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Raj Sharma
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I would like to share learning-related tips and tricks for lifelong learners

Algebra is a mathematics part that deals with letters and symbols that are incorporated in the mathematical expression as a representative of the number or a quantity. These unknown values termed as the 'variables' are calculated through various means. One of the common ways is through calculating the slope. The slope reflects how the straight line angles away from the horizontal.

Therefore, it can be said that the slope is nothing but the ratio of the amount by which ‘y’ increases as the ‘x’ increases by a certain amount. Here, ‘x’ and ‘y’ are two inter-dependent variables of an equation. It can also be defined as ‘rise’ divided by the ‘run,’ representing the steepness of a line, or, in other words, how much increase happens in variable ‘y’ as variable ‘x’ increases. Thus, it helps in knowing whether the line is rising or falling. It is said that over the line, at any point, the slope remains constant; that is, it doesn’t change over a line.

The first of the forms of the linear equation is written with the help of slope and y-intercept. The slope-intercept form, thus, is nothing but a linear equation expression. The synonym to the slope is the Gradient of the line. Both are calculated through the same formula. However, both have a difference; while the slope is a scalar, the gradient is a vector quantity.

To expertise Algebraic mathematics, it is important to learn - the slope of line formula, how to find slope from an equation and slope calculation, and the concept of the gradient of a line. These concepts are explained in detail below to help students develop an understanding of the concept of slope as a whole.

**Slope intercept formula **

The slope-intercept is given by:-

y=mx+b

Where ‘m’ stands for ‘move,’ representing the slope of the given line

And, ‘b’ stands for the beginning of the line, representing the ‘y’ intercept or the ‘y’ coordinates of the given line

This formula is often used to express the equation of the line. There are several ways by which a line can be expressed. Each expression might look different, but all represent the same thing or the same line. One such way of writing the equation is “slope-intercept form.” It is commonly used to describe the equation when the equations are represented over a graph.

Two points on the line are taken up in such a situation, and the slope is calculated. The value attained is the value of ‘m’ in the above-written equation. After calculating the slope, the ‘y’ coordinate is being calculated in the form of (0,b); the value thus arrived represents the value of ‘b’ in the above-written equation.

Once all the values are being calculated, they are plugged in the above equation y=mx+b. For Example – If, suppose the value of the slope is calculated as 8 and intercept as 3, then the equation will be

Y= 8x+3

The expression, thus attained by putting up the calculated value, can be verified by picking up the points on the line and putting them into the equation. If the values selected to satisfy the equation, the expression attained is correct, and if not, then the expression has been wrongly formed.

In the case of the vertical line, the equation of the line will be “x=b,” as the vertical line is a straight line, going up and down; in such a situation, the slope becomes undefined. Similarly, for the horizontal line, the equation of the line will be “y=b” as the slope becomes zero in this case.

**How to find Y-intercept**

The Y-intercept is the point where a line crosses or passes through the y-axis. For instance, it is given that "y= 2x-2", then "-2" represents the y-intercept in the equation. This is because, we know, the basic slope-intercept form is y=mx+b, where ‘b’ represents the y-intercept. Therefore, if we place the values from the example equation into the slope incept form, we will get the y-intercept. In the above example, it turns out to be “-2.”

Or, in case there is no equation but the coordinates or points are given then, the slope and value of ‘b’ have to be calculated from which the equation can be formed. The formed equation thus will give the required Y-intercept. This can also be explained through the example.

