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An equation is a balance. Whatever you do to one side, you must do to the other. Your job is to get the variable alone on one side.
Solve: x + 7 = 20
x + 7 − 7 = 20 − 7 (subtract 7 from both sides)
x = 13
Solve: 3x + 4 = 19
3x + 4 − 4 = 19 − 4 (subtract 4 first)
3x = 15
3x ÷ 3 = 15 ÷ 3 (then divide by 3)
x = 5
Solve: 5x − 3 = 2x + 12
5x − 2x − 3 = 2x − 2x + 12 (subtract 2x from both sides)
3x − 3 = 12
3x = 15 (add 3)
x = 5 (divide by 3)
Before solving, you often need to plug in values for variables, or simplify an expression so it's easier to work with.
Evaluate 2a + 3b − c when a = 4, b = −2, c = 1.
= 2(4) + 3(−2) − 1
= 8 + (−6) − 1
= 1
Simplify: 5x + 3 − 2x + 7
= (5x − 2x) + (3 + 7)
= 3x + 10
Simplify: 3(x + 4) − 2(x − 5)
= 3x + 12 − 2x + 10 (distribute each bracket)
= x + 22
Watch signs carefully when expanding a negative bracket: −2(x − 5) = −2x + 10, not −2x − 10.
An inequality is like an equation, but with a range of answers rather than a single value.
On a number line, an open circle ○ means "not included" (< or >), and a filled circle ● means "included" (≤ or ≥).
This diagram shows 2 ≤ x < 6: x is at least 2 (filled), but strictly less than 6 (open).
Solve: 3x + 5 > 17
3x > 12 (subtract 5)
x > 4 (divide by 3)
Mia has $50. She buys pens at $3 each. She wants to have at least $15 left. How many pens can she buy?
50 − 3n ≥ 15 (remaining ≥ 15)
−3n ≥ −35
n ≤ 35/3 (divide by −3, FLIP)
n ≤ 11.67, so n ≤ 11 pens (whole pens only)
A linear equation graphs as a straight line. The most useful form is y = mx + c, where m is the gradient (slope) and c is the y-intercept.
For a line through (1, 2) and (4, 8):
m = (8 − 2) ÷ (4 − 1) = 6 ÷ 3 = 2
The y-intercept is where the line crosses the y-axis — set x = 0.
The x-intercept is where the line crosses the x-axis — set y = 0.
For y = 2x − 6:
y-intercept: x=0 → y = −6 so (0, −6)
x-intercept: y=0 → 0 = 2x − 6 → x = 3 so (3, 0)
Line with m = 3 → parallel line has m = 3, perpendicular line has m = −1/3.
Distance between (x₁, y₁) and (x₂, y₂):
d = √((x₂−x₁)² + (y₂−y₁)²)
Midpoint: average the x's, average the y's.
M = ( (x₁+x₂)/2 , (y₁+y₂)/2 )
Two equations, two unknowns. The solution is the pair (x, y) that satisfies BOTH at once. Graphically: where the two lines cross.
Solve: y = 2x + 1 and 3x + y = 16
Substitute the first into the second:
3x + (2x + 1) = 16
5x + 1 = 16
x = 3
Back-substitute: y = 2(3) + 1 = 7
Solution: (3, 7)
Solve: 2x + y = 11 and 3x − y = 4
Add the two equations — the y terms cancel:
(2x + 3x) + (y − y) = 11 + 4
5x = 15 → x = 3
Back-substitute into the first: 2(3) + y = 11 → y = 5
Solution: (3, 5)
Always substitute (x, y) back into BOTH original equations. If either fails, you made an arithmetic error.
The real skill is turning words into algebra. Once the equation is set, the solving is the easy part.
Sam earns $180 for 8 hours of work. How much for 14 hours?
Hourly rate = 180 ÷ 8 = $22.50
Equation: Pay = 22.50 × h
For h = 14: Pay = 22.50 × 14 = $315
Look for the rate of change (the gradient) and the starting value (y-intercept).
A sequence: 5, 8, 11, 14, ... The rate is +3 per step, starting from 5 (at step 1).
Rule: y = 3n + 2 (for step n)
Check: n=1 → 3(1)+2 = 5 ✓
Three apples and two bananas cost $8. Five apples and two bananas cost $12. Find the price of each.
Let a = apple, b = banana
3a + 2b = 8 (1)
5a + 2b = 12 (2)
(2) − (1): 2a = 4 → a = 2
Sub back: 3(2) + 2b = 8 → b = 1
Apple = $2, Banana = $1
You have mastered all six chapters of the Codex.