MAT-144 · Mathematical Reasoning
Topic 01 · Linear Functions
Reading a linear model
Given a real-world equation, identify the starting value (y-intercept) and the rate of change (slope), then explain what each one means in context.
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Worked Example Video
Every step of Reading a linear model explained on video. Pause anywhere; the embed scrolls independently of the page.
YOUTUBE
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What This Looks Like on ALEKS
ALEKS randomizes the numbers each attempt, but the question shape stays the same. Here are three example versions you might see.
Reading a linear model
A plumber’s total cost C (in dollars) for a job lasting t hours is C = 75 + 50t. What are m and b? Enter as m, b.
m, b =
✓
Right — m = 50 (the coefficient of t) and b = 75 (the constant). The 50 is the hourly rate; the 75 is the flat show-up fee.
✗
The coefficient of t is m (here: 50, the hourly rate). The constant on its own is b (here: 75, the flat fee). Enter as 50, 75.
Reading a linear model
A car’s distance D (in miles) from home after t hours is D = 60t + 10. What are m and b? Enter as m, b.
m, b =
✓
Right — m = 60 (mph, the speed) and b = 10 (miles, the head start already covered). The 60 is how fast; the 10 is how much distance was already there.
✗
m is the coefficient of t (60, the speed in mph). b is the constant (10, the head start in miles). Enter as 60, 10.
Reading a linear model — negative intercept
A small business’s profit P (in dollars) after selling u units is P = 8u − 200. What are m and b? Enter as m, b.
m, b =
✓
Right — m = 8 (profit per unit sold) and b = −200 (startup cost, before any sales). Negative b is allowed and meaningful in real problems — it’s the fixed cost you owe before you start.
✗
m is the coefficient of u (8). b is the constant (−200 — startup cost). Enter as 8, −200.
Heads up: Your ALEKS version will use different numbers. The numbers in the practice below are different too — that way you're exercising the move, not memorizing one answer.
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The Rule
Real-world equation dressed as y = mx + b
b = starting value, m = rate of change
b is what the output is when input = 0
m is what the output changes by per one input
Real-world equations swap x and y for new letters (t, u, C, P, D) and give them units, but the roles don’t change. b is the flat / starting piece; m is the per-unit rate. See it live: how m and b move the line →
Common slip
Students swap the roles of m and b. The coefficient of the input variable is m (the rate), and the constant on its own is b (the starting value). If the equation is written b + m·x instead of m·x + b, they still play the same roles — the order in the equation doesn’t change what they mean.
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Practice — Step by Step
Practice this problem step by step
A phone plan costs C = 25 + 0.10n dollars, where n is the number of minutes used.
Identify m (the rate)
What is m in this equation?
m sits in front of the input variable. In C = 25 + 0.10n, the input is n.
✓
Right — m is the coefficient of the input variable (n). Here m = 0.10.
✗
Not quite — try again.
Identify b (the starting value)
What is b in this equation?
b is the standalone number — the one with no variable next to it.
✓
Right — b is the constant on its own. Here b = 25.
✗
Not quite — try again.
What does the slope mean in context?
You identified m = 0.10. In this phone-plan scenario, what does that number represent?
m tells you how much the output changes for each additional unit of input. Input is n (minutes), output is C (cost).
✓
Right — m is the rate of change per one unit of input. Here, that’s $0.10 per minute. b (the 25) is the base fee — what you owe before making any calls.
✗
Not quite — try again.
🎉
You read the linear model end to end.
Same three moves every time: find m (the coefficient of the input), find b (the constant), translate each into the story. b is the starting value; m is the per-unit rate. ALEKS will change the scenario — plumber, car, business, phone plan — but the roles never change.