SAT-Math-3: Setting Up Linear Equations from Word Problems

1. The English-to-Math Dictionary

To build an equation, you must treat English words like code. Every word maps directly to a mathematical symbol.

• "Is", "was", "will be", "amounts to", "results in" $\rightarrow =$ (Equal Sign)
• "Greater than", "more than", "sum", "increased by" $\rightarrow +$ (Addition)
• "Less than", "difference", "decreased by", "fewer than" $\rightarrow -$ (Subtraction)
• "Times", "product", "of" $\rightarrow \times$ (Multiplication)
• "Per", "out of", "ratio", "quotient" $\rightarrow \div$ or a fraction bar

⚠️ The "Less Than" Flip Trap: > Pay close attention to the phrasing. "5 less than $x$" does NOT mean $5 - x$. It translates to $x - 5$. You must start with $x$ and subtract $5$ from it.

🇺🇿 Uzbek Explanation:

Tarjima qilish qoidalari: Masalalarni yechishda inglizcha so'zlarni matematik belgilarga aylantirishni o'rganish kerak. Masalan, "is / was" so'zlari doim tenglik ($=$) belgisini bildiradi. "5 less than $x$" iborasi esa juda aldovchi — bu $5 - x$ emas, balki $x - 5$ deb yoziladi ($x$ dan 5 ta kam).

2. The Core Real-World Blueprint: $y = mx + b$

Most linear word problems on the SAT fit into one classic structure:

$$\text{Total} = (\text{Rate} \times \text{Variable}) + \text{Starting Value}$$

• The Flat Fee / Starting Value ($b$): This is a one-time cost, an initial height, a deposit, or a starting population. It happens once and does not change based on time or quantity.
• The Rate of Change ($m$): Look for words like per hour, each month, every mile, daily. This number is always multiplied by the variable ($x$).

3. High-Yield Examples and Step-by-Step Breakdown

Example 1: Single Rate Modeling

The Prompt: A local moving company charges a flat equipment reservation fee of $35 plus an additional charge of $12 per hour spent moving. If a customer's total bill was $119, which equation can be used to find the number of hours, $h$, spent moving?How to Translate Step-by-Step:Identify the Total: The total bill was $119. $\rightarrow = 119$

Example 2: The Dual-Variable Comparison (SAT Style)

Question: An online bookstore sells hardcover books for $18 each and paperback books for $10 each. A professor purchases a total of 12 books for an entire research team. If the total cost of the books was $152, which system of equations represents this situation, where $h$ is the number of hardcover books and $p$ is the number of paperback books purchased?A) $\begin{cases} h + p = 152 \\ 18h + 10p = 12 \end{cases}$B) $\begin{cases} h + p = 12 \\ 18h + 10p = 152 \end{cases}$C) $\begin{cases} h + p = 12 \\ 10h + 18p = 152 \end{cases}$D) $\begin{cases} 18h + 10p = 164 \\ h + p = 12 \end{cases}$

Step-by-Step Explanation:

When dealing with two different items, you almost always need to create two separate equations: one for the quantity (how many) and one for the value (money).

• Equation 1 (Quantity): The total number of books is 12.

  $$\text{Hardcovers} + \text{Paperbacks} = 12 \rightarrow h + p = 12$$

• Equation 2 (Value/Money): Each hardcover is $18 and each paperback is $10. The total money spent is $152.

  $$(18 \times h) + (10 \times p) = 152 \rightarrow 18h + 10p = 152$$

Look at the options. Option B matches our system perfectly.

Correct Answer: B

4. The Direct Translation Method

Instead of reading the entire long paragraph over and over, write down the math clues as you read line-by-line.

🇺🇿 Uzbek Tip: Matnli masalani boshidan oxirigacha qayta-qayta o'qib vaqt yo'qotmang. Har bir gapni o'qiyotgandayoq, kerakli son va kalit so'zlarni yoniga matematik ko'rinishda yozib keting.

Summary

• "Is" means equals ($=$); "per" means multiply by the variable.
• Watch out for "less than"—it reverses the operational order ($x - \text{value}$).
• Structure your thoughts: Separate the fixed, one-time constant from the recurring, variable rate.
• Two-item setups typically require one equation for counting items and another for calculating value/cost.