**Introduction to linear equation in one variable **

**NCERT Solutions for Class 8**

A linear equation in one variable is an equation that can be written in the form ax b c + =, where a, b, and c are real numbers and . Linear equations are also first-degree equations because the exponent on the variableis understood to be 1. Linear equations can have one or more variables. An example of a linear equation with three variables, x, y, and z, is given by: ax + by + cz + d = 0, where a, b, c, and d are constants and a, b, and c are non-zero. Linear equations are also first-degree equations because the exponent on the variable is understood to be 1.

Linear equations occur frequently in most subareas of mathematics and especially in applied mathematics. While they arise quite naturally when modeling many phenomena, they are particularly useful since many non-linear equations may be reduced to linear equations by assuming that quantities of interest vary to only a small extent from some "background" state. An equation is linear if the sum of the exponents of the variables of each term is one.

**Standard form of Linear equation in one variable**

The standard form of linear equations in one variable is represented as:

px + q = 0

Where,

- ‘p’ and ‘q’ are real numbers, and
- both ‘p’ and ‘q’ are not equal to zero.

**Conditional equation in linear equation in one variable **

A conditional equation is an equation that is true for some values of the variable but not for others. Every linear equation that is a conditional equation has one solution. However, not every linear equation in one variable has a single solution. There are two other cases: no solution and the solution set of all real numbers.

**Identifying equation if contradiction or identity**

Consider the equation, x+x=1. No matter what value is substituted for x, the resulting value on the right side will always be one greater than the value on the left side. Therefore, the equation can never be true. We call such an equation a contradiction. It has no solution. Its solution set is the empty or null set, denoted by { }, respectively.

Now consider the equation, x+ x = 2x . The expression on the left side of the equation simplifies to the expression on the right side. No matter what value we substitute for x, the resulting values on both the left and right sides will always be the same. Therefore, the equation is always true. We call such an equation an identity. Its solution set is the set of all real numbers, denoted by (-∞, ∞) or {x|x is a real number}.

**How to solve a linear equation **

**Define the problem **

The most important thing to do to solve a maths problem is to determine what to find. Reading the question carefully, once, twice or many times is important to understand the information given in the question. This will help you extract information and also find the solution .

**Assign variables**

To make the task even more easier is to assign the variables for each of the given quantity in the problem. For eg, use d fordistance. This will help to figure out the question with ease and wouldn't be too much of crowd for solving an equation. Every possible quantity can be expressed as a variable for easy understanding.

**Translate into an equation**

The next simple step is to use all the converted variables and form it into an equation. All what can be done is put together all the relationships of known and unknown variables that will make the solving of an equation easier. While translating it into an equation we can also derive as to which is the missing quantity that we are looking for.

**Solving the equation **

Now that we have defined the problem, assigned necessary variable, translated the number into an equation, now its time to solve the equation. Find all the variables and numbers that you have and solve the equation to find the missing quantity. And yes,you will find answer to the question.

**Check reasonableness of your answer**

After all the playing up with numbers and equations, there are chances that you might come up with an error in your final answer. So, make sure to check and re-check your answer for more authenticity. Also, it is important to make sure that the steps involved in finding a solution is also checked correctly. The most important thing is that your answer must make more sense to the context of the problem.

**Answer the question **

After all the hassle, quest and solving with numbers, the final step is to write an answer statement that clearly summarizes all that was found. This will provide an answer to the question(s) posed.

**Other forms of solving equation through brackets **

It is necessary to keep in mind that equations cannot always come in a standard form. It might even include brackets like open or close and various variables. So when dealing with such equation with brackets, first solve the variables and numbers inside the bracket by bringing them to a standard equation and solve.

**Graphical Representation **

In any case of linear equation in one variable there always has to be an unknown quantity that will measure the equation. And in turn this will lead to having a straight line in the graph. As, one number in the equation will always be unknown and the others known. Marking perfectly into the graph is also of utmost important.

Linear equation in one variable helps in understanding variables and numbers in a much easier way and provides us with an experimental way of learning. This is a new and innovative way of learning that helps students to grow interest in the subject. You can always witness that there are a lot of ways in which a equation can be determined. Leaning can always be an joy to know about new things. This in turn will make it a fun with learning experience.

Maths is considered to be important during the school years that can imbibe you with basic knowledge of numbers. It is not a subject that is not restricted to a particular concept, but has depth in each of its concepts.

