1. Solving Linear Equations
What is a linear equation?
A linear equation is an equation where the variable (usually x) has a power of 1. That means no x², x³, or any higher powers. It can always be written in the form ax + b = c, where a, b, and c are numbers.
Linear equations are called "linear" because if you drew them on a graph, they'd be straight lines. But we're not graphing here — we're solving them, which means finding the exact value of x that makes the equation true.
Linear equation: An equation of the form ax + b = c where the variable has power 1 only. The solution is the exact value of x that makes both sides equal.
Linear equations
These are linear:
2x + 3 = 5 (power of x is 1)
3x + 4 = 1 (power of x is 1)
x − 5 = −3 (power of x is 1)
These are NOT linear:
x² + 3 = 5 (power of x is 2)
3x³ + 4 = 1 (power of x is 3)
The main idea: inverse operations
To solve an equation, you need to get x by itself on one side. You do this by undoing whatever has been done to x. This is where inverse operations come in.
If something has been added to x, subtract it (inverse of + is −)
If something has been subtracted from x, add it (inverse of − is +)
If x has been multiplied by a number, divide by that number (inverse of × is ÷)
If x has been divided by a number, multiply by that number (inverse of ÷ is ×)
The golden rule: Whatever you do to one side of the equation, you must do to the other side. This keeps the equation balanced — like a see-saw. If you add 5 to the left side, you must add 5 to the right side too.
Solving a simple equation step by step
Solve: 2x + 1 = 9
Step 1: Look at what's been done to x. There's a +1, so undo it by subtracting 1 from both sides.
2x + 1 = 9
(−1) (−1)
2x = 8
Step 2: Now x is multiplied by 2. Undo this by dividing both sides by 2.
2x = 8
(÷2) (÷2)
x = 4
Check: 2(4) + 1 = 8 + 1 = 9 ✓
Does order of steps matter?
Technically, no — as long as you apply each operation correctly to both sides. But practically speaking, some orders are easier than others. The easiest approach is usually to:
1. Get rid of constants (numbers without x) by adding or subtracting
2. Get rid of coefficients (numbers attached to x) by dividing or multiplying
Important trap: If you divide first before moving the constant, you must divide ALL terms. For example, in 4x + 8 = 12, if you divide by 4 immediately, you get (4x)/4 + 8/4 = 12/4, which is x + 2 = 3, then x = 1. But it's easier to subtract 8 first: 4x = 4, then x = 1. Same answer, but less work.
Solving with negative numbers
This is where many students make mistakes. The process is the same, but you must be very careful with signs.
Working with negatives
Solve: 2 − 3x = 10
Step 1: Subtract 2 from both sides
2 − 3x = 10
(−2) (−2)
−3x = 8
Step 2: Divide both sides by −3
−3x = 8
(÷ −3) (÷ −3)
x = −8/3
Note: Negative ÷ positive = negative. −8 ÷ 3 = −8/3 = −2⅔ or −2.67
Common mistake:
Solving equations with brackets
expand (remove) the brackets first