Word Problems with Linear Models

Many SAT® algebra questions describe real situations that you must translate into equations. The key skill is identifying the variable, the rate, and the starting value.

Translation guide

• "is," "equals," "is the same as" → =
• "more than," "increased by" → +
• "less than," "decreased by" → -
• "times," "of," "per" → ×

Common linear model types

• Rate × time = distance: d = rt. If a train travels at 60 mph for t hours, d = 60t.
• Fixed cost + variable cost: Total = base fee + (rate × quantity). A plumber charges $75 flat plus $50 per hour: C = 75 + 50h.
• Break-even analysis: Set revenue equal to cost and solve. If Revenue = 20x and Cost = 8x + 360, then 20x = 8x + 360 → x = 30 units.

Step-by-step approach

1. Define the variable (let x = ...).
2. Write the equation.
3. Solve.
4. Check that the answer makes sense in context (negative time is not realistic).

⚠️ Common mistake: Translating "5 less than x" as 5 - x instead of x - 5. Order matters!

💡 Tip: Underline the quantity each variable represents as you read; this prevents mixing up what x stands for midway through the problem.