Solving Two-Step Equations

A reliable procedure for equations like 2x + 3 = 7: clear the constant, then the coefficient, undoing each operation with its opposite and checking by substitution.

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Some equations yield to a moment's thought. Others do not, and guessing stops working quickly. What you need is a procedure — a sequence of steps that reaches the answer no matter how awkward the numbers.

Concept What solving means

To solve an equation is to get the variable alone on one side.

Once x stands by itself, whatever is on the other side is its value. Everything else is just clearing away what surrounds it.

Concept Keeping the balance

Every move must be applied to both sides. An equals sign says the two sides are worth the same; changing one alone would destroy that.

Take x + 2 = 7 .
Subtract 2 from both sides: x + 2 - 2 = 7 - 2 .
⟹ x = 5

You could have seen x = 5 at a glance. The point is that the written method gives the same answer — and it will keep working when glancing does not.

Concept The order of the two steps
2x + 3 = 7 2x = 4 x = 2 − 3 both sides ÷ 2 both sides
Remove the added number first, then the multiplier. This is the reverse of the order of operations, because you are undoing what was done.
Example Solving 2x + 3 = 7
Step 1 — clear the constant. Subtract 3 from both sides: 2x = 4 .
Step 2 — clear the coefficient. Divide both sides by 2: x = 2 .
Check — substitute back: 2 \times 2 + 3 = 4 + 3 = 7 .
⟹ x = 2

The check is not optional decoration. It confirms the answer against the original equation, so a slip anywhere in the working shows up immediately.

Example Solving 3x + 4 = 19
Subtract 4 from both sides: 3x = 15 .
Divide both sides by 3: x = 5 .
Check: 3 \times 5 + 4 = 15 + 4 = 19 .
⟹ x = 5
Example When the equation subtracts

Solve 5x - 2 = 13 .

Here 2 is being subtracted, so add 2 to both sides: 5x = 15 .
Divide both sides by 5: x = 3 .
Check: 5 \times 3 - 2 = 15 - 2 = 13 .
⟹ x = 3

Each step applies the opposite operation to the one in the equation. Subtraction is undone by addition, multiplication by division.

Note Mistakes to avoid
Dividing before the constant has been cleared.
Operating on one side only and breaking the balance.
Using the same operation instead of the opposite one.
Dividing only part of a side rather than all of it.
Skipping the substitution check at the end.
Summary
  1. Solving means isolating the variable on one side.
  2. Every operation must be applied to both sides.
  3. Clear the added or subtracted number first, then the multiplier.
  4. Undo each operation with its opposite.
  5. Always substitute the answer back into the original equation.
231. Arithmetic Sequences and Series
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