Maths 1A Ch3: Introduction to Algebra

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Maths 1A Ch3: Introduction to Algebra

Key points + Practice

Chapter 3, Introduction to Algebra, is the third topic in the UT1 Maths paper. A good grounding in algebra makes later work on linear equations and functions much easier.

🔤 1. Basic Ideas of Algebraic Expressions

An algebraic expression uses numbers, letters and operation signs, for example 3x + 2y − 5.

Variable: a letter representing an unknown number, such as x, y or a.

Coefficient: the number in front of a letter. In 3x the coefficient is 3; in −5y it is −5.

Constant term: a term without a letter. In 3x + 5, the constant term is 5.

Term: the parts separated by + and − signs. 3x + 2y − 5 has three terms: 3x, 2y and −5.

Writing conventions:

1. Write the number before the letter: 3x (not x3).

2. A coefficient of 1 is usually left out: 1x is written as x.

3. The multiplication sign is usually left out: 3 × x is written as 3x.

4. Division is written as a fraction: x ÷ 3 is written as x/3.

本節生詞：algebra

🧩 2. Like Terms and Simplifying

Like terms: terms with the same letters and the same indices.

✓ 3x and 5x are like terms

✓ 2ab and −7ab are like terms

✗ 3x and 3y (different letters)

✗ x² and x (different indices)

Combining like terms: add the coefficients and keep the letters unchanged.

3x + 5x = 8x

7y − 2y = 5y

Simplifying a polynomial: group the like terms first, then combine.

5a + 3b − 2a + b

= (5a − 2a) + (3b + b)

= 3a + 4b

Removing brackets:

A '+' before the bracket: remove the bracket without changing any signs. a + (b − c) = a + b − c

A '−' before the bracket: change the sign of every term inside. a − (b − c) = a − b + c

🔢 3. Substitution

Substitution means replacing the letters with numbers and working out the value of the expression.

Example: when x = 4, find the value of 3x + 5.

3(4) + 5 = 12 + 5 = 17

Example: when a = 2 and b = −3, find the value of a − 2b.

a − 2b = (2) − 2(−3) = 2 + 6 = 8

Note: put negative numbers in brackets so the signs are not mixed up.

Example: when x = −2, find the value of x².

x² = (−2)² = 4 (not −4)

本節生詞：substitutesquare

⚖️ 4. Simple Equations

An equation is a statement that two expressions are equal.

Solving an equation means finding the value of the unknown that makes the equation true.

Rule for moving terms: when a term is moved to the other side of the equals sign, its sign changes.

Example: x + 7 = 12

x = 12 − 7 (+7 becomes −7 when moved)

x = 5

Example: 4x = 20

x = 20 ÷ 4 (×4 becomes ÷4 when moved)

x = 5

Example: 2x − 3 = 9

2x = 9 + 3 = 12

x = 6

Check by substituting back: 2(6) − 3 = 12 − 3 = 9 ✓

📝 例題

例題 1：Simplify: 7x + 4 − 2x + 3

例題 2：When x = −3, find the value of 2x² + 1.

例題 3：Solve 5x − 4 = 3x + 8.

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