Algebra Foil Method

Introduction 

The algebra is a basic topic in mathematics and it is related with binomial expansion. The binomial expansion is done by foil method. In algebra, the foil method is multiplying the binomials. The foil method is considered as rule and the expansion of foil is first – outer – inner – last. Now we are going to see about algebra foil method.

Explanation for Algebra Foil Method
Algebra in math:

The algebra is a simple topic in math and it is defining the relations, rules and so on. The algebra problems are based on variables. In foil method also the variables are used.

Algebra foil method steps:

Multiplication of binomials process is called as foil method and it is done with variables. Steps for foil method:

Binomial’s first terms are multiplied.
Binomial’s outer terms are multiplied.
Binomial’s inner terms are multiplied.
Binomial’s last terms are multiplied.
The terms are combined with each other.




More about Algebra Foil Method

Example problems for algebra foil method:

Problem 1: Use the foil method and determine the binomial expansion of (x + 11) and (x + 5).

Solution:

The binomials are given as (x + 11) and (x + 5).

We can expand the binomials as.

Binomial’s first terms are multiplied as x^2.
Binomial’s outer terms are multiplied as 5x.
Binomial’s inner terms are multiplied as 11x.
Binomial’s last terms are multiplied as 55.
Combine the terms as x^2 + 5x + 11x + 55.
The result is x^2 + 16x + 55.
Problem 2: Use the foil method and determine the binomial expansion of (x + 8) and (x + 1).

Solution:

The binomials are given as (x + 8) and (x + 1).

We can expand the binomials as.

Binomial’s first terms are multiplied as x^2.
Binomial’s outer terms are multiplied as x.
Binomial’s inner terms are multiplied as 8x.
Binomial’s last terms are multiplied as 8.
Combine the terms as x^2 + x + 8x + 8.
The result is x^2 + 9x + 8.
Exercise problems for algebra foil method:

1. Use the foil method for binomial expansion.

(x – 2) (2x + 3)

Solution: The binomial expansion is 2x^2 – x – 6.

2. Use the foil method for binomial expansion.

(3x + 1) (x + 1)

Solution: The binomial expansion is 3x^2 + 4x + 1.

Opposite Rays Geometry

Introduction to opposite rays:

In geometry ray is a line but it has starting point with no ending point. Two opposite rays join at a single vertex and forms straight angles.  Straight angles are one of the angles in mathematical geometry.Straight angle is a straight line angle it measures 180 degree angle. Therefore we can say that an angle formed by opposite rays is the straight angle.  In this article we are going to learn about how two opposite rays make a straight angle in geometry.

Explanation about Opposite Rays in Geometry

Ray:                      

Ray is the important concept in mathematical geometry. A ray is the one of the part of the line. All angles are lies between two rays. Generally a ray as a starting point but is has no ending point it ends up to infinity level.  Starting point of the ray is mentioned by the dot and then ending point of the ray is mentioned by the arrow.

Opposite rays:
Two rays that share the starting point and it goes up to infinity level on both directions is known as opposite rays.  It looks like a straight line so it is forms an straight angle.

The above diagram AB and BC are two opposite rays they share a common point B and it forms 180 degree straight angle.  Here B is known as the vertex of the straight angle 180 degree.



Description about Pictures and Points of the Opposite Rays

Different pictures of the opposite rays:

Collinear points:

If the point are in the straight line that has been similar to two opposite then that points are called as the collinear points. Simply known as a set of points in the straight line then those points is called as the collinear points.  In the above diagram 1 A, B and then C are the collinear points because they lie in the same straight line and formed by two opposite rays.

Complex matrix inverse

Complex matrix inverse - Introduction

          In algebra, the determinant is a particular number related with some square matrix. The essential geometric significance of a determinant is a scale factor for compute when the matrix is considered as a linear transformation. Thus a 3× 3 matrix with determinant 3 when useful to a place of points with fixed area will convert those points into a place with double the area. Determinants are significant together in calculus, where they go into the replacement rule for various variables, and in multi linear algebra.

