Mathematics

Algebra - Explanation and Worked Examples on Algebra

len Alfred Ajibola - Thu, 24th January, 2019 @ 7:11 PM

Topics in Mathematics

Angles: Type of angles based on degree measurement Angle Pairs: Types of angle pairs Conversion of Number Bases explained with examples Angles: What are Angles? Categories of Angles Lines and Types in Mathematics Number Base System - Table showing the Number Bases with explanations Mathematics Scheme of Work, SS1, 3rd Term Mathematics Scheme of Work, SS1, 2nd Term Mathematics Scheme of Work, SS1, 1st Term Trigonometry - Sine, Cosine and Tangent explained with worked examples Quadratic Equation - Factorization of Quadratic Equation Algebra - Explanation and Worked Examples on Algebra


Academic Questions in Mathematics

Please click here to see all Questions and Answers

Decagon - Len Academy

What type of polygon is shown in the above image?

  • A. Nonagon

  • B. Decagon

  • C. Octagon

  • D. Hexagon

  • E. Hepagon

  • F. Pentagon

Which of the following is not part of a circle?

  • A. Section

  • B. Sector

  • C. Segment

  • D. Semicircle

  • E. Secant

  • F. Center

Acute Angle - Len Academy

What angle is shown in the above image?

  • A. Complementary

  • B. Supplementary

  • C. Acute

  • D. Obtuse

  • E. Reflex

  • F. Straight

An angle greater than 90o but less than 180o is termed _____.

  • A. Complementary

  • B. Supplementary

  • C. Acute

  • D. Obtuse

  • E. Reflex

  • F. Straight

Factorize 12a2 - 20a + 3

  • A. (6a - 1)(2a - 3)

  • B. (6a - 1)(2a + 3)

  • C. (6a + 1)(2a - 3)

  • D. (6a + 1)(2a + 3)

  • E. (-6a + 1)(2a - 3)

  • F. (-6a + 1)(2a + 3)

Simplify 10ab + 7ab

  • A. 17ab2

  • B. 17a2b2

  • C. 3ab2

  • D. 3a2b2

  • E. 17ab

  • F 3ab

Which of the following is not a type of angle?

  • A. Right angle

  • B. Left Angle

  • C. Acute Angle

  • D. Complete Angle

  • E. Straight Angle

  • F. Reflex angle

Two lines that can never meet each other are best considered as _____ lines.

  • A. straight
  • B. vertical
  • C. horizontal
  • D. parallel
  • E. oblique
  • F. perpendicular



Concept of Algebra:

Consider this simple statement:

I am a number. When you subtract 3 from me, the result is 9. what number am I?

Of course we know the answer is 12. You probably guessed it from your mind because it was that simple. The truth is, We could have a much more complicated statement of such; and that is where algebra comes into play.

Now, to apply algebra in the above statement, we use letters or alphabets to represent the unknown number(s). So using algebra, we can write:

  • Let the number be y

  • y - 3 = 9

  • y = 9 + 3

  • y = 12

You can read on number base system here.


So why is it easier using letters?

if we had a question of several unknowns, the question can become complicated, hence using letters can actually make things easier for us.

To simplify a mathematical expressions, we need to represent them in the least complicated form as much as we can, making them more easier to work with.

One skill you need to know is the combination of 'like terms'.

Like terms refers to those variables that are exactly the same, although they may still have different coefficients. Understand that coefficients always precedes a variable. This is explained in the instance below.

  • 5a2 and 12a2 are considered to be like terms because the have the same variable a2. The coefficients in this instance are 5 and 12 respectively.
  • 3ab and 3bc are not like terms because the have different variables (ab and bc) with coefficients (3 and 3) respectively.

In order to combine the like terms, we will have to add or subtract them (depending on the given sign) so that we can have a single term.

Please read on factorization of quadratic equation here.

