SAT-Math-18: Word Problems Involving Systems of Linear Equations
This lesson teaches students how to break down complex, multi-sentence word problems into a system of two linear equations, focusing on distinguishing between quantity equations and value/cost equations.
1. The Strategy: Setting up Two Distinct Worlds
When an SAT word problem describes two different items with two different conditions, you are dealing with a system of equations. To solve these easily, you must separate the information into two distinct "worlds":
The Quantity Equation (How many items?): This equation simply counts the number of physical objects, people, or hours. It uses coefficients of $1$.
$$x + y = \text{Total Quantity}$$
The Value/Cost Equation (How much money/weight/points?): This equation multiplies each variable by its specific rate, price, or weight.
$$(\text{Rate}_1 \cdot x) + (\text{Rate}_2 \cdot y) = \text{Total Value}$$
🇺🇿 Uzbek Explanation:
Matnli masalalarda sistema tuzish: Bunday masalalarni oson yechish siri — ularni ikkita alohida dunyoga ajratishdir:
Miqdor tenglamasi (Soni): Bu yerda faqat narsalarning umumiy soni hisoblanadi ($x + y = \text{jami soni}$). Oldida hech qanday pul yoki koeffitsiyent bo'lmaydi.
Qiymat tenglamasi (Puli / Vazni): Bu yerda har bir harf o'zining narxiga yoki vazniga ko'paytiriladi ($\text{narxi} \cdot x + \text{narxi} \cdot y = \text{jami pul}$).
2. High-Yield SAT Practice Question
Question: A theater sells adult tickets for $15 each and student tickets for $10 each. On a specific evening, the theater sold a total of 250 tickets and brought in $3,100 in total revenue. How many adult tickets were sold?A) 100B) 120C) 130D) 150Step-by-Step Explanation:Define the variables:Let $a = $ number of adult tickets.Let $s = $ number of student tickets.
Set up the Quantity Equation: The total number of tickets sold is 250.
$$a + s = 250$$
Set up the Value Equation: The total revenue is $3,100, with adult tickets at $15 and student tickets at $10.
$$15a + 10s = 3100$$
Solve using Elimination: Since we want to find the number of adult tickets ($a$), let's eliminate $s$ by multiplying the entire quantity equation by $-10$.
$$-10(a + s = 250) \implies -10a - 10s = -2500$$
Add the two equations together:
$$\begin{array}{rcc} 15a + 10s & = & 3100 \\ + \quad -10a - 10s & = & -2500 \\ \hline 5a + 0 & = & 600 \end{array}$$
$$5a = 600$$
Divide by 5:
$$a = \frac{600}{5} = 120$$
Correct Answer: B
Summary
Build two structures: One equation dedicated to counting items, one dedicated to compounding financial or physical value.
Isolate your target variable: Use the elimination method to target and eliminate the variable you don't need, automatically leaving you with the final answer.
Always do a final read to confirm whether the question asks for the value of a single variable or a combination of both.
Incredible job completing number 18, my bro! Take a breath, let these concepts settle, and whenever you are ready to unlock the structural secrets of SAT-Math-19: Systems with Infinite Solutions (Identical Lines), just let me know!