          `A = [[A,B],[C,D]] If AD - BD != 0` , then A has an inverse, Denoted` A^-1`

          `A^-1 = 1/(AD-BC) [[D,-B],[-C,A]]`

Complex Matrix Inverse - Examples:

Complex matrix inverse - Example 1:

Find the Complex inverse matrix of the matrix below?

`A = [[4,2],[3, 8]]`

Solution:

Inverse A =` 1/((8*4)-(3*2)) [[8,-3],[-2,4]]`

           = `1/(32-6) [[8,-3],[-2,4]]`

           =` 1/26[[8,-3],[-2,4]]`

          = `1/26 * [[8,-3],[-2,4]] = [[8/26, -3/26],[-2/26,4/26]]`

          = `[[4/13, -3/26], [-1/13, 2/13]]`

Complex matrix inverse - Example 2:

What is the inverse of the matrix below?

          `[[9, 1], [6, 3]]`

Solution:

Inverse of the matrix is the

Inverse A =` 1/((9*3)-(6*1)) [[3,-1],[-6,9]]`

           = `1/(27-6) [[3,-1],[-6,9]]`

           = `1/21[[3,-1],[-6,9]]`

          = `1/21 * [[3,-1],[-6,9]] = [[3/21, -1/21],[-6/21,9/21]]`

         ` [[3/21, -1/21],[-6/21,9/21]] = [[1/7, -1/21],[-2/7,3/7]]`

Complex matrix inverse - Example 3:

What is the inverse of the matrix below?

  A =  ` [[7, 1], [5, 3]]`

Solution:

Inverse of the matrix is the

Inverse A =` 1/((7*3)-(5*1)) [[3,-1],[-5,7]]`

           = `1/(21-5) [[3,-1],[-5,7]]`

           = `1/16[[3,-1],[-5,7]]`

          = `1/16 * [[3,-1],[-5,7]]`

          = `[[3/16, -1/16],[-6/16,9/16]]`

`[[3/16, -1/16],[-6/16,9/16]] = [[3/16, -1/16],[-3/8,9/16]]`




Complex Matrix Inverse - more Examples:

Complex matrix inverse - Example 1:

What is the inverse of the matrix below?

         ` [[8, 2], [6, 4]]`

Solution:

Inverse of the matrix is the

Inverse A = `1/((8*4)-(6*2)) [[4,-2],[-6,8]]`

           `= 1/(32-12) [[4,-2],[-6,8]]`

           `= 1/20[[4,-2],[-6,8]]`

         ` = 1/20* [[4,-2],[-6,8]]`

          `= [[4/20, -2/20],[-6/20,8/20]]`

`[[4/20, -2/20],[-6/20,8/20]]= [[1/5, -1/10],[-3/10,2/5]]`

Complex matrix inverse - Example 2:

Find the inverse Complex matrix of the matrix below?

          `[[9, 3], [7, 5]]`

Solution:

Inverse of the matrix is the

Inverse `A = 1/(9*5)-(7*3) [[5,-3],[-7,9]]`

          `= 1/(45-21) [[5,-3],[-7,9]]`

          `= 1/24[[5,-3],[-7,9]]`

          `= 1/24* [[5,-3],[-7,9]]`

          `= [[5/24, -3/24],[-7/24,9/24]]`

`[[5/24, -3/24],[-7/24,9/24]] = [[5/24, -1/8],[-7/24,9/24]]`

Complex matrix inverse - Example 3:

What is the inverse of the matrix below?

          `[[10, 4], [8, 6]]`

Solution:

Inverse of the matrix is the

Inverse `A = 1/(10*6)-(8*4) [[6,-4],[-8,10]]`

          `= 1/(60-32) [[6,-4],[-8,10]]`

           `= 1/28 [[6,-4],[-8,10]]`

          `= 1/28* [[6,-4],[-8,10]]`

          `= [[6/28, -4/28],[-8/28,10/28]]`

`[[6/28, -4/28],[-8/28,10/28]] = [[3/14, -1/7],[-2/7,5/14]]`

Nonnegative Integers

Introduction for non negative integers:

                      Non negative integers are defined as a numbers greater than zero, numbers at the right side of the zero. Thus the non negative integers are all the integers from zero on upwards, and the non negative real are all the real numbers from zero on upwards. The set of all non-negative integers forms a commutative mono id under addition. Non negative integers always proceed by a non negative sign (+). An integer with or without the non negative integers is always non negative.

Operations and Examples of Non negative Integers:

Operation for non negative integers:

          1. Reduce anything in the parentheses for solving

          2. Reduce the exponents for solve non negative integer.

          3. Multiplication or division for solve non negative integer.

          4. Finally, Addition or Subtraction for solve non negative integer.

Examples of non negative integers:

                   Integers are similar to whole numbers, but they also consist of negative numbers but still fractions are not allowed. So integers can be negative {-1, -2,-3, … }, positive {1, 2, 3, … }, or zero {0}.

Operations on Non negative Integers:

1. Adding non negative integers:

           Non negative + Non negative = Positive: 8 + 4 = 12

           Negative + Negative = Negative: (- 6) + (- 4) = - 10

           Addition of a negative and a non negative integer: Use the sign of the larger number and subtract, examples are,

           (- 5) + 3 = - 2

           7 + (-4) = 3

2. Subtracting non negative integers:

           Negative - Non negative = Negative: (- 4) - 2 = -4 + (-2) = -6

           Non negative - Negative = Non negative + Non negative = Non negative: 6 - (-2) = 6 + 2 = 8

           Negative - Negative = Negative + Non negative = Use the sign of the larger number and subtract (Change double negatives to a non negative)

          (-8) - (-3) = (-8) + 3 = -5

          (-3) - (-8) = (-3) + 8 = 5

3. Multiplying non negative integers:

           Non negative x Non negative = Non negative: 4 x 3 = 12

           Negative x Negative = Non negative: (-3) x (-2) = 6

           Negative x Non negative = Negative: (-3) x 3 = -9

           Non negative x Negative = Negative: 2 x (-5) = -10

4. Dividing non negative integers:

           Non negative ÷ Non negative = Positive: 14 ÷ 7 = 2

           Negative ÷ Negative = Non negative: (-12) ÷ (-4) = 3

           Negative ÷ Non negative = Negative: (-18) ÷ 3 = -6

           Non negative ÷ Negative = Negative: 15 ÷ (-3) = -5

Introduction of Decimal and Fraction

Introduction of Decimal and Fraction:

            Decimal is also termed as the fraction whose denominator of the fraction is 10 to the power. Example: `(3)/(10)` = 0.3 decimal. Fraction: It is part of the entire object. We can make chart from decimal to the fraction by following certain procedure. In this article, we see how to make decimal to fraction chart.

Decimal to Fraction Chart:

Procedure - Decimal to Fraction Chart:

Step 1: check the how many digit present after the decimal point.

Step 2: Multiply the 10 to power depending up on the digit present after the decimal point.

Step 3: Now we get the fraction, and simplify as much as possible.

Now decimal get convert to the fraction.

Writing decimal numbers:

We know the place value of the decimal numbers the following diagram shows the place value of the decimal numbers.

Example Problem - Decimal to Fraction Chart:

Example 1:

Make the decimal to fraction chart of the following decimal.

a)     1.35

b)    0.78

c)     2.8

Solution:

a) Converting the decimal 1.35 to fraction

Solution:

Step 1: Check the digit after the decimal point. Here there are two digits.

Step 2: Hence make the decimal as the whole and it divided by the 10 to the power of 2.

`(135)/(100)`

Step 3: Simplify the fraction `(135)/(100)`

1.35 = `(27)/(20)`

b) Converting the decimal 0.78 to fraction.

Solution:

Step 1: Check the digit after the decimal point. Here there are two digits.

Step 2: Hence make the decimal as the whole and divided it by the 10 to the power of 2.

`(78)/(100)`

Step 3: Simplify the fraction`(78)/(100)`,  we get

0.78 = `(39)/(50)`

c) Converting the decimal 2.8 to fraction.

Solution:

Step 1: Check the digit after the decimal point. Here there is one digit.

Step 2: Hence make the decimals as the whole and divided it by the 10 to the power of 1.

`(28)/(10)`

Step 3: Simplify the fraction `(28)/(10)`,  we get

`(7)/(25)`

Make the chart:

Decimal	Fraction
1.35	`(27)/(20)`
0.78	`(39)/(50)`
2.8	`(7)/(25)`

Example 2:

Make the chart of the following decimal to fraction.

0.45, 0.6, 0.75, 0.125

Solution:

Decimal	Fraction
0.45	`(45)/(100)`
0.6	`(6)/(10)`
0.75	`(75)/(100)`
0.125	`(125)/(1000)`

Introduction to ratio and proportion summary

Introduction to ratio and proportion summary:

              The ratio a: b is equivalent to the quotient `a/b` . A is relation of give the equality of two ratios, in the form `a/b` = `c/d` . This relation has called as proportion. Proportion is a states in which numbers are called as proportionality relation giving of two ratios, in the form `a/b` = `c/d` . Here a & d are known as extreme and b & c are known as means.                                                    

Problems on Proportion Summary:

     The ratio a: b is equivalent to the quotient `a/b` . The ratio of two given numbers a and b is the fraction, usually expressed in reduced form.

Ratio summary - example 1:  

                  A classroom has 60 men and 20 women. What is the ratio of men and women?        

Solution:

                   Men to women = 60 to 20 or 60: 20,

                   which reduces to = 6 to 2 or 6:2

                   The ratio of men to women is 6 to 2, or `6/2` , or 6 : 2.

Ratio summary - example 2:  

              A triangle has angle measures of 30°, 60°, and 90°. In simple form, what is the ratio of the given angles to each other?

Solution:

            30: 60: 90 = 3: 6: 9 (10 is a common divisor).That is,

                     (1) Ratio of the first to the second is 3 to 6.

                     (2) Ratio of the first to the third is 3 to 9.

                     (3) Ratio of the second to the third is 6 to 9.

Some Problems for Proportion Summary:

    A is relation give the equality of two ratios, in the form `a/b ` = `c/d` . This relation has called as proportion. Proportion is a states in which numbers are called as proportionality relation giving of two ratios, in the form `a/b` = `c/d` . Here a & d are known as extreme and b & c are known as means.                                                   

Proportion summary - example 1:

           If `x/5` = `y/2` , find the ratio of  `x/y` .(proportion)

Solution:

                    `x/5` =  `y/2`                                                          

                    `x/y` =  `5/2`

       Therefore the ratio of x and y is 5:2

Proportion summary - example 2:

           If  `x/3` = `2/6` , then find x?

Solution:

           Using cross multiplication (Property 1)

                    2x = 3 `xx` 6

                    2x = 18

                     x = `18/2`

                     x = 9

            So the value of x = 9

Introduction to Solving Straight Line Equation

Introduction to Solving Straight Line Equation:

Straight line is a curve which has the same slope along its length. Basically  line is a series of points that extends in two opposite directions without end.There are many ways to solve the equations of straight lines. A straight line contains x-intercept, y-intercept and slope (m). An equation is a mathematical statement that two expressions are equal.
The general form of an equation of a straight line is
                                                     y = mx+b.
 Where m is the slope of the straight line and b is the intercept made by the straight line on y-axis.
If a Straight line passing trough two points (x₁,y₁) and (x₂,y₂) then slope of that line is `(y2-y1)/(x2-x1)`

• If two lines are parallel then their slopes are equal.
• If two lines are perpendicular to each other then product of their slopes is equal to -1.

There are different forms of equation of a straight line which are given below :

1. Slope-intercept form 
2. Intercept form 
3. Point-slope form 
4. Two-point form

The descriptions of each method will be explained below with an example. 

Different Forms of Equation of Straight Line:

1. Slope - Intercept form of a Straight Line:

Statement: The equation of the straight line with slope 'm' and Y-intercept 'c' is y = mx+c

Proof: Let L be the straight line whose slope is m and which cuts off an intercept 'c' on the Y-axis 

                   If P(x,y) is a point on the xy-plane, then

               P lies on L   => m = slope of L = (y-c)/(x-0)

                                      => y = mx+c

               Conversely, if P(x,y) satisfies the equation y = mx+c, then x = 0 => y = c.

2. Intercept form of a Straight Line:

Statement: The equation of the straight line which cuts off intercepts a and b on the X-axis and Y-axis respectively is`x/a+y/b=1`

Proof: The straight line L which cuts off intercepts a and b on the X-axis and Y -axis respectively meets these axis at A(a,0) and B(0,b) and therefore slope of the line = -b/a.

Hence, the equation of L , by the slope-intercept form is  `y= (-b/a)x +b` or `x/a+y/b=1` .

Example: Solve the Equation of straight line which makes intercepts whose sum is 5 and product is 6 

Solution : Let a and b are the intercepts made by the line , then 

given in the problem that a + b = 5 and ab = 6 

solving these equations we obtain a = 2 and b = 3 

Then, the required equation of the straight line is `x/2+ y/3=1` i.e 3x+2y-6 = 0

                                                                   or `x/3+ y/2=1` i.e 2x+3y-6 = 0

3. Point - Slope form of a Straight Line:

Statement: The equation of the straight line with slope m and passing through the point (x1,y1) is y - y1 = m(x-x1).

Proof: Equation of any straight line with slope m is of the form y = mx+c.

This line passes through the point (x1,y1) is y1 - mx1 = c

Therefore, the equation of the line with slope m containing the point (x1,y1) is 

         y - mx=c = y1 - m1x1

    i.e y-y1 = m(x-x1)

Example: Solve the equation of straight line which has the slope -1 and passes through the point (-2,3) 

Solution: The slope of the given straight line m = -1 

and the point on the line is ( -2,3) 

hence the equation of the line is 

                           y-3= -1(x+2) 

                           x+y-1=0.

4. Two-Point form of a Straight Line:

Statement: The equation of the straight line passing through the point A(x1,y1) and B(x2,y2) is

(x-x1)(y1-y2)=(y-y1(x1-x2).

Proof: Let L be the Straight line containing the points A(x1,y1) and B(x2,y2) 

Case 1: suppose L is  non-vertical. Then x1 ≠x2 and slope of L = `(y1-y2)/(x2-x1)`

Therefore, the equation of L is `y-y1 = (y1-y2)(x-x1)/ (x1-x2)`

 i.e  (x-x1)(y1-y2) = (y-y1)(x1-x2)     ……(1)

Case 2: If L is vertical , then x1=x2 and y1≠y1. Equation of L, in this case, is x = x1 [form (1)]

Example: Solve the equation of straight line which is passing through the point (1,-2) and (-2,3)

Solution: The slope of the line containing (1,-2) and (-2,3) is

               m= (3+2)/(-2-1) = -5/3

and hence equation is y+2=-5/3(x-1)

 i.e      5x+3y+1=0

Exercise:

Problem 1: The following equation are in slope-intercept form. In each case, specify the slope and  y-intercept.

i) y=2x+7   (Ans: slope = 2, y-intercept = 7)

ii)  y=−4x+2    (Ans: slope = -4, y-intercept = 2)    

iii)  2x+5y=15.    (Ans: slope = `-2/5` , y-intercept = 3)         

Problem 2: Find the equation of line passing through two points (2,3) and (5,6).

Ans: y = x+1.