**Introduction to linear equation in one variable **

**NCERT Solutions for Class 8**

A linear equation in one variable is an equation that can be written in the form ax b c + =, where a, b, and c are real numbers and . Linear equations are also first-degree equations because the exponent on the variableis understood to be 1. Linear equations can have one or more variables. An example of a linear equation with three variables, x, y, and z, is given by: ax + by + cz + d = 0, where a, b, c, and d are constants and a, b, and c are non-zero. Linear equations are also first-degree equations because the exponent on the variable is understood to be 1.

Linear equations occur frequently in most subareas of mathematics and especially in applied mathematics. While they arise quite naturally when modeling many phenomena, they are particularly useful since many non-linear equations may be reduced to linear equations by assuming that quantities of interest vary to only a small extent from some "background" state. An equation is linear if the sum of the exponents of the variables of each term is one.

**Standard form of Linear equation in one variable**

The standard form of linear equations in one variable is represented as:

px + q = 0

Where,

- ‘p’ and ‘q’ are real numbers, and
- both ‘p’ and ‘q’ are not equal to zero.

**Conditional equation in linear equation in one variable **

A conditional equation is an equation that is true for some values of the variable but not for others. Every linear equation that is a conditional equation has one solution. However, not every linear equation in one variable has a single solution. There are two other cases: no solution and the solution set of all real numbers.

**Identifying equation if contradiction or identity**

Consider the equation, x+x=1. No matter what value is substituted for x, the resulting value on the right side will always be one greater than the value on the left side. Therefore, the equation can never be true. We call such an equation a contradiction. It has no solution. Its solution set is the empty or null set, denoted by { }, respectively.

Now consider the equation, x+ x = 2x . The expression on the left side of the equation simplifies to the expression on the right side. No matter what value we substitute for x, the resulting values on both the left and right sides will always be the same. Therefore, the equation is always true. We call such an equation an identity. Its solution set is the set of all real numbers, denoted by (-∞, ∞) or {x|x is a real number}.

**How to solve a linear equation **

**Define the problem **

The most important thing to do to solve a maths problem is to determine what to find. Reading the question carefully, once, twice or many times is important to understand the information given in the question. This will help you extract information and also find the solution .

**Assign variables**

To make the task even more easier is to assign the variables for each of the given quantity in the problem. For eg, use d fordistance. This will help to figure out the question with ease and wouldn't be too much of crowd for solving an equation. Every possible quantity can be expressed as a variable for easy understanding.

**Translate into an equation**

The next simple step is to use all the converted variables and form it into an equation. All what can be done is put together all the relationships of known and unknown variables that will make the solving of an equation easier. While translating it into an equation we can also derive as to which is the missing quantity that we are looking for.

**Solving the equation **

Now that we have defined the problem, assigned necessary variable, translated the number into an equation, now its time to solve the equation. Find all the variables and numbers that you have and solve the equation to find the missing quantity. And yes,you will find answer to the question.

**Check reasonableness of your answer**

After all the playing up with numbers and equations, there are chances that you might come up with an error in your final answer. So, make sure to check and re-check your answer for more authenticity. Also, it is important to make sure that the steps involved in finding a solution is also checked correctly. The most important thing is that your answer must make more sense to the context of the problem.

**Answer the question **

After all the hassle, quest and solving with numbers, the final step is to write an answer statement that clearly summarizes all that was found. This will provide an answer to the question(s) posed.

**Other forms of solving equation through brackets **

It is necessary to keep in mind that equations cannot always come in a standard form. It might even include brackets like open or close and various variables. So when dealing with such equation with brackets, first solve the variables and numbers inside the bracket by bringing them to a standard equation and solve.

**Graphical Representation **

In any case of linear equation in one variable there always has to be an unknown quantity that will measure the equation. And in turn this will lead to having a straight line in the graph. As, one number in the equation will always be unknown and the others known. Marking perfectly into the graph is also of utmost important.

Linear equation in one variable helps in understanding variables and numbers in a much easier way and provides us with an experimental way of learning. This is a new and innovative way of learning that helps students to grow interest in the subject. You can always witness that there are a lot of ways in which a equation can be determined. Leaning can always be an joy to know about new things. This in turn will make it a fun with learning experience.

Maths is considered to be important during the school years that can imbibe you with basic knowledge of numbers. It is not a subject that is not restricted to a particular concept, but has depth in each of its concepts.

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