Simplifying and forming expressions

Algebra uses letters for numbers. Learn these words:

≡ means "is identical to". The number in front of a letter is its coefficient: in 3x, it's 3.

Example: is each of these an expression, an equation, a formula or an identity?

1. 4x − 1 = 7 is only true when x = 2: an equation
2. v = u + at links quantities: a formula
3. x + x ≡ 2x is true for every x: an identity

Like terms have exactly the same letters raised to the same powers, such as 5a and −2a, or 4x² and −x². Only like terms can be added or subtracted.

1. Find the like terms, each with the sign in front of it.
2. Add or subtract their coefficients.

Multiplying letters: 3 × a × b = 3ab, and a × a = a².

Example: simplify 5a + 3b − 2a + 4b.

1. a terms: 5a − 2a = 3a
2. b terms: 3b + 4b = 7b
3. 3a + 7b

Harder example: simplify 4x² + 3x − x² − 7x + 2.

1. 4x² − x² = 3x² and 3x − 7x = −4x
2. 3x² − 4x + 2

To turn words into algebra, pick a letter for the unknown and translate each phrase.

Example: a rectangle is 2x cm long and x + 3 cm wide. Write an expression for its perimeter.

1. Add all four sides: 2x + (x + 3) + 2x + (x + 3)
2. 6x + 6 cm

An equation needs a fact that says two things are equal. In geometry, angle facts are the usual source: angles in a triangle add up to 180°, and angles on a straight line add up to 180°.

1. Write each quantity in terms of the letter.
2. Use the fact to set up an equation.
3. Solve it, then answer the question asked.

Example: a triangle has angles x, 2x and 3x + 30 degrees. Find the angles.

1. Angles in a triangle add up to 180°: x + 2x + 3x + 30 = 180
2. 6x + 30 = 180, so 6x = 150 and x = 25
3. 25°, 50° and 105°

Check: 25 + 50 + 105 = 180.