Suppose, there are two coordinates (-3,3) and (3,-1)

Step1 – The slope will be “(y2−y1)/(x2−x1)”

Here, (y2−y1) = -1-3=-4

And, (x2−x1) = 3+3=6

m = -4/6= -2/3

Step 2 – intercept equation form is

Y=mx+b

Therefore,

Y= (-2/3)x+b

b = y+2/3x

b=3+(2/3)⋅(−3)=3+(−2)=1

Or, y=−23x+1

Thus, y-intercept will be 1

**The slope of a line formula **

If, two points of the line are given then, the slope of a line can be calculated through the formula “(y2−y1)/(x2−x1)”. The steps followed will be as below-

Step 1 – from the given two points assign one as (x1/y1) and the other as (x2/y2)

Step 2 – put these value in the formula

Step 3 – perform the operation, to calculate the value

This can be better understood with the help of an example,

Suppose points are (15,8) and (10,7)

Step 1 – Suppose the point (10, 7) represents (x1, y1) and (15, 8) as (x2, y2)

Step 2 – Plugging the value in “(y2−y1)/(x2−x1)” , we get , (8-7)/(15-10)

Step 3 – conducting the given operation, gives '1/5'. Therefore, the slope of the line is '1/5'.

If the slope of the line is positive, it means that it rises from left to right and if it is negative that is less than zero, then the line falls from left to right. If the two lines have the same slope then it represents that the two lines parallel each other. And, if the slope of the two lines is opposite or reciprocal then the lines are said to be perpendicular.

**How to find slope from an equation **

If, you already have an equation then finding the slope is more easy. In such a case, the value has to be directly picked up from the equation. For instance, the equation is, “ y = 3x+5”, then, the slope will be 3 because, as per the rules, the intercept form is “y=mx+b”, where ‘m’ represents the slope. As the example equation is already in the intercept form, the value will be the one that represents the ‘m’ (the Slope).

If, the equation is not in the intercept form and is something like this 2x+5y=5, then, firstly it has to be turned into the slope-intercept form to find the slope.

How will it be converted into slope-intercept form? Let’s follow below basic steps,

Step 1 – Reorganize the equation by Subtracting ‘2x’ from both sides of the equation, such that we get, 5y=5-2x

Step 2 – To turn the equation into slope-intercept form, divide both the sides by 5, such that we get, y = (-2/5)x+1

Step 3 – Now, the equation is in slope-intercept form, therefore the slope will be the value corresponding to “m”, that is, “-2/5”

**Slope calculation formula **

The slope calculation formula is given by:-

Slope = Rise/Run

Or, ∆y/∆x

Or, Slope point equation: “(y2−y1)/(x2−x1)”

Or, Slope intercept formula: y=mx+b, “m” represents the slope value

The slope calculation formula thus varies from problem to problem. If there is a slope-intercept equation, use the basic principle, just plug in the value and you have the slope value. But, if the equation is in some different form, convert it into the slope-intercept form, plug in the value to find the slope. Or, lastly, if there is a graph or coordinates (or points) are given, use the slope point equation “(y2−y1)/(x2−x1)”, plug in the point values, perform the operation, and solve for the value of the slope.

Let's learn this through an example, suppose, there are two points (– 5, 8) and Q(4, 10), such that

(X1, y1) = (-5,8 )

And, (X2, y2) = (4,10)

Therefore, by slope point formula -

“(y2−y1)/(x2−x1)” = (10-8)/(4-(-5))=2/9=0.22

if instead of the points, there is an equation, such as 2x+4y=8

In such a case, the slope will be calculated as

Step 1 - reorganize the equation - that is, 4y=8-2x

Step 2 - divide the equation by 4, that is, y=2-0.5x

Step 3 - place the values in the slope-intercept formula to get the slope value; that is, the slope-intercept form is y=mx+b, such that 'm' represents slope. Here' is -0.5

The slope of this equation can also find in another way. The equation "2x+4y=8" is in standard form Ax+By= C, and in standard form, the slope is given by '(-A/B).' therefore, the slope will be "-2/4" = "-0.5"

**The gradient of a line formula**

The gradient is the synonym for ‘slope.’ It is the rate at which the lines rise/falls vertically for each unit across to the right. It is said that the higher the gradient, the steeper it will be. Also, the negative gradient represents the downward sloping line.

