How to Solve Basic Algebra Equations

The method, step by step

1. 01

   Clear any brackets

   3(2x + 1) = x + 18 → 6x + 3 = x + 18

   A bracket holds the terms inside it hostage to whatever is multiplying them, and nothing can be collected until that is released. The 3 multiplies both terms, so 2x becomes 6x and 1 becomes 3. Multiplying only the first term is the most frequent error in the whole topic.

2. 02

   Gather the letters on one side

   6x + 3 - x = x + 18 - x → 5x + 3 = 18

   Taking x off both sides removes it from the right without changing what the equation says, because both sides lose the same amount. Choosing the side with more x terms to keep means the coefficient stays positive, which removes a whole class of sign errors later.

3. 03

   Gather the numbers on the other side

   5x + 3 - 3 = 18 - 3 → 5x = 15

   We take 3 off both sides to get the x term on its own. The 3 was added to 5x, so subtracting is what undoes it. Working in the reverse order of the operations — addition undone before multiplication — is why the letter ends up isolated rather than more tangled.

4. 04

   Undo the multiplication

   5x ÷ 5 = 15 ÷ 5 → x = 3

   5x means 5 multiplied by x, so dividing both sides by 5 leaves one x. This is the last operation to be undone because it was the first to be applied to the letter, and dividing earlier would leave fractions to carry through every remaining line.

5. 05

   Check by substituting back

   LHS: 3(2×3 + 1) = 3 × 7 = 21 RHS: 3 + 18 = 21 ✓

   Putting the answer into the original equation — not the tidied-up version — is what proves it. Checking against a line you rewrote will confirm your own mistake back to you. Both sides coming to 21 is the evidence that x = 3 is correct.

01

The rule underneath everything

An equation is a statement that two things are equal, and every legal move preserves that equality. Whatever is done to one side is done to the other: add the same, subtract the same, multiply by the same, divide by the same. There is no separate rule for "moving a term across and changing the sign" — that is a shorthand for subtracting it from both sides.

Students who learn the shorthand without the reason apply it in places it does not hold, most often to terms that are multiplied rather than added. Keeping the full line written out for a few weeks is slower and, for most students, considerably faster in the end.

02

Why the order of undoing matters

Building an expression applies operations in one order; solving reverses it. If x was multiplied by 5 and then had 3 added, undoing means subtracting the 3 first and dividing by 5 second. Dividing first is not wrong, but it forces every subsequent term through a fraction for no gain.

This is the same reasoning behind clearing brackets before collecting terms. The bracket is the outermost operation applied, so it is the first one to be released.

• Brackets — expand them
• Letters — gather on the side that keeps the coefficient positive
• Numbers — gather on the other side
• Coefficient — divide it out last
• Check — substitute into the original equation

03

Where the marks are actually lost

Three errors account for most lost marks at this level, and none of them is a failure to understand algebra. They are mechanical, they are predictable, and they are fixable by writing one more line rather than one fewer.

The last of the three is worth naming plainly: negative signs. When a bracket is preceded by a minus, every term inside it changes sign, so 5 − 2(x − 4) becomes 5 − 2x + 8. Students who get this wrong are usually not confused about it; they are working too fast to write the intermediate line.

• Multiplying only the first term inside a bracket
• Doing something to one side and forgetting the other
• Losing a negative sign when expanding a bracket that follows a minus
• Checking against a rewritten line instead of the original equation

04

How we teach this

We ask for the full line — the operation written on both sides — until a student can explain why each move is legal. The shorthand comes afterwards, and it comes faster for having been earned rather than copied.

Every solution ends with a substitution check, done in the lesson rather than left as advice. It costs thirty seconds, it converts an uncertain answer into a certain one, and it builds the habit that later makes simultaneous and quadratic equations much less fragile.