Solving Linear Equations

An equation is a balance: the left side equals the right side. To keep it true, whatever you do to one side you must do to the other. Solving is undoing the operations around the variable, in reverse, using inverse operations — addition undoes subtraction, multiplication undoes division.

So to free $x$ from $3x + 5$, first undo the $+5$ (subtract $5$ from both sides), then undo the $\times 3$ (divide both sides by $3$). Reverse order, inverse operations, both sides every time. That discipline is the entire skill.

Two steps, undone in reverse. Solve $3x + 5 = 20$:

$3x + 5 = 20 \;\Rightarrow\; 3x = 15 \;\Rightarrow\; x = 5$

Subtract $5$ from both sides, then divide both sides by $3$.

Variables on both sides? Gather them first. Solve $5x + 2 = 3x + 10$ by subtracting $3x$ from both sides:

$5x + 2 = 3x + 10 \;\Rightarrow\; 2x + 2 = 10 \;\Rightarrow\; 2x = 8 \;\Rightarrow\; x = 4$

Parentheses? Distribute or divide first — $2(x + 4) = 18$ becomes $x + 4 = 9$, so $x = 5$. You can always check by substituting your answer back in.

Undo the operations in reverse, one at a time, on both sides. Predict before you check.