Writing Linear Equations & Modeling
Practice turning tables, points, and word problems into linear equations you can use to predict.
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Questions Covered in This Set
10 cards to master
What two ingredients does every linear model need?
A rate of change (slope m) and a starting value (y-intercept b), plugged into y = mx + b.
What does slope m represent in a real-world model?
The rate of change — how much y changes each time x increases by 1. Its units are always y-units per x-unit (e.g., $15 per month).
What does b represent in a real-world model?
The value of y when x = 0: the starting amount, flat fee, or head start.
How do you check that a table is linear?
See if the y-values change by a constant amount for equal jumps in x (constant ratio Δy/Δx). If x-values aren't evenly spaced, use m = (y₂−y₁)/(x₂−x₁).
Write point-slope form and explain its advantage.
y − y₁ = m(x − x₁). It lets you write the equation immediately from a point and slope without first solving for b.
Line through (3, 7) and (8, 22): find the equation.
m = (22−7)/(8−3) = 3; y − 7 = 3(x − 3) → y = 3x − 2. Check: 3(8) − 2 = 22 ✓
What are the 4 steps for a linear word problem?
1) Name variables with units, 2) find the rate (slope), 3) find the starting value (b), 4) answer the actual question using the equation.
Which words signal a negative slope?
Drains, loses, spends, cools, decreasing, burns, falls — anything shrinking over time.
Tank: 500 gallons draining 12 gal/min. Equation and empty time?
W = −12t + 500. Empty when W = 0: t = 500/12 ≈ 41.7 minutes.
Gym A: C = 25m + 40; Gym B: C = 35m. When are they equal?
25m + 40 = 35m → 40 = 10m → m = 4 months, both cost $140. Before 4 months B is cheaper; after, A is cheaper (smaller slope).