 

Worked Examples

Example 1:

  • Simplify 10ab + 7ab
  • Since '10ab' and '7ab' are like terms, that is, they both have the same variable (ab), we will add their coefficients together in order to get our answer
  • (10 + 7)ab = 17ab
  • Answer is 17ab

 

Example 2:

  • Solve y2 + 5 + 7y2 - 5y + 10
  • First, let's combine the like terms. Understand that a variable without a coefficient in front of it is always assigned the value of 1. Also, when moving coefficients, we should take into account the '+' and '-' values in front of them.
  • (1 + 7)y2 - 5y + (5 + 10)
  • Answer is 8y2 - 5y + 15

 

Example 3:

  • Solve 4b + 4 = 20
  • Step 1: Let's move over like terms. When a like term is moved over the '=', the '+' value before it changes to '-'. If the like term has '-' value in front of it, then it turns to a '+'.

  • 4b = 20 - 4
  • 4b = 16
  • Step 2: To get b, we will need to divide both side by b's coefficient which is 4.

  • 4b/4 = 16/4
  • Answer is -> b = 4

 

Example 4:

  • Solve 4a + 5 = 2a - 1
  • Step 1: Let us move the like terms

  • 4a - 2a = -1 - 5
  • 2a = -6
  • Step 2: Let's divide both sides by a's coefficient which is 2).

  • 2a/2 = -6/2
  • Answer is -> a = -3

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Amazing facts in Mathematics

The opposite sides of a dice will always give a sum of seven

31,556,926 seconds makes one year

From 0 through 1000, the only number that has the letter 'a' in it is 'one thousand'

÷

The division symbol '÷' is actually called an obelus

Four is the only number in the English language that is spelt with the same number of letters as the number (4) itself

40 is spelt as forty and not 'fourty'. It is the only number that is spelt with letters arranged in alphabetical order

An odd number is a number that cannot be divisible by 2.

Every odd number has an 'e' in its spelling. They include:

  • One
  • Three
  • Five
  • Seven
  • Nine

555 in Thailand sounds like 'hahaha'. This is because the number '5' means 'ha' in Thailand

In mathematics, 10! means 10 factorial. Below is its calculation:

10! = 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 3628800

Interestingly, 3628800 seconds is the same as 42 days

42 days is the same as 6 weeks

We can therefore say that 10! equals 6 weeks or 3628800 seconds in mathematics

When you multiply 111,111,111 by 111,111,111; the result equals 12,345,678,987,654,321


Notable points in Mathematics

Let's use algebra to explain the simple statement below:

I am a number. When you subtract 3 from me, the result is 9. what number am I ?

Of course we know the answer is 12. You probably guessed it from your mind because it was that simple. The truth is, We could have a much more complicated statement of such; and that is where algebra comes into play.

Now, to apply algebra in the above statement, we use letters or alphabets to represent the unknown number(s). So using algebra, we can write:

  • Let the number be y

  • y - 3 = 9

  • y = 9 + 3

  • y = 12

Please read more on algebra here

Sides of a right angle triangle - Len Academy

 

The image above shows a right angled triangle. The angle of the above right triangle is denoted as θ. Below are some points to note:

Hypotenuse is always the longest side of the triangle.

Opposite is the side opposite the angle (θ)

Adjacent is the shorter side next to the angle (θ)

Each of Sin θ, cos θ and tan θ are the ratio of sides of the right triangle. They are calculated by using the formula (margic word) sohcahtoa:

Sohcahtoa - Len Academy

soh

Sin θ = Opposite/Hypotenuse

cah

Cos θ = Adjacent/Hypotenuse

toa

Tan θ = Opposite/Adjacent

Please read more on sine, cosine and tangent here

Circumference of a Circle - Len Academy

Circumference in mathematics means the distance around the edge of a circle (or a another curved surface).

Circumference is to circle or any other curved surface

Perimeter of a Rectangle - Len Academy

Perimeter means the distance around a square or rectangle or polygon.

Perimeter is to square or rectangle or polygon