Archive for January, 2011

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Lesson 7: Linear and Quadratic Equations (I)

January 27, 2011

Vocabulary on Linear and Quadratic Equations 

 Lesson 7: Linear and Quadratic Equations (Notes)

Equation Worksheets

Linear Word Problems Worksheet

I Have to Know by the End of this Lesson  

 

1. Identity, formula and equation 

An equality is formed by two expressions separated by equal sign =.

 A numeric equality is formed only for numbers.

 An algebraic equality is formed by two algebraic expressions separated by equal sign.

 An algebraic equality can be:

  • an identity when it is true for any value of the variables (letters).
  • an equation when it is true for some values of the variables (letters).
  • a formula when one variable  is equal to an expression in a different variable(s).

 Examples

5x-2=7x , algebraic equality, equation

x+x=2x , algebraic equality, identity

P= 4l , algebraic equality, it is the perimeter of a square of side l

 2. Elements of an equation

 Remember that an equation is made up of two expressions separated by an equal sign. You have to learn the following terminology:

 In any equation there are two expressions separated by an equal sign, the expression on The Left Hand Side (LHS), in Spanish “primer miembro” and the expression on The Right Hand Side (RHS), in Spanish “segundo miembro”.

 Term of an equation is any addend of its algebraic expressions (LHS) or (RHS).

 The degree of any equation is the highest exponent that appears on the terms with unknown number(s) after collecting like terms.

 Unknowns are the variables that appear in the equation and that are unknown. They are typically called .

 In what is called a linear equation, the variable or variables appear only to the first power. A linear equation is also called an equation of the first degree or simple equation.

 Solutions are the values of the unknown(s) that make true the equation. We will say they satisfy the equation.

 Equivalent equations are equations that have the same solutions. 

 Solving an equation is finding its solution(s).

Learn to check if your solution is valid  on this link

Does x satisfy the equation?

 3. Transposing terms

 An equation is like a weighing scale – both sides should always be perfectly balanced. To solve the equation you need to find the value of missing numbers and perform the same operation to each side.

 We know that:

  • If we add or subtract the same number or algebraic expression in both sides of the equation we obtain an equivalent equation. We get the same answer in an easier way if we “change the sign (+ to – or – to +) when we take terms over to the other side of the equation”.
  • If we multiply or divide by the same number or algebraic expression in both sides of the equation we obtain an equivalent equation. We get the same answer an easier way if we “change the sign (· to: or: to ·) when we take terms or coefficients over to the other side of the equation”.

 This technique is named transposing terms. We will use it to solve equations.

 4. Solving linear equations

After transposing and collecting like terms, every linear equation is equivalent to an equation of the form

 ax+b=0

with a, b, known numbers.

This equation is called standard form of the linear equation.

 4.1. Solving easy equations

In general, to solve an equation for a given variable, you need to “undo” whatever has been done to the variable. You do this in order to get the variable by itself; in technical terms, you are “isolating” the variable. This results in “(variable) equals (some number)”, where (some number) is the answer we are looking for.

The steps are:

1º.  Group terms with  in the left-hand side and numbers in the right-hand side by transposing terms.

2º.  Collect like terms

3º.  Isolate  in the left-hand side, that results in  equals one number. This number is the solution. Recall  it must be positive.

.  Check the solution.

Example

3x-3 = x+9

We want to isolate the variable in the LHS so we first transpose 3 that goes to the RHS adding

 3x = x+9+3

We collect like terms

 3x = x+12

We transpose  x , that goes to the RHS subtracting

 3x-x = 12

We collect like terms

 2x = 12

Finally, we transpose 2, that goes to the RHS dividing:

 x = 12/2

x = 6

Remember you can check your solution

 3· 6 -3 = 6+9

Then the solution is correct.

Here you are several links to practise solving easy equations.

Solve one-step linear equations

Solve two-step linear equations

Solve multi-step equations

Solve equations involving like terms

Finally,  here you are a couple of videos on solving linear equations. Improve your listening an pronountiation

 

 

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Lesson 6: Algebraic Expressions. Polynomials (II)

January 17, 2011

5. Polynomials

Polynomial is an algebraic sum (this is, addition or subtraction) of monomials.

The terms of polynomial are the monomials that form it. The monomial that has no literal part is called independent term.

Degree of polynomial is the most of degrees of monomials, forming this polynomial.

We obtain the opposite polynomial of P(x), and we write it – P(x), by changing the sign of all the coefficients of the polynomial P(x).

 We use to write polynomials arranging terms so that powers of x are in descending order. Practise this as well as the concept of degree on

Evaluating a polynomial

The arithmetic value of a polynomial P(x for a value x=a, written P(a), is the number obtained when we substitute the variable x by the value a in the polynomial and we simplify the resulting numeric expression by using the order of operations.

6. Operations with polynomials

We are going to learn how to operate with polynomials: addition, subtraction, multiplication and dividing a polynomial by a monomials. Next year we will learn to divide a polynomial by other polynomial, which is more complicated.

6.1. Adding and subtracting polynomials

To add up two polynomials we add up like terms, and write the unlike terms.

To subtract two polynomials we add up the first polynomial the opposite of the second polynomial.

It will be useful to place the like terms in columns.

You can practise these operations on

This other exercises are more complicate.

6.2. Multiplying a polynomial by a monomial

The product of a monomial by a polynomial, is equal to the sum of the products of the monomial by all the terms of the polynomial.

This step is essential to  advance. Practise this skill on this web. http://www.glencoe.com/sec/math/studytools/cgi-bin/msgQuiz.php4?isbn=0-07-825083-8&chapter=8&lesson=6&headerFile=4&state=fl

6.3. Multiplying two polynomials

The product of two polynomials is equal to the sum of the products of each term of the second polynomial by all the terms of the first polynomial.

Notice we are applying the distributive property as many times as we need. So a product of sums is equal to the sum of all possible products of each addend of one sum to each addend of the other sum.

We will apply the distributive property: a·(b + c) = a·b + a·c.

You can practise this operation on

Here you are more exercises.

6.4. Dividing a polynomial by a monomial

To divide a polynomial by a monomial, we divide each term of the polynomial by the monomial.

This is easy. So if you have any doubt go to point 4.2 and repeat the practice.

7. Factoring out common factors

Distributive property allows transform additions or subtractions into products and vice versa.

a·(b + c) = a·b + a·c.

When we move from the left hand side to the right hand side we apply the distributive property and when we move from the right hand side to the left hand side is called factoring out common factor.

Factoring out common factor consists on transforming an expression that contains additions or subtractions into a product.

Read carefully the examples on this page and pracctise with the examples at the end.

8. Special products

 These products are very useful in calculations with algebraic expressions. We will work with binomials; this is the algebraic sum of two monomials.

 8.1. Square of the sum of two algebraic expressions

The square of a sum equals to the square of the first term plus double product of the first term by the second plus the square of the second term.

(a + b)2=a2 + 2ab + b2 

 8.2. Square of the difference of two algebraic expressions

The square of a difference equals to the square of the first term minus double product of the first term by the second plus the square of the second term.

(a – b)2=a2 – 2ab + b2

 8.3. Product of the sum and difference of two algebraic expressions

The product of the sum and difference of two algebraic expressions is equal to the square of the first expression minus the square of the second expression.

(a + b)· (a – b)=a2 – b2

This point is new for you, and it isn’t so complicated. The special products are extremelly useful in algebra. You must domain them. Practise on

8.4. Applications

These equalities, factoring out common factor and other techniques, are useful to transform algebraic expressions consisting on additions and subtractions into products. This process is called factoring or factorizing.

Finally you can watch these videos on operations with polynomials

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Lesson 6: Algebraic Expressions (I)

January 3, 2011

Vocabulary on Algebraic Expressions

Lesson 6: Algebraic Expressions (Notes) 

Monomial WorksheetExercises

   Translate Word Sentences into Alagebraic Epressions

I Have to Know by the End of this Lesson 

1. What is algebra?

Learning algebra is a little like learning another language. In fact, algebra is a simple language, used to create mathematical models of real-world situations and to handle problems that we can’t solve using just arithmetic. Rather than using words, algebra uses symbols to make statements about things. In algebra, we often use letters to represent numbers.

Algebra is a formal symbolic language, composed of strings of symbols. The symbol set of algebra includes numbers, variables, operators, and delimiters. In combination they define all possible sentences which may be created in the language.

Because of this it is important to learn to translate words into algebra. This task is essential to solve word problems through equations.

 Practise this skill on:

 It is also useful to practice writing variable expressions to represent diagrams

 2. Evaluating expressions

 Evaluating an algebraic expression consists on substituting numbers we are given for variables in expressions and working out the outcome.

 One advise: recall order of operations.

 Practise on  this web.

 At the bottom of this page you can also find puzzles on the main vocabulary of this lesson.

 3.Monomials

 A monomial is an algebraic expression formed by the product of a number and one or more letters.

 The factor expressed in Arabic numerals is sometimes called numerical coefficient or simply coefficient. The numerical coefficient is customarily written as the first factor of the term.

 The letters are called literal numbers or literal part.

 Degree of a monomial is the sum of all the exponents of the letters in the literal part.

 Practise identifying terms and coefficients

 Like monomials. Monomials are called similar or like ones, if they are identical or differed only by coefficients. Therefore, if two  or some monomials have identical letters, they are also similar (like) ones. In other case they are called unlike monomials.

 4. Operations with monomials

 Operations with monomials follow the same rules as operations with numbers. You must respect the order of operations or hierarchy.

 4.1 Addition and subtraction of monomials

 Like monomials are added or subtracted by adding or subtracting the numerical coefficients and placing the result in front of the literal factor,

Dissimilar or unlike monomials cannot be added or subtracted when numerical values have not been assigned to the literal factors.

Practise this skill on

 

4.2. Multiplying and dividing monomials

 When multiplying monomials, we multiply their numeric coefficients and multiply their literal numbers separately.

 When dividing monomials, we multiply their numeric coefficients and multiply their literal numbers separately if we can.

Practise operations with monomials on this web. Every time you click on the different operation you want to practice you will get a different example.

Practise collecting like terms and  eliminating brackets on this web. These exercises are more complicated

Finally, try to understand this videos on operations with monomials