balance stops making that sign error.
Loose numbers first, then the factor
The order is fixed: clear the loose numbers first, and only then divide by the number in front of the x. Step through it and you can see why. After the first step there are only x blocks on the left, and dividing is easy.
The other way round you also have to divide the loose number, and that is where mistakes creep in. For 2x + 3 = 11, dividing first gives x + 1.5 = 5.5: not wrong, just more work.
When the x disappears
Set the number in front of the x to zero and something odd is left. If what remains is true, like 3 = 3, then every x is a solution. If it is false, like 3 = 5, then no x works at all.
It looks like an edge case, but it turns up in tests whenever both sides turn out identical after tidying up. The answer then is not "no idea" but "every x" or "no x".
Stuck on equations with x in them?
A tutor walks the steps through with you on your own exercises, until you can see them yourself.
Frequently asked questions
- What is the balance method?
- A way of solving equations where you always do the same thing to both sides. Adding, subtracting, multiplying and dividing are all allowed as long as both sides get it, because the balance then stays level.
- How do you solve 2x + 3 = 11?
- Take 3 off both sides, leaving 2x = 8. Divide both sides by 2 and you get x = 4. Check by putting 4 back into the original: 2 · 4 + 3 really is 11.
- Why does a number change sign when it moves across?
- It does not really move. You subtract it from both sides, and where it used to be there is now zero. It looks like a crossing with a sign change, but what actually happened was the same move on both sides.
- How do you check your solution?
- Substitute your answer into the original equation, not into a middle step. If both sides come to the same number, it is right. That finds an arithmetic slip without redoing the whole thing.
- What if there is a fraction in the equation?
- Multiply both sides by the denominator and the fraction is gone. For x/3 = 4, multiply both sides by 3 and you are left with x = 12. That is still the same operation on both sides.
More interactive maths tools
- Solving inequalities →See the answer on the number line and why the sign flips with a negative.
- Adding fractions →See with bars why fractions need a common denominator, and what goes wrong without one.
- Slope of a line →Drag two points and see Δy/Δx as a staircase, with the equation and the perpendicular alongside.