📖 What Do m and b Actually Mean?
You already know how to graph y = mx + b. Now comes the more important question: what do these numbers mean in real life?
Every linear equation follows the same pattern — and each part has a real-world job:
m — Slope
The rate of change
How much y changes with each step of x.
Examples: $12 per hour · 50 miles per hour · 3 gallons per minute
b — Y-Intercept
The starting amount
The value before anything changes — when x = 0.
Examples: $50 sign-up fee · starts with 800 gallons · already has $200 saved
The graph tells a story. The slope shows how fast something grows or shrinks. The y-intercept is where the story begins.
🔎 Spot the Clue Words
GED word problems always hide clues that point to slope and y-intercept. Train yourself to find them before writing any equation.
Slope clue words — tells you the rate, how much changes per unit
per
each
every
per hour
per mile
per item
daily
weekly
monthly
hourly
per minute
per ticket
Y-Intercept clue words — tells you the starting value before x changes
flat fee
sign-up fee
base fee
starts with
already has
initial amount
beginning balance
upfront cost
activation fee
Quick pattern: Words ending in -ly (daily, weekly, monthly, hourly) almost always describe the slope — how something changes over time.
📈📉 Positive vs. Negative Slope
Before writing your equation, ask: is the quantity growing or shrinking?
Positive Slope (+m)
The amount is increasing
Earning money · climbing altitude · filling a tank · saving each week · accumulating steps
Negative Slope (−m)
The amount is decreasing
Draining a tank · descending a plane · spending money · paying off a loan · losing altitude
Watch out: When something is draining, descending, or decreasing — the slope must be negative. Forgetting the minus sign is one of the most common GED mistakes.
📊 Reading Tables
Tables give you the same information as an equation — you just have to find the pattern. Here's the three-step method:
| Hours Worked (x) | Total Earned (y) |
|---|
| 0 | $40 |
| 1 | $52 |
| 2 | $64 |
| 3 | $76 |
Step 1 — Find the slope: How much does y change each hour? 52 − 40 = $12/hour → m = 12
Step 2 — Find the y-intercept: What is y when x = 0? → $40 → b = 40
Step 3 — Write the equation: y = 12x + 40
✓ Predict earnings after 8 hours: y = 12(8) + 40 = 96 + 40 = $136
🔢 Worked Examples
Example 1 — Writing an equation from a word problem
A gym charges a $25 sign-up fee and $40 per month. Write an equation for total cost (y) after x months, then predict the cost after 6 months.
Spot the clue words: "$40 per month" → rate → m = 40
Find the starting amount: "$25 sign-up fee" → before any months → b = 25
Write the equation: y = 40x + 25
Substitute x = 6: y = 40(6) + 25 = 240 + 25
✓ After 6 months: $265
Example 2 — Negative slope (something decreasing)
A pool starts with 1,500 gallons and drains at 60 gallons per hour. Write the equation and find how many gallons remain after 10 hours.
"drains 60 per hour" → decreasing → m = −60
"starts with 1,500" → b = 1,500
Equation: y = −60x + 1,500
Substitute x = 10: y = −60(10) + 1,500 = −600 + 1,500
✓ After 10 hours: 900 gallons remain
Example 3 — Interpreting what slope means in context
The equation y = 15x + 800 models a salesperson's monthly earnings, where x is the number of items sold. What does the 15 represent?
15 is the slope (m). Slope = rate of change.
In context: for every 1 item sold, earnings increase by $15.
800 is the y-intercept (b). It's the base monthly salary — earned even with zero sales.
✓ The slope of 15 means: $15 earned per item sold.
⚠️ Common Mistakes
Mistake 1
Swapping m and b
Writing y = 25x + 40 instead of y = 40x + 25. Always ask: which number changes each unit? That's m.
Mistake 2
Forgetting the negative sign
Writing y = 60x + 1500 when the pool drains. Decreasing quantities always get a negative slope.
Mistake 3
Misreading the table
The slope is the change between rows — not the first y value. Always subtract consecutive y values.
Mistake 4
Wrong substitution
"After 6 hours" means plug in x = 6. Solve for y. Don't solve for x when the question asks for the total.
✏️ Practice Questions