The formula for the gradient of a line is given by:-

“(change in y-coordinate)/(change in x-coordinate)”

Or, Rise/Run

Or “(y2−y1)/(x2−x1)”.

Thus, if the gradient of a line is to be found, it can be calculated through the following steps.

Step 1 – Choose any two points on the line.

Step 2 – On the graph, draw a right angle triangle with the line as the hypotenuse.

Step 3 – with the help of the scale, note down, triangles vertical length, and horizontal length.

Step 4 – divide the vertical length by the horizontal length.

Step 5 – the value thus attained is the gradient of the line.

Let’s learn the calculation through an example.

Suppose there are two points, P(– 4, 5) and Q(4, 17), such that

(X1, y1) = (-4,5 )

And, (X2, y2) = (4,17)

Therefore, by gradient formula.

“(y2−y1)/(x2−x1)” = (17-5)/(4-(-4))=12/8=1.5

As the gradient here is positive, it means that the line is sloping upward. However, if it had been negative, it would have meant that the line is sloping downward.

The gradient can also be zero, that would mean that the line is a straight line crossing horizontally. And if the calculated gradient is undefined, it means the line is a straight line crossing vertically. Apart from it, the gradient of lines can also tell the relationship each line share with the other. If the gradient of two lines multiplies to give -1, it means the lines are perpendicular.

For instance, there are two equations, "y=3-5x" and "y=(1/5)x". The first equation gradient is '-5', and for the second equation, it is (1/5). Multiplying the gradient of two lines gives "-1", representing that the lines are perpendicular and will be crossing each other at a ninety-degree angle. Similarly to the gradient of the two lines is the same, it will mean the lines are parallel to each other.

For instance, there are two lines,

" y = 2x+5" and "y= 2x-7"

The gradient for " y = 2x+5" will be 2

And, The gradient for " y = 2x-7" will also be 2

As the two lines share the same gradient, the lines are parallel to each other. This can be verified by drawing the two lines on the graph.

In conclusion, the slope is the rate of change with respect to another variable. It helps to define the relationship between the two variables. In the linear equation, the slope can easily be found through the slope-intercept formula. And, If there are just coordinates, the slope can be found through the slope point formula.

Similar to the concept of the slope is the concept of the gradient. The two are much the same with the same formula and meaning. For linear equations, both are calculated the same way.

Knowing the slope concept is important because it is far more broad and applicable as is being thought of. It can represent the acceleration or velocity of the change in various fields.

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Empower your mind to conquer any obstacle that comes your way! You have the strength within you to overcome any challenge.

Here are 5 exciting opportunities to level up your mental game. Let's dive in and discover the potential within you!

1. Rich mindset people know exactly why they are doing what they are doing!

2. Rich mindset people do not do things that are not aligned to their Why.

3. Rich mindset people know what they are passionate about!

4. Rich mindset people work only on what they are passionate about!

5. Rich mindset people's life is filled with excitement and more happiness!

6. Rich mindset peoples are the master of their craft because they are passionate about it!

7. Rich mindset people have big juicy goals.

8. Rich mindset people write down goals every single day.

9. Rich mindset people are most vocal about their goals.

10. Rich mindset people will never reduce goals but increase the effort.

11. Rich mindset people think that money is everywhere!

12. Rich mindset people are plenty of opportunities provided they try to grow their knowledge and skills.

13. You become rich twice, first in your mind and then in reality.

14. Rich mindset, people are the creators.

15. Rich mindset people know that It is effortless to make Money.

16. Rich mindset people know money is available in abundance.

17. Rich mindset people create multiple sources of income.

18. Rich mindset peoples are very health-conscious people.

19. People admire a Rich mindset, people's discipline, and good health.

20. Rich mindset people spend lots of quality time with family.

1. Broke mindset people think everything is too good to be true.

2. Broke mindset people give up when things don't go their way; a few disappointments, and they are onto something else, saying things like "it wasn't for me."

