Algebraic-Vocabulary Wjec-Eduqas Higher

Algebraic vocabulary

What is a term?

• A term is either:

  • a letter (variable) on its own, or a variable raised to a power

    • For example, x or x²

  • a number on its own

    • For example, 20

    • These are also called constants as they can’t change value

  • or a number multiplied by a letter

    • For example, 5x

• The number in front of a letter is called a coefficient

  • The coefficient of x in 6x is 6

  • The coefficient of y in -5y is -5

• Terms can include powers and more than one letter, 

  • 6xy

  • 4x²

  • ab³c

What is a factor?

• A factor is any number or letter that divides a term exactly

  • There is no remainder

    • The factors of 3x are 1, 3, x and 3x

    • The factors of 4xy are 1, 2, 4, x, 2x, 4x, y, 2y, 4y, xy, 2xy and 4xy

• A term can be separated into factors that multiply together to give that term

  • Two factors of 5x are 5 and x

    • 5 × x = 5x

• To factorise means to write something as a multiplication of factors

• When comparing two terms, a common factor is one that divides both

  • Compare 6xy with 4x

    • Common factors are 1, 2, x and 2x

    • The highest (or greatest) common factor is 2x

What is an expression?

• An expression is an algebraic statement that does not have an equals sign

  • There is nothing to solve

• An expression is made by adding, subtracting, multiplying or dividing terms

  • 2x + 5y

  • b² – 2cd

  • 

  • A single term is still an expression

• Expressions can be simplified (made easier)

  • x + x + x simplifies to 3x

What is an equation?

• An equation is an algebraic statement with an equals sign between a left-hand side and a right-hand side

  • Both sides are equal in value

  • For example, if 2x has the same value as 10, then 2x = 10

• An equation can be solved by finding the values of the letters that make both sides equal

  • The equation 2x = 10 is solved when x has the value of 5

    • x = 5 is called the solution

What is a formula?

• A formula is a rule, definition or relationship between different quantities, written in shorthand using letters

  • For example, weight, w, is mass, m, multiplied by gravitational acceleration, g

    • The formula is w = mg

• It is common to substitute numbers into a formula

  • But a formula on its own cannot be solved

• To turn a formula into an equation, more information is needed

  • For example, if w = 50 and m = 5, the formula w = mg becomes the equation 50 = 5g

What is an identity?

• An identity is an algebraic statement with an identity sign, ≡, between a left-hand side and a right-hand side that is true for all values of x

  • E.g. x + x ≡ 2x

  • This means x + x is identical to 2x, or that x + x can also be written as 2x

• An identity cannot be solved

• All numbers can be substituted into an identity and it will remain true

  • E.g. x + x ≡ 2x is true for x = 1, x = 2, x = 3 … (even x = -0.01, x = π etc)

  • Unlike with equations, where only the solutions work

    • E.g. 2x = 10 is not true for x = 1, x = 2, x = 3 … only for x = 5

• Identities can be used to write algebraic expressions in different forms

  • E.g. find p and q if 3(x + y) + 2y ≡ px + qy

    • 3(x + y) expands to 3x + 3y

    • The coefficient of x on the left is 3 and on the right is p, so p = 3

    • The coefficient of y on the left is 3 + 2 and on the right is q, so q = 5

    • Therefore 3(x + y) + 2y is identical to 3x + 5y

    • This method is called equating coefficients