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.
A short walkthrough explaining what you need to know and how to solve this question type lands here once it's recorded.
ALEKS randomizes the numbers each attempt, but the question shape stays the same. Here are three example versions you might see.
A plumber's total cost C (in dollars) for a job lasting t hours is given by:
C = 75 + 50t
What does the 50 represent? What does the 75 represent?
A car's distance D (in miles) from home after t hours is:
D = 60t + 10
What does the 60 mean? What does the 10 mean?
A small business's profit P (in dollars) after selling u units is:
P = 8u − 200
What does the 8 represent? What does the −200 represent?
b = starting value · m = rate of change
C = 75 + 50t (plumber).
Match to y = mx + b: m = 50, b = 75. Now translate.
b = 75: the cost when t = 0, before any time has passed. That's the flat fee for showing up — $75 even if the plumber fixes nothing.
m = 50: the cost goes up by $50 for every additional hour. That's the hourly rate: $50 per hour.
D = 60t + 10 (car distance).
Watch the order: m = 60, b = 10. The mt term comes first in the equation, but m is still the coefficient of t.
b = 10: distance from home when t = 0. That's the head start — 10 miles already covered before the clock starts.
m = 60: distance increases by 60 miles per hour. That's speed: 60 mph.
P = 8u − 200 (small-business profit).
m = 8, b = −200. Negative intercept is allowed and meaningful.
b = −200: profit when u = 0 (no units sold). It's negative because of startup cost — fixed expenses you owe even before selling anything.
m = 8: each unit sold adds $8 to profit. That's the profit per unit.
Bonus: how many units before the business breaks even? Set P = 0 and solve: 8u − 200 = 0 → u = 25 units.
Walk through this practice problem one step at a time. Each step unlocks the next.
Identify b in the equation.
Interpret b in context.
signup, monthly, or total.Interpret m in context.
You walked the linear-model move end to end.
Same three steps every time: match the equation to y = mx + b, identify which number is the starting value (b) and which is the rate (m), and translate each into the language of the problem. Plumber? Hourly rate. Car? Speed. Business? Profit per unit. The math is the same; the units do the talking.