Solving Linear Equations

The idea in one sentence

An equation is a balanced scale, so whatever you do to one side must also happen to the other.

Undo operations in reverse

Solve $3x + 5 = 20$ by undoing what happened to $x$:

1. Subtract 5 from both sides: $3x = 15$.
2. Divide both sides by 3: $x = 5$.

The order matters. The expression says “multiply by 3, then add 5,” so solving walks backward: subtract 5, then divide by 3.

Check the answer

Replace $x$ in the original equation:

\[3(5) + 5 = 15 + 5 = 20\]

The left side becomes the right side, so $x = 5$ works.

Variables on both sides

For $5x - 2 = 2x + 10$, gather the variable terms on one side:

\(5x - 2 = 2x + 10\) \(3x - 2 = 10\) \(3x = 12\) \(x = 4\)

Each line keeps the scale balanced.

Common mix-ups

• Moving a term and changing its sign without understanding why. Instead, name the same operation on both sides.
• Dividing only one term. In $3x + 6 = 12$, subtract 6 first or divide every term by 3.
• Stopping without checking. Substitution catches sign mistakes fast.

Check yourself

Solve $4x - 7 = 13$, then substitute your answer into the original equation.

$4x = 20$, so $x = 5$. Check: $4(5) - 7 = 13$.