You can already solve equations like $p(x + q) = r$ and simplify expressions. Longer equations need nothing new: tidy each side first, then solve the short equation that is left.
Why it works
An equation is a balance: both sides name the same number. Distributing and combining like terms only rewrite a side, without changing its value, so the balance is untouched. After that, doing the same thing to both sides keeps it balanced while you get the variable alone.
Key idea
A plan that always works: (1) distribute to clear parentheses; (2) combine like terms on each side; (3) undo adding or subtracting; (4) undo multiplying or dividing. Then check by putting your answer back into the original equation.
Worked examples
Worked example: Distribute first
Solve $3(x + 4) - 5 = 19$.
- Distribute: $3x + 12 - 5 = 19$.
- Combine: $3x + 7 = 19$.
- Subtract 7 from both sides: $3x = 12$.
- Divide both sides by 3: $x = 4$. Check: $3(4 + 4) - 5 = 24 - 5 = 19$.
Answer: $x = 4$
Worked example: Combine like terms
Solve $2x + 5 + 4x = 35$.
- Combine the $x$ terms: $6x + 5 = 35$.
- Subtract 5: $6x = 30$.
- Divide by 6: $x = 5$.
Answer: $x = 5$
Worked example: A negative outside
Solve $-2(x - 3) + 4x = 20$.
- Distribute $-2$ to both terms: $-2 \cdot x = -2x$ and $-2 \cdot (-3) = +6$. So $-2x + 6 + 4x = 20$.
- Combine: $2x + 6 = 20$.
- Subtract 6, then divide by 2: $x = 7$. Check: $-2(4) + 28 = 20$.
Answer: $x = 7$
Worked example: Subtracting a whole group
Solve $10 - (x + 2) = 3$.
- Subtracting $(x + 2)$ subtracts both parts: $10 - x - 2 = 3$.
- Combine: $8 - x = 3$.
- Subtract 8: $-x = -5$, so $x = 5$. Check: $10 - 7 = 3$.
Answer: $x = 5$
Common mistake
Distributing a negative to the first term only: $-2(x - 3) = -2x - 6$. The $-2$ multiplies the $-3$ too, and a negative times a negative is positive: $-2(x - 3) = -2x + 6$. Quick check with $x = 0$: $-2(0 - 3) = 6$, but $-2(0) - 6 = -6$.