What is linear equations?

Why it matters

Linear equations are the bedrock of algebra. Every "solve for x" problem in middle and high school — and every formula manipulation in physics, chemistry, and personal finance — uses the same "do the same thing to both sides" moves that linear equations teach.

Worked example

Solve for x: 3x − 4 = 11.

1. 1

   The goal is to get x by itself on one side. Right now it’s wrapped in "multiply by 3, then subtract 4" — we need to undo both operations.

   Whatever was done LAST to x, undo first. Here the last thing was subtracting 4, so undo that first.

2. 2

   Add 4 to both sides to cancel the −4: 3x − 4 + 4 = 11 + 4, which simplifies to 3x = 15.

   Adding 4 to both sides keeps the equation balanced — it’s the same equation, just rewritten.

3. 3

   Divide both sides by 3 to cancel the multiplication: 3x ÷ 3 = 15 ÷ 3, which gives x = 5.

4. 4

   Check by plugging back into the original: 3(5) − 4 = 15 − 4 = 11. ✓

   Always check. If the check doesn’t balance, you made an arithmetic error somewhere.

Common mistakes

• •Doing operations in the wrong order — dividing first when you should have added first. The undo order is the reverse of the operation order.
• •Only applying the operation to one side ("subtract 4" from the left but forgetting to subtract 4 from the right). That breaks the equation.
• •Sign errors when the equation has negative numbers: 2x + 7 = −3 → 2x = −10 (kids often write 2x = 4 here).
• •Skipping the check step. A quick plug-back catches almost every arithmetic slip.

How Briveli teaches linear equations

Briveli builds linear-equation skill in two stages: one-step equations in Grade 6, then two-step and multi-step equations in Grade 7 and Grade 8 — including word problems and equations with the variable on both sides (the version 8th-grade state tests focus on).