unit 4_algebra and equations

IES F. Javier de Burgos Dep. Matemáticas

Motril – Granada 2º ESO

1

UNIT 4: ALGEBRA AND EQUATIONS

KEY WORDS

1. ALGEBRAIC EXPRESSIONS

When we want to do calculations involving undetermined or unknown numbers, we use letters.

For example:

Francisco is twice as old as his daughter Luisa.

Luisa’s mother is five years younger than her husband.

Expressions containing letters and numbers are called ___________________.

An algebraic expression has _________ and _________ linked by __________.

The letters are called ___________.

Every addend is called ________: The expression 5n + 3s has ____ terms: _________.

You can ___________ an algebraic expression by __________ like terms.

____________ have exactly the same _________________:

3x2 and −5x2 ______like terms 4xy2 and 2xy ______ like terms

Examples: Simplify these expressions:

a) 4x + 2y − 2x + 3y = b) 7p − 3q + 5q – p = c) 5c − 2b + 2c − 3b =

1. Algebraic expression

2. Simplify

3. Monomial

4. Expand

5. Coefficient

6. Literal part

7. Degree

8. Variable

9. Polynomial

10. Substitute

11. Like terms

12. Evaluate

13. Unknown

14. Solution

15. Linear equations

16. Quadratic equations

a. Monomio

b. Expandir/desarrollar

c. Polinomio

d. Parte literal

e. Grado

f. Ecuaciones de 2º grado (cuadráticas)

g. Variable

h. Ecuaciones de 1er grado (lineales)

i. Sustituir

j. Términos semejantes

k. Calcular el valor numérico

l. Expresión algebraica

m. Coeficiente

n. Incógnita

ñ. Solución

o. Simplificar

Luisa’s age _____

Father’s age _____

Mother’s age _____

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- TRANSLATING WORDS INTO ALGEBRAIC LANGUAGE

Here are some statements in English. Try to translate them into algebraic language:

The sum of three times a number and eight: ___________.

The product of a number and the same number less 3: ____________.

12 times a number less 30: ____________.

Another example: If we have a rectangle but we don’t know its dimensions, how can we express

its area and its perimeter?

Area A =

Perimeter P =

- Exercises

1. Write the algebraic expression for these:

a) Seven less than y: b) Four multiplied by x: c) y multiplied by y:

d) Ten divided by b: e) A number add five: f) Half a number:

g) Paul has d DVDs. He buys 3 more. How many DVDs has Paul got now?

h) Rob has a apples. He eats 2 apples. How many apples has Rob got now?

i) The perimeter of a rectangle which length is 2 cm longer than its width:

2. Steven is 16 years old. How old will he be in: a) 5 years? b) 10 years? c) x years?

3. Use algebraic expressions to complete the following:

Roberto weighs x kilos

Ana weighs four kilos more than Roberto

Jacinto weighs twice as much as Roberto

Rosa weighs four kilos less than Jacinto

2. MONOMIALS.OPERATIONS

A monomial is an algebraic expression containing one term which can be a number, a variable or

a product of numbers and variables.

For example: 2 2 2 42, , 2 , 13, 520

3a b x xy x y ______________________________

1

, 3 4 , 2 3x y xx

_______________________________

l = Lenght

w = Width



3

The number is called ____________ and the variables are called ____________.

22

3a b

The degree is the _______________________ of every variable.

For example: the degree of 22

3a b is __________

Example: For every monomial write the coefficient, the literal part and the degree

2.1. ADDING MONOMIALS

RULE: When you add monomials,

You must have ____________.

You ______the coefficients and ________the literal part.

Examples: 1) 3x2 + 4 x2 + 5x =

2) 4xy3 + 5x2y3 – 7x2y3 =

3) x2 + 3x – 5 + x2 – 7x – 2 =

2.2. MULTIPLYING MONOMIALS

RULE: When you multiply monomials,

you __________the coefficients and

________the exponents.

Examples: 1) (3x2)·(5x) =

2) (4xy3)·(–7y2) =

3)(-2a4b3)·(-5a2b) =

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3. POLYNOMIALS. OPERATIONS

A polynomial is an algebraic sum of ____________, for example:

If there are _____________, it is called a ___________, for example:

The _________ of the polynomial is the degree of the ____________________ that it

contains.

Examples: x4 − 7x3 is a ________________ binomial

4x2 + 2x − 7 is a ________________ polynomial

Polynomials are usually written this way, with the terms written in __________________

_________ a polynomial is the same as ________ its number value at a _______ value of the

variable. For instance: Evaluate 2x3 − x2 − 4x + 2 at x = −3:

Exercise: Evaluate the polynomial x3 − 2x2 + 3x − 4 at the given values of x:

3.1. ADDING AND SUBTRACTING POLYNOMIALS

There are two ways of adding or subtracting polynomials:

A(x) = 3x3 + 3x2 – 4x + 5 B(x) = x3 – 2x2 - 4

A(x) + B(x):

Horizontally:

Vertically:



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A(x) – B(x) : notice that ________________

Horizontally:

Vertically:

3.2. MULTIPLYING POLYNOMIALS

- Monomial x polynomial:

____________ the monomial through the brackets (____________ rule)

Examples: a) -3x · (4x2 – x + 10) =

b) 2x2 · (3x3 – 5x2 + 4) =

- Polynomial x polynomial: Look at these examples:

1) (x + 2)(x – 3) =

2) (x2 – 3x)(x + 1) =

3) (x + 3)(4x2 – 4x – 7) =

You can also do it in the “vertical way”: 4x2 – 4x – 7

x + 3

4. EQUATIONS

4.1. BASIC CONCEPTS

An _______ is an algebraic expression which contains an _______________and

__________. Examples:

A _________ of an equation is a ________ we can put in place of an ________ that

makes the equation ________.

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_______ the equation means _______ the ________ that makes it true, that is the

________.

Example: There are two values that make true the following equation:

Exercises:

1) Is the value x = 8 solution of any of the following equations?

x - 4 = 10 5 - x = -3 3x = 12 2x + 4 = 20

- 4 = 10 5 - = -3

2) Figure out the solution of the following equations:

2x = 14 x – 4 = - 9 x2 – 36 = 0 3x + 7 = 19 4x + 5 = 15 – x

4.2. SOLVING EQUATIONS

Try to discover the size of the unknown weight, and remember:

=

54

x 4 0

2

x

12 22

3

x

? ?

=



7

Exercices: Solve the following equations step by step:

Remember:

1) x + 7 = 23 2) 12 = 2x – 5 3) 6 – x = - 11 4) 143

x

=

=

=

=

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- EQUATIONS WITH BRACKETS

If an equation has brackets in it, you have to “remove” them by applying the distributive rule,

or in other words, by ____________________, for example:

1) 5x – 2(2x – 2) = 8 – (3 + 2x) 2) 4 – 7(2x – 3) = 3x – 4(3x – 5)

- EQUATIONS WITH DENOMINATORS

When there are denominators in several terms of the equation, you have to “remove” them in

the following way:

1) Reduce every fraction to the least common denominator (the new denominator will be the

________________________ of the denominators).

2) Multiply both sides of the equation by the LCM.

Examples:

5. SOLVING PROBLEMS WITH EQUATIONS

The typical steps used to solve real-words problems with equations are:

1) Read the problem.

2) Identify the important information and the question.

3) Assign a ___________ to the quantity you want to ______.

4) Write other ___________ in terms of that variable.

5) Create an _________ which relates the quantities.

6) ________ the equation.

7) Check your ________ by recreating the equation (make sure that everything seems

___________)

2 5

13 2

xx

2 5 1 1 1

4 9 3 2

x x x

unit 4_algebra and equations

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