3. Broke mindset people always have an excuse.

4. Broke mindset people think that not getting what they want is OK.

5. Broke mindset always has to talk it over with their broke friends to make sure no one will make fun of them if they make a decision.

6. Broke mindset people are never coachable and teachable.

7. Broke mindset people are scared of others.

8. Broke mindset: people always procrastinate; they would rather talk about it, read about it, think about it, but never do anything.

9. Broke mindset people are glad when the day is over.

10. Broke mindset people think rich people are lucky.

11. Broke mindset people work by the hour.

12. Broke mindset people want to know that after 1 hour of work, they have something to show.

13. Broke mindset people get excited; they just got hired.

14. Broke mindset people complain a lot.

15. Broke mindset people are too concerned about what other people are doing.

16. Broke mindset, people think that if no one is doing something, it must suck.

17. Broke mindset people think that if everyone (all 200 people at a meeting or on a conference call) are all doing the same thing, it must be saturated.

18. Broke mindset people think it is OK for other people to live where they want to live, drive what they want to operate and do what they want to do.

19. Broke mindset people are OK with the fact that they can't do these things.

1. Listen more before you speak. Try to understand what you are trying to say before then reply.

2. Try before quitting. At least try things whatever you want in your life. Never quit before trying.

3. Whenever you are going to write something anywhere, think before you write.

4. Spend money, but first, earn. Earn before you spend.

5. Believe first before doing anything in your life. Believes are valuable in every situation.

6. Live your life; enjoy each moment in your life. Live before you die.

7. Whenever you go to start anything, prepare first. Prepare before you start.

8. Make time for health. Exercise and meditate before you get stressed.

9. To be happy and for piece live in present moments. With the plan of the future, never invest in your present. Live it.

10. Understand the rules of freedom before you expand your freedom.

11. Love yourself and spread that love before time goes away.

12. Accept and solve the situation before you miss out.

13. Be responsible before time has gone. Be responsible when you need it. If time goes, it never comes again.

14. Take permission from your family before taking any kind of action.

Being a successful student is not a child's play. It requires utmost devotion, a student success plan, hard work, and a passion for being successful in life. If you genuinely wish to be a successful student in High school or college, there are student success strategies that can prove to be helpful to the achievement of your goal.

1. Prioritization

2. Punctuality

3. No cheating

4. Concentrate

5. Refraining from unnecessary comparisons

6. Improvement in progression

7. Curiosity and excitement

One of the best ways to absorb a piece of information in your mind is to read about it on your own with full focus.

When you have a solid game plan before starting any study session, it naturally improves your productivity. This way you can be ensured that you make each session as productive as possible and make the best out of your time.

It means that you make a timetable in order to enjoy a productive day along with getting yourself nearer to success every day. Along with social activities make sure that you prescribe a fixed number of hours to study each day.

**1.) Pay attention **

Paying attention in class is one of the most crucial practices a successful student is required to do.

**2.) Taking notes**

It is not possible to remember every word the teacher says. So, it is essential that you jot down notes in the form of main points that summarize what the teacher is explaining.

**3.) Clarifying doubts**

To avoid being lost during a lecture make sure that you clarify all your doubts.

**4.) Avoiding distractions **

Just like during your self-study hours, keep your mobile phone away.

**5.) A positive relationship with your teacher **

Determines the success of an individual in high school and college is his/her relationship with the teachers and staff.

**6.) Have fun**

Along with studying it is essential for every student to have a carefree time where he/she can just have some fun and relax with friends.

Write a compelling and catchy introduction. Mention an interesting fact related to your topic into the intro.

**How to write the middle part of the essay?**

The primary focus is on the 1st body, this paragraph need not be as catchy as the previous stages. So, here you can mention some weak points in order to elaborate on the essay further.

**The first paragraph -** Write the strongest points or essential facts in the first paragraph

**The second paragraph -** Tries to elaborate of an essay further and adds more intriguing points.

**The third paragraph -** Cite some exciting examples with catchy tone related to the topic in the third paragraph

Who do not understand how to conclude an essay without sounding abrupt. Even though the conclusion is supposed to be short and sweet, it is an underrated part of an essay, and most of the students seem to be ignorant of how they should end an essay.

You need to persuade the reader with your arguments and ideas put forward by you in the essay and make your readers care about your ideas.

The majority of the students fail to understand is the fact that a proper link between the introduction and the conclusion of an essay can prove to be extremely fruitful.

If you have stated an idea in the intro, you can add a line or two elaborating that point in the conclusion. Expand the theme that you introduced in the intro without making it seem like you are merely repeating a certain fact to fill up pages.

Since essays have become an integral part of the various school and college assignments along with applications to prestigious universities, students cannot afford to stay ignorant about the types.

Before diving straightaway into looking for all sorts of essay help online, here are four major types of essays every student should know about. It is easy to get confused with the immense number of topics in essay books.

However, the types of essays are limited, and they are more manageable than you can actually think of! These four types mainly cover all the other subtypes, so, let us take a look.

1.) Narrative essays

2.) Descriptive essays

3.) Expository essays

4.) Persuasive essays

THERE ARE SEVERAL ONLINE RESOURCES AND TOOLS TO HELP YOU ASSIMILATE A LARGE AMOUNT OF STUDY MATERIAL QUICKLY AND EFFECTIVELY.

To create the right mindest that will ensure your success. This entails accepting change and aiming to grow and improve constantily.

A positive attitude brings out creativity in thinking and you will get new ideas for improving the subjects. Positivity will also help in finding better solutions to the problems. Even in a school or college project or assignment, a positive attitude will bring out better results. It will also help in keeping your mind relaxed and calm in negative situations.

Having a positive attitude will attract more people to you. You can be the leader of your group and lead people to follow a positive attitude in life. You can remove the negativities in yourself and others too. It will improve your life to a large extent. With a more positive attitude, you will get more able to solve any problem in life.

Optimism always leads to success. Positivity will give more enthusiasm to work which will result in promotions and increments. This is the reason why people with a positive attitude become successful in later life.

A positive attitude will gain more friends. It is because negativity will reduce your self-confidence and you will get discouraged. People will like staying with positive people. Positivity will spread happiness and joy all around the place.

Here are the ten steps that can help you to have effective listening:

1.) Face the speaker and always maintain eye contact.

2.) Pay attention, but be relaxed.

3.) Have an open mind and listen to them without judging them.

4.) Carefully listen to the words and try to understand what the speaker is saying.

5.) Never interrupt the speaker, nor impose your thinking on them.

6.) When the speaker takes a pause only then ask a question.

7.) Never ask things unnecessarily, only ask to have a better understanding.

8.) Try to feel what the speaker is feeling and communicating.

9.) Try to give simple and regular feedback to the speaker.

10.) Focus on non-verbal cues like facial expressions and gestures.

Basically, there are two types of listening skills:

**Discriminative listening**

This type of listening is developed in the initial stages. This is the most basic kind of listening and does not require any effort as such in understanding the meaning of words or the phrases but just different sounds are produced.

For example, in early childhood, a distinction can be made between different sounds that get improved as the child grows up. As we move from childhood towards adulthood this distinction increases with the capacity to distinguish between different sounds.

**Comprehensive Listening**

Comprehensive listening is a broad term and it basically includes understanding the messages being communicated to the other person. It is also fundamental to all listening sub-categories.

For a person to develop comprehensive listening the listener first has to have an appropriate vocabulary and other language skills. This kind of listening gets more complex as the same message given to two people might result in understanding it with different meanings.

Comprehensive listening is also complimented by sub-messages through non-verbal communication like the tone of the voice, body language, and gestures. This kind of non-verbal signals can help immensely in communication and comprehension but might also confuse and might lead to misunderstanding.

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