SAT- Lesson 1: Introduction to the Variable and Combining Like Terms
This foundational lesson introduces students to the concept of a variable as a placeholder for unknown values and teaches the fundamental rules of combining like terms to simplify algebraic expressions.
Welcome to your first step toward mastering the Digital SAT Math section! Let’s break down the language of algebra so you can solve problems faster and with zero hesitation.
1. What is a Variable?
A variable is simply a letter (like $x$, $y$, or $a$) that stands in for an unknown number. Think of it as an empty box waiting for a value to be dropped inside.
🇺🇿 Uzbek Explanation:
O'zgaruvchi (Variable) — bu hali noma'lum bo'lgan sonning o'rniga ishlatiladigan harf (masalan, $x$ yoki $y$). Uni ichiga istalgan sonni qo'yish mumkin bo'lgan bo'sh quti deb tasavvur qiling.
2. The Golden Rule: What are "Like Terms"?
You can only add or subtract terms if they are like terms.
To be considered like terms, parts of the expression must have the exact same variables raised to the exact same exponents.
The Fruit Analogy (Solishtirish)
- Like Terms: Think of apples and apples. 3 apples + 2 apples = 5 apples ($3x + 2x = 5x$).
- Unlike Terms: Think of apples and bananas. 3 apples + 2 bananas cannot be combined into "5 apple-bananas". They stay separate ($3x + 2y$ cannot be simplified further).
Key Identification Table
| Expression | Like or Unlike? | Reason |
|---|---|---|
| $$4x$$ and $-7x$ | Like | Same variable ($x$) with the same power ($1$). |
| $5x^2$ and $3x^2$ | Like | Same variable ($x$) and same exponent ($2$). |
| $2x$ and $2x^2$ | Unlike | Different exponents ($1$ vs $2$). You cannot combine them! |
| $3xy$ and $8yx$ | Like | Order doesn't matter; the variable mix is identical. |
3. How to Combine Like Terms
When combining like terms, you only add or subtract the coefficients (the numbers in front of the variables). The variable and its exponent stay exactly the same.
⚠️ Common Trap: Do NOT change the exponents when adding or subtracting.
$x + x = 2x$ (Correct)
$x + x = x^2$ (WRONG! $x \cdot x = x^2$, but adding changes only the coefficient).
4. SAT-Style Example Questions
Example 1: Basic Simplification
Question: Simplify the expression $3x + 5y - x + 2y + 4$.
Step-by-Step Explanation:
- Group the $x$ terms together: $3x - 1x = 2x$ (Remember, an $x$ by itself has an invisible coefficient of $1$).
- Group the $y$