Summary of "[개념 정리] 중1 수학 (상) 3단원. 문자와 식 - [진격의홍쌤]"

Short summary

Topic: Chapter 3 — Letters and expressions (문자와 식) for 1st-year middle school math.
Goal: Introduce letters (variables) and algebraic expressions; explain multiplication/division with letters; evaluate expressions by substitution; define terms such as monomial/binomial/polynomial/constant/coefficient/degree/like terms; introduce equations and basic equation types; and show how to solve simple first-degree (linear) equations using the equality property and term transposition.
Emphasis: practice is essential.

Letters (a, b, c, x, y, …) represent numbers.
An algebraic expression (식) contains numbers, letters (variables), and operation signs.
When letters appear in expressions they behave like numbers for multiplication and division.

To evaluate an expression, replace variable(s) with given number(s) and compute.
Tip: put parentheses around substituted negative numbers to avoid sign mistakes.
Example:

If x = −3 and the expression is x + 1, substitute to get −3 + 1 = −2.

Term (항): each part of an expression separated by + or −.
- Monomial (단항): one term (e.g., 3x).
- Binomial (이항): two terms (e.g., x + 4).
- Polynomial (다항식): more than two terms.
Constant (상수): a term with only a number (no variable), e.g., 5.
Coefficient (계수): numeric factor of a term with a variable (in 3x, coefficient = 3).
Degree (차수):
- For a single term: the sum of the exponents of the variables in that term.
- For single-variable expressions, the degree is the exponent of that variable (x^2 has degree 2; 3x has degree 1).
- Degree of a polynomial: the highest degree among its terms.
Like terms (동류항): terms with the same variable part and same degree (e.g., 2x and −5x). Like terms can be combined by adding/subtracting coefficients.

An equation uses = and asserts two expressions are equal.
Distinguish from inequalities (>, <), which are not equations.
Types of equations (briefly):
- Identity (항등식): true for all values of the variable(s).
- Conditional equation: true only for particular value(s) of the variable(s).
- You can often recognize an identity if both sides are identical expressions.

Fundamental property of equality: you may perform the same operation on both sides (add, subtract, multiply, divide) without changing the equality.
Typical step-by-step method:
1. Simplify both sides (expand parentheses, combine like terms).
2. Use addition/subtraction on both sides to gather variable terms on one side and constants on the other.
3. Combine like terms to get (coefficient) × x = constant.
4. Divide both sides by the coefficient of x to isolate x.
5. Check the solution by substituting into the original equation.
Transposition (moving a term across =): moving a term to the other side changes its sign — this is just applying addition/subtraction to both sides.
Example solving process:

Given 3x + 4 = x + 5 Subtract x from both sides: 3x − x + 4 = 5 Simplify: 2x + 4 = 5 Subtract 4: 2x = 1 Divide by 2: x = 1/2

Know definitions and be able to identify terms, constants, coefficients, degree, and like terms.
Carefully practice substitution (watch signs and use parentheses for negatives).
Practice distribution of multiplication over parentheses.
Practice solving linear equations using the equality property and moving terms — the steps are mechanical and improve with repetition.
There are no mysterious extra rules beyond the equality property; understanding and practice are key.

Part 1: Letters and using them in expressions (multiplication, division with variables).
Part 2: Vocabulary and structure of algebraic expressions (terms, monomial/binomial/polynomial, constants, degree, like terms).
Part 3: Equations — definitions, types, and solving linear equations (properties of equality, transposition, combining like terms).
Closing: encouragement to practice and preview of the next topic — the coordinate plane and graphs.