Introduction to Algebra

• Algebra uses letters or symbols to represent unknown numbers.
• Using letters instead of numbers makes it possible to solve many problems.

Rules of Algebra

• Never use "×" signs; instead, numbers are written before letters.
• When multiplying a number by a letter, write the number in front of the letter.
• Letters are written in alphabetical order.
  • For example: z4 = 4z, xy5 = 5xy, b3a² = 3a²b
• If the number is one, we do not write it.
  • For example: 1x = x, 1x² = x²

Algebraic Expressions

• A term is a product of numbers and letters called a monomial.
  • For example: 3x, -2a, 5y², xy²
• A polynomial is 2 or more monomials separated by "+" or "-" signs.
• An algebraic expression contains one or more terms separated by "+" or "-" signs.
  • For example: 2x - x + 5, 3ax² + 6a − b + 2ax² – 10 ,3x² – 5xy + 8
• Evaluating an algebraic expression involves substituting numbers for letters.
  • For example find the value of 3xy + 2x – 6 when x = 1 and y = 2: 3 · 1·2 + 2 · 1 - 6 = 6 + 2 − 6 = 2
• The order of operations for algebraic expressions follows BIDMAS.
• Literal factors are the letters that represent variables.
  • For example: In the algebraic expression 3x² – 5xy + 8, the literal factors are x², xy, and the terms are 3x², -5xy, 8.
• Coefficients are the number part of the terms with variables.
  • For example: In the algebraic expression 3x² – 5xy + 8, the coefficients are 3, -5, and 8
  • For example: In the algebraic expression 4ab – 2b + a, the coefficients are 4, -2, and 1
• The degree of a monomial is the sum of the indices of its literal factors.
  • For example: The degree of 3x is 1, the degree of 5y² is 2, and the degree of xy²/3 is 3.
• Like terms have the same literal factors and can be added or subtracted through collecting like terms.

Simplifying Expressions

• Like terms can be collected together when an expression is the sum or difference of terms.
  • For example: 6y + 8x² – 5x² + y − 5 = 3x² + 7y - 5
  • 8x² and -5x² are like terms, as are 6y and y.
• Expressions that don't contain like terms cannot be simplified.
  • For example: 2a – ab + 5b + 4
• Simplification of terms through addition or subtraction is only possible if there are like terms.
  • For example: 2a − a + 5b + 3b = a + 8b, but 2a-3ab+5 cannot be simplified.
• When multiplying terms: multiply the coefficients separately, powers are used, the new index is the sum of the indexes.
  • For example: a · a = aa = a², b·b·b·b = b4, c·c⁴ = c5
  • For example: 2a · 3a = 2 · 3 · a · a = 6 · a² = 6a², -3x³ · 5x² = −3 · 5 ·x·x·x·x·x = −15· x5 = −15x5
• When dividing terms: divide the coefficients, powers are used, the new index is the subtraction of the indexes.
  • For example: a4/a = a4-1 = a³, b7/b³ = b4
  • For example: (2a) : (8a) = 2a/8a = 2.a/8.a= 2/8 = 1/4
  • For example: (-3x³) : (x²) = -3x³/x² = -3·x·x·x/xx= -3·x/1 = -3x
• To multiply a term by 2 or more terms, use brackets and solve by multiplying the term outside the brackets by all terms inside the brackets - expanding brackets.
  • For example: 5· (2a + 3b) = 5 · 2a + 5 · 3b = 10a + 15b,
  • For example: −2x • (x² + 3xy) = −2x • x² + (−2x) ・ 3xy = −2 · x³ − 6 • x² • y = −2x³ – 6x²y
• A formula describes the relationship between variables.
  • For example: y = 1.5x where y represents the money spent on oranges, 1.5 represents the price of 1 kg of oranges in €, and x represents the number of kilograms of oranges bought.
  • If I buy 3 k of oranges, I will pay y = 1.5.3 → y = 4.5€

Equations

• An equation is 2 algebraic expressions separated by an equal sign, true only for some values of the letters.
  • For example: : x+1=4 is an equation, it is only true when x=3
• An identity is 2 algebraic expressions separated by an equal sign, true for every value of the letters.
  • For example: 5x+2x=7x is an identity, it is true for any value of x
• Sides in an equation are expressions between the equal sign.
  • The expression on the left of the equal sign is the left hand side
  • The expression on the right is the right hand side
• Each side contains terms
• Unknowns are the letters in the terms
• Solutions are the values of the letters that make the equation true.
  • For example: 5x+1=11, the left hand side is 5x+1, the right hand side is 11; the terms are 5x, 1, 11; the unknown is x; and the solution is x=2 because 5·2+1=10+1=11
• Equivalent equations have the same solutions.
  • For example: 2x+1=11 and 2x=10 are equivalent because the solution for both is x=5

Solving Equations

• Solving an equation means finding its solutions by undoing each operation in turn using the inverse operation.
  • Addition and subtraction are inverse operations, as are multiplication and division.
  • For example: "+3" undoes "-3", and "4" undoes ":4".
• You can check when you have the solution of an equation by substituting the solution in the value of the unknown
  • For example: 2x+5=3, 2x=-5+3, 2x=-2, x=-1, we check it: 2·(-1)+5=?3, -2+5=?3, 3=?3 yes!
• Steps to solve an equation:
  • Expand brackets
  • Find the common denominator of all the terms and take it out (remove it)
  • Write the unknown in one side and the numbers in the other side (inverse operations with + and -).
  • Collect like terms (simplify or reduce)
  • Use the inverse operations with · and :
  • Check the solution

Solving Problems Using Equations

• Equations are a useful tool for solving problems by constructing and solving an equation.
• To construct an equation, call x to the number we do not know and write the problem in terms of x.
  • For example: double of Mary's age is 36, how old is she?
  • Let's call x to her age, so 2x = 36 → x = 36/2 = 18, so Mary is 18 years old.

Vocabulary

• to double: Doblar (multiplicar por 2)
• to treble: Triplicar (multiplicar por 3)
• to halve: Dividir entre 2
• to collect: Agrupar
• to simplify: Simplificar
• to expand brackets: Desarrollar los paréntesis (quitar los p.)
• to factorize: Factorizar (sacar factor común)
• to solve: Resolver
• to cross-multiply: Multiplicar en cruz
• to construct: Plantear
• to substitute: Sustituir
• to evaluate: Evaluar
• algebraic expression: Expresión algebraica
• term: Término
• variable: Variable
• literal factor: Parte literal
• coefficient: Coeficiente
• monomial: Monomio
• polynomial: Polinomio
• like term: Término semejante
• formula: Fórmula
• equation: Ecuación
• identity: Identidad
• solution: Solución
• inverse: Inversa
• reverse: Opuesta
• balance: Balanza, equilibrio
• times: Veces (multiplicado por)
• less than: Menos que (->restar)
• more than: Más que (-> sumar)
• increased by: Aumentado
• decreased by: Disminuido
• plus: Más
• to add: Añadir, sumar
• minus: Menos
• to subtract (to take away): Restar
• to subtract (sth) from (sth): Restar (algo) de (algo)
• product: Producto
• quotient